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Our common notational system has had a very long evolution. Not only does it store and convey information but we can also use it to
understand the theoretical and acoustic foundations of our musical system. Pitch, key, rhythm, performance instructions and so on, are all
captured upon the page. It is easy to point out certain notational shortcomings, but with enough clever manipulations we can usually make our
intent clear enough. The exact rhythm and pitch inflection of say spoken or sung text however is almost impossible to capture accurately,
which comes down to a simple maxim; if it is possible to notate the subtle complexities of say, spoken or sung text or a free improvisation, it
would also be almost impossible to read and realize it. Notationally we can sketch an outline and our musical sense will realize what is
missing. For some music, say keyboard music, our notational system is excellent. For other music, like a free solo improvisation or a musical
recitation, the challenges to capture or compose such are very difficult. As well there are not many notational alternatives. Overall though, our
notational system is excellent, concise and clear and it has carried well the burdens put upon it through time.
12 Notes per Octave
We have a complete notational system for 12 notes per octave. Between naturals, sharps, flats, double sharps and flats we have no less
than 35 possible ways to notate the 12 pitches of the octave. Each pitch can be written in three possible ways except G#/Ab. For example, the
three enharmonic pitches of C are C, B# and Dbb. We draw the line at triple sharps and flats. We nowadays don’t think much of the difference
between say G# and Ab, but historically that has not always been the case. In Meantone Temperaments, G# and Ab are distinct pitches, to
the point that organs and harpsichords could have one or more split keys. Though most historical instruments split only a few keys, on rare
instruments there can be up to 19 or even 21 keys per octave. The demands on the performer and instrument maker can though become
unrealistic. This is dealt with in depth in the chapter on Equal Meantone Temperaments.
As tuning systems evolved through Well-Temperament to Equal Temperament it became possible to modulate out through the sharps and
back through the flats or visa-versa. While we only have 12 notes to the octave there are 15 major and 15 minor keys. Three major keys can
be written in two different ways; B/Cb, F#/Gb and C#/Db, as well as three minor keys; G#m/Abm, D#m/Ebm and A#m/Bbm. And so we need
now as well a number of double sharps and flats to realize these keys. Every new set of musical demands, for example microtonal fluctuation,
requires a new set of symbols but the underlying system of 7 letter names with sharps and flats remains unaltered. The diatonic scale and by
extension the chromatic scale are almost more laws of nature than humanly created and so are universal across all cultures no matter how
remote in geography or time.
And if the 12 notes in the octave are pretty much a law of nature is our beginning also our end? Every relation of every degree of the
chromatic scale to every other degree has by now been thoroughly explored and exhausted. But it is obvious to anyone that music does
evolve. In what way does it evolve then? Largely it evolves through the creation of new machines and the subtle tuning systems inherent to
them. The piano and electric guitar are both machines that evolved out of what came before them and the current state of technology that they
were created in. Yet the overall system of 12 chromatic notes to the octave remains largely the same even though instruments themselves
continue to undergo evolution. 12 notes to the octave are eminently manageable and nicely spaced. We hear different keys, chords, intervals
and modes as tonal, harmonic and balanced. When tuning evolved from Meantone Temperament to Well and Equal Temperament not only
was the scale evening itself out, albeit with an overall loss of harmonic purity, but also the need for the few split keys disappeared as well. And
so for the last 400 years instruments with 12 notes to the octave continue to be the only norm.
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It is hard to see how it can be possible to evolve comprehensively beyond our twelve notes to the octave. The few instruments capable of
more than twelve notes to the octave are rare, expensive (both to purchase and learn time-wise) and overall ergonomically cumbersome, in no
way allowing the same virtuosic potential of our regular instruments. Further, an overall theoretical and notational foundation is completely
lacking and most microtonal music in the end just sounds like poorly played music on a completely out of tune instrument. There is a huge
upside and advantage that was lacking as recently as a few decades ago. We no longer have to build instruments to realize possible
microtonal systems, we can use the machine of the computer to assist us in our exploration and realization.
Like all explorations many people will pursue many avenues in many different ways. Some are happy to take any temperament whatsoever
(17Et, 19Et, 22Et etc) and write music in it regardless how out of tune it sounds harmonically. For others, a blind doubling or tripling of the
number of notes in the octave is the way to go, again completely oblivious that doubling or tripling the number of notes in the octave in no way
improves over 12Et tuning wise.
As humans, we enjoy when two notes combine in a pleasing manner. Notes that combine in a pleasing manner as well have a simple
physical relationship to one another which we delineate by the simple use of just intonation ratios. We can bend and pull these simple
consonances slightly out of tune as we do in our many different tempering systems but we never lose sight or hearing of what that simple
interval is. Overall we can say that almost all the intervals of our listening experience fall within what we call the 2, 3 and 5 Limit ratios.
Argument can be made as well for the 7 Limit ratio of 7/4 also known as the “harmonic seventh”. Pretty much everything else though (7, 11
and 13 Limit ratios) falls into the cracks and is heard to be out of tune.
Foregoing 11 and 13 Limit ratios for now, can we one day as well become acclimatized to 7 Limit ratios and harmonies and naturally accept
them as consonant, as we accept the combinations of 2, 3 and 5 Limit ratios. As 12Et in no way approximates 7, 11 and 13 Limit ratios it won’t
be through our equal tempered scale that we will ever become familiar with these intervals. It is sort of a catch-22. Only by creating new scales
and instruments that support these higher Limit ratios can we become familiar with them. Maybe though no amount of familiarity will suffice
and higher Limit ratio will always sound generally out of tune. Maybe a start would be to combine and bury higher Limit ratios into familiar
chords as Harry Partch does with his 4:5:6:7:9:11 chords or Otonalities, though the same cannot be said of his Utonalities.
If we have exhausted all the relationships of note and interval in our 12 note chromatic scale then the only thing possible to do is add more
notes to the octave. We could add 7 Limit 7/4 ratios to each note and double the number of notes in the octave to 24 notes. Maybe not the
most elegant or thought through solution!
In adding more notes to the octave we want to be able to maintain and build upon the system or tonality that has been used since time
immemorial. We want to expand or extend the tonality that forms the basis of our experience, not replace it. We want the system that we are
used to, to sit perfectly and indistinguishably within the new system completely. The new system must contain within itself the familiar system
and at the same time extend the familiar system with completely new relationships. Three equal temperaments allow us to meet this criterion:
31Et, 43Et and 53Et.
31 and 43 Notes per Octave
With 31Et and 43Et we have the first two viable options to move beyond 12Et. 12Et, 31Et and 43Et can be considered a set as they share
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similar characteristics. Firstly, all three temper out the 10/9 and 9/8 ratio small and large major seconds to a single pitch. What this means is
that we can create the 6 major/minor triads of a key using only 7 diatonic notes. As well three intervals of a perfect fifth (C G D A), is equal
exactly to the major sixth note and four intervals of a perfect fifth (C G D A E), is equal exactly to the major third note. This may seem
completely self evident but none of these will hold when dealing with the almost perfect 5 Limit just intonation of 53Et. All three equal
temperaments as well temper their intervals in the preferred direction; flat for perfect fifths, minor thirds/sixths and sharp for perfect fourths,
major thirds/sixths
Because of the above characteristics, 12 note subsets of 31Et and 43Et can as well form their own Meantone Temperaments. We don’t
need an instrument of 31 or 43 notes to hear the tonal characteristics of each temperament. A regular 12 note to the octave instrument
suffices to familiarize ourselves with a limited subset of their characteristics.
When we have only seven diatonic notes to a scale we must by necessity choose the balance between purer perfect fifths and purer major
thirds (C G D A E). The flatter the fifths are tempered from just the less sharp the major third becomes. When four perfect fifths are each
tempered by 5.377 cents or one quarter Syntonic Comma 81/80 (21.51 cents), the major third will be pure. The major thirds in 31Et are almost
pure and so the perfect fifths are tempered by almost one quarter Syntonic Comma each. At the other end of our set of three equal
temperaments, the fifths in 12Et are close to just, being only around 2 cents flat, but the major thirds are now almost 14 cents or more than a
seventh of a semitone sharp. Right in the middle lies 43Et with perfect fifths tempered flat (4.28 cents) by almost the same amount the major
thirds are tempered sharp (4.38 cents). Very nicely these three temperaments encompass the range of choice we have to make when
balancing the perfect fifths and major thirds!
In what way now does 31Et and 43Et expand or extend upon the tonality inherent in 12Et? For starters 12Et is one of the worst
temperaments to approximate 7, 11 and 13 limit ratios. Any approximation to these Limits by 31Et and 43Et in and of itself opens a new world
of harmonic possibility whether dissonant or with the possibility of future acclimatization. Of all equal temperaments under 53Et, 31Et
approximates the best to 7 and 11 Limit ratios though some ratios in these Limits approximate closer than others. We can also say that in 43Et
some ratios in the 7,11 and 13 Limits approximate better than others, but overall 43Et poorly approximates 7 Limit ratios and is only middling
at approximating 11 and 13 Limit ratios. If we are of the belief that higher Limit ratios will always be dissonant in our ears then the nice balance
and tempering of 43Et makes it an excellent temperament to work in.
To be able to add higher Limit ratios is one possible way to extend or expand upon our familiar tonality. The other is to increase the number
of equally spaced notes in the octave which of course increases the number of possible keys and available intervals between the notes and
keys. A whole new set of subtle interval and key relations immediately become available to us. In 31Et and 43Et we would have 31 and 43
different possible keys or modal centers. In 12Et we have a set of 12 unique intervals, all of which are familiar to us and which we have long
utilized. In 31Et and 43Et there would be 31 and 43 unique intervals but whether all of them are usable or not is another matter. Chromatically
our music would have incredibly subtle shades of pitch.
53 Notes per Octave
Much that is written above holds as well for 53Et, but 53Et belongs to a different class completely. 53Et and 5 Limit just intonation are pretty
much synonymous with one another. Even to high number 5 Limit ratios, 53Et differs from 5 Limit just intonation by only 1 or 2 cents. They are
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almost identical to one another and so share the same characteristics. 53Et however is much easier to work with than having to manipulate a
plethora of ratios. The interval relationships between different degrees in 53Et are easily determined just by measuring steps.
In 53Et the 10/9 and 9/8 small and large major seconds are untempered from one another and now exist independently. That means in order
to have all 6 major/minor triads in a key we now need 8 diatonic notes per major scale instead of 7. That means we have 8 modes instead of
the regular 7. There are two Dorian modes on the two different major seconds. We have as well 53 key or mode centers and 53 unique
intervals and even subtler shades of chromaticism. Having an 8 note diatonic scale as well means we no longer have to balance the major
thirds and perfect fifths for purity of tuning as we must in 12Et, 31Et and 43Et. Both the perfect fifths and major thirds are almost just. For
higher Limit ratios, 53Et has excellent 13 Limit ratios, very good 7 Limit ratios, though the 11 Limit ratios could be better.
The perfect fifths of 53Et are almost identical to pure Pythagorean 3/2 ratios. That means four perfect fifths added together are almost
identical to the Pythagorean Ditone of 81/64 (407.82 cents). The just major third (5/4) we are looking for though lies a Syntonic Comma (21.51
cents) below the Ditone which is almost 1 step in 53Et (22.64 cents). The almost just Ditone (81/64) is then 18 steps in 53Et while the almost
just major third (5/4) is 17 steps.
Unlike 12Et, 31Et and 43Et, in 53Et we can’t cycle up through four perfect fifths to find the major third. In 53Et, as in the Pythagorean Just
Fifth Tuning we have to cycle down 8 perfect fifths to reach the diminished fourth (Fb Cb Gb Db Ab Eb Bb F C) which is the enharmonic major
third we are looking for. In Pythagorean Just Fifth Tuning this diminished fourth/enharmonic major third is 1.95 cents flat from just while in 53Et
it is 1.41 cents flat from just. This goes to show how almost identical these two systems are, one a temperament, the other a tuning. 53Et can
be thought of as a miracle temperament due to its almost identical approximation to 5 Limit just intonation.
Notation - Extended Cycle of Perfect Fifths
In order to utilize a musical system an equivalent notational system is absolutely necessary. We have seen that for 12 notes per octave we
have 35 possible ways to notate the 12 different pitches. Further the cycle of perfect fifths nicely cycles through the double flats, flats, naturals,
sharps and double sharps:
| Fbb Cbb Gbb Dbb Abb Ebb Bbb | Fb Cb Gb Db Ab Eb Bb | F C G D A E B | F# C# G# D# A# E# B# | Fx Cx Gx Dx Ax Ex Bx |
With 12Et we only have to cycle through 12 perfect fifths before we end up back where we started, albeit with a different letter name. For
example: C G D A E B F# C# G# D# A# E# B#. With 31Et, 43Et and 53Et we have to cycle through 31, 43 and 53 perfect fifths respectively to
end up back where we started.
Even with 7 or 8 notes to a diatonic scale, 12Et, 31Et, 43Et and 53Et overall are all similar musical systems both diatonically and
chromatically. Anything that is written in 12Et can be played in 31Et, 43Et and 53Et but with a greater accuracy of tuning towards just
intonation. Notationally, 31Et and 43Et follow directly the system and cycle of fifths in 12Et, in the same manner and with the same clarity. In
53Et however our scale has to be written enharmonically to follow the cycle of fifths and so immediately becomes a mind-boggling mess:
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F
C G
D
Ebb Bbb Fb Cb
Logically and soundly though, we can use an extended cycle of perfect fifths to form the basis of a notational system for 12Et, 31Et, 43Et
and also 53Et. In all four equal temperaments we can as well modulate out through the sharps and back through the flats, except these now
will include triple and even quadruple flats and sharps. As we look at the notation for 12Et we can always understand where we are key wise
etc and the same goes for 31Et, 43Et and 53Et. The notational system for 31Et, 43Et and 53Et by the extended cycle of perfect fifths is not an
abstract, obtuse system. It is an exact extension of 12Et and all the note relations hold true even with 31, 43 and 53 keys! We just have a few
more keys to deal with and a notational mess for 53Et!
As we have 35 possible ways to notate pitches with sharps, flats and double sharps and flats, we could completely notate 31Et not going
beyond double sharps and flats. We could notate any equal temperament up to 35Et this way and 35Et then would have only one distinct
written note per pitch. However in 31Et to have 15 sharp keys and 15 flat keys plus the one key with no sharps and flats, and not mix sharps
and flats, there must be 6 pitches that are enharmonic equivalents and so 37 written notes are required to notate the 31 pitches of 31Et. 31Et
then also requires one triple sharp and one triple flat. In this way we can logically and clearly move through the keys in perfect fifths from Gx
major to Fbb major without mixing double sharps and flats, in precisely the same way we move from sharps to flats or flats to sharps going
around the keys in 12Et. We could also add more enharmonic triple sharps or flats if musically required, for example as leading notes in minor
keys or flat seconds etc.
The same is also true for 43Et where we need a complete set of triple sharps and flats for the 21 sharp keys and 21 flat keys and the one
key without sharps or flats. There will then be 6 enharmonic equivalents for a total of 49 written notes. For 53Et we need almost a full set of
quadruple sharps and flats for the 26 sharp keys and 26 flat keys and the one key without sharps and flats. There will as well be 6 enharmonic
equivalents between quadruple sharps and triple and quadruple flats. 59 written notes then are required to notate 53 pitches using the
extended cycle of perfect fifths.
In 31Et, 43Et and 53Et it is important to realize that in these systems there are no enharmonic equivalents except for the ones laid out
above. D is no longer the same as Cx as it is in 12Et. Before anyone decries all these triple and quadruple sharps and flats one must realize
that we are dealing with 31, 43 and 53 pitches now. There is no easy way, yet the above systems operate exactly in the same way and with
the same theoretical understanding and acoustic foundations as in 12Et. This makes complete sense as one of our criteria was that any new
system must contain within itself completely the familiar system.
There is now though one strange thing we have to deal with. The ultra “chromatic” scales of 31Et, 43Et and 53Et can start looking a little
strange and confusing. For example, to move chromatically from C to C# in 53 Et we would have to write: C B# Ax# Ebbb Db C#. Strange
indeed but there is no way around this using notation based on the extend cycle of perfect fifths. While that adds further to our confusion,
especially in 53Et!, it is however clear which keys are higher and lower pitch-wise which can be somewhat advantageous.
We are now however swimming in a morass of accidentals. In our common musical system we have a single symbol for a natural, sharp, flat
and double sharp. Unfortunately we do not have a single symbol for a double flat as we use two flats. What is vitally necessary then is to have
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as well a single symbol for double flats and triple and quadruple sharps and flats in an attempt to clarify our notation. We unfortunately have
no end of new symbols that look pretty much like Martian and are as hard to distinguish, write and memorize!
One way around this mess of accidentals, strange chromatic progressions and enharmonic chord writing, has been to introduce simple
microtonal inflections to a simple set of notes that go no further than one sharp or one flat. This works very well in 31Et and 43Et with our
familiar keys from Cb major through to C# major. This system is a good simplification but is incomplete. It needs expansion so every single
degree in 31Et and 43Et can be utilized as a home key and read consistently without mixing sharps and flats and microtonal inflections.
Using a simplified system of microtonal inflections is also possible in 53Et though now we need both single and double microtonal
inflections. We can also use the simple microtonal inflections in 53Et to nicely clarify the confusing enharmonic writing of chords!
Polychromatic Notation and Extended Tonality
It would be an understatement to say that a fair bit of time and thought went into figuring this all out! How is it possible to create a notation
for high number octave divisions like 31Et, 43Et, 53Et or even 108Et and yet keep the same clarity and theoretical comprehension we have
with our regular notational system of 12Et? Every single scale degree as well has to be notationally completely consistent key and interval
wise. We as well have to be able to move logically and understandably through all the scale degrees right around to where we started from.
We also have to be able to extend our familiar tonality without compromise through a high number of scale degrees without a morass of
symbols, always immediately being able to understand where we are both notationally and theoretically.
We can effect this all in two simple ways. We simply start with our entire familiar musical system of 15 sharp and flat keys. To that we add a
simple system of upward and downward inflections (!,!!,!!!,!!!! and ","",""",""""). The second way is to substitute the upward and
downward inflections by different coloured noteheads, hence Polychromatic Notation. The whole system is as simple and easy as that, though
it might take just a little more explaining for a complete understanding!
An early use of the term “Polychromatic Notation” can be found in an edition of Bach Sinfonias edited by Bern Boekelman in 1904. The
concept of different coloured noteheads however goes right back to music written in the fourteenth century. Besides figuring out how to create
the above system one of the first challenges in developing a Polychromatic Notation was how to choose and organize the colours. As for most
things, one always comes back to the simplest and most familiar construct. The colours simply proceed from lowest frequency to highest
frequency like we find in a rainbow: Red, Orange, Yellow, Green, Blue and Violet. Orange and Green are between the primary colours of Red,
Yellow and Blue. It is strange why in nature Violet being the highest frequency is between the primary colours Blue, a higher frequency, and
Red which is the lowest frequency. We have eight inflections though and so as Violet fades out of sight into ultra-violet we use a light mauve.
Similarly as Red fades into infra-red we use a light pink. Now we have equated our eight inflection symbols with eight colours. What about our
regular uninflected normal notation? We are going to put that right in the middle of the eight colours, simple. The Polychromatic Notation
charts colour this off-white to match the lightness of the yellow below and the green above, but in writing our notation these will just be the
normal black notes. Kind of like writing regular black notes on off-white paper!
We find we can use this Polychromatic Notation for a large number of high number note systems. Green, Blue, Violet and Ultra-violet
proceed upwards pitch wise and Yellow, Orange, Red and Infra-red proceed downwards pitch wise from our normal notation pitch. All three
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Polychromatic systems for 31Et, 43Et and 53Et follow the same layout even tough 53Et differs slightly from 31Et and 43Et. This works
because all three systems are based upon the cycle of perfect fifths. For all three systems, miraculously if we add together 12 perfect fifths
from C, when we arrive back at the first note (B#), we have simply moved down one step for 31Et and 43Et or up one step for 53Et. This also
goes to show how the pitch center of systems that have perfect fifths smaller than 700 cents descend while the pitch center of systems that
have perfect fifths larger than 700 cents ascend.
Further in trying to come to terms with the complexities of creating our layout we see that all three systems of 31Et, 43Et and 53Et in the end
produce a similar layout with exactly the same crossover points between the different colours or inflections. These crossover or shared pitches
between different levels exist exactly where we find them in our familiar notation; on C#/Db, F#/Gb and B/Cb. This is also somewhat
miraculous. In the end everything looks so logical, simple, self-obvious and normal that it is hard to believe any effort had to be taken at all to
find the system. To reach what seems to be this self-consistent natural system however took a lot of time and effort to create or discover!
Looking at the polychromatic layout as well we can see how it can easily become the basis of an “Extended Tonality Keyboard” colours and
all!
And so we can now see how we can simply extend our familiar tonality to a high number of divisions of the octave, never losing simple
comprehension of where we are, with just a simple set of inflections or polychromatic notation and a simple layout. Each scale degree can
belong to a number of polychromatic levels and so moving from one level to another means no more than the equivalent of changing from say
F# major to Gb major in our familiar system. Every single scale degree of the octave for the three layouts for 31Et, 43Et and 53Et is as well its
own key center with of course dual keys on C#/Db, F#/Gb and B/Cb. The systems are perfect and complete and every note of every key can
be found. Not only the major keys but also all the minor keys including the remote minor keys on A#/Bb, D#/Eb and G#/Ab.
We aren’t finished yet though! For every single scale degree we can perfectly also find the notation for all seven modes in 31Et and 43Et
and all eight modes for 53Et. In order to make available the minor keys of G#, D# and A# minor found in our familiar notation, we have to
extend three perfect fifths beyond C#. To keep everything symmetrical we will also then extend three perfect fifths below Cb giving as well Fb,
Bbb and Ebb as key or modal centers. Our entire range of complete modes or keys then extends from Ebb all the way A#. On each degree of
this range we can find all the modes from Ebb Locrian right through to A# Lydian. The entire range of each polychromatic colour or inflection
then is:
Bbbb | Fbb Cbb Gbb Dbb Abb Ebb Bbb | Fb Cb Gb Db Ab Eb Bb | F C G D A E B | F# C# G# D# A# E# B# | Fx Cx Gx Dx
The system is entirely complete right across every scale degree and key. This is how we extend our familiar tonality to higher number divisions
of the octave! Each colour or inflection consists of a cycle of 32 perfect fifths and 33 notes. We can also see in the polychromatic layout
multiple chromatic scales, one per colour or inflection, hence a polychromatic layout for our polychromatic notation!
We have mentioned above in detail the differences between 12Et, 31Et, 43Et as a group and 53Et. In 12Et, 31Et and 43Et each system of
keys or modes is contained within a single cycle of perfect fifths and its corresponding colour or inflection. We can simply cross over levels of
colours or inflections as need be to keep the notation of our extended tonality as simple as possible. For the 8 note diatonic scale in 53Et to be
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contained in a single colour or inflection requires two completely separate groups of four notes (separated by fifths), with the two groups
themselves separated by sixth fifths:
Ebb Bbb Fb Cb ( Gb Db Ab Eb Bb ) F C G D
This though once again gives us the mind-boggling mess of enharmonic scale and chord writing. We can get around this simply by notating
the eight note diatonic scale in 53Et using two different adjacent levels and their corresponding colour or inflection:
F C G D
D! A! E! B!
Things look normal once again besides needing two different colours or inflections. To write or analyze music now in just intonation becomes
incredibly simple and visually intuitive. Perfect fifths are always of the same colour or inflection. Major thirds are always one colour downwards
or one additional downward inflection more. Minor thirds are always one colour upwards or one additional upwards inflection more. Things
could not be simpler or clearer. We could simply analyze a Renaissance motet in just intonation and see where it comes out if all the
relationships between voices must always be perfectly just. In this perfect 5 Limit just intonation world, which we can almost perfectly
approximate by using 53Et, and completely and simply notate using our Polychromatic Notation, it would be more than interesting to see
where each piece might drift to and end up in comparison to it’s starting point. As well, composing in 5 Limit just intonation through 53Et with
Polychromatic Notation is notationally as clear and easy as day!
With 31Et and 43Et we can see how the pitch center drops through successive fifths, while with 53Et the pitch center rises with successive
fifths. Right in the middle is the polychromatic layout for the 12nEt system which includes 12Et, 24Et, 36Et, 48Et, 60Et, 72Et, 84Et, 96Et and
108Et, where the pitch center neither rises nor falls. To put it simply I am not at all a fan of this system. 24Et and 36Et present equal
temperaments that are no more in tune than 12Et. 48Et can be said to be more in tune but the fifth and thirds are tempered in the wrong
direction which makes it even worse. Sure 96Et has good 5, 7, 11 and 13 Limit ratios, but with that many notes to the octave how can it not
approximate well these Limits. The blind usage of 24Et or 36Et to expand our tonal resources completely belies a lack of understanding of
temperaments and tunings. 24Et and 36Et seem to be a default mode when little or no thought has gone into microtonal systems except to
think that doubling or tripling the number of notes in the octave must be what it means to write or play microtonal music. Further I don’t include
the 12nEt systems in the category of “Extended Tonality”. What we are really dealing with here with are parallel 12Et systems separated by
some division of the semitone without any crossover points to move from one level through to another level. I instead class these
temperaments (along with 34Et (2 x17Et)) under the category “Parallel Tonalities”.
For 31Et, 43Et and 53Et the charts also show the “Generalised Isomorphic Layout” for each temperament along with their “Isomorphic
Keyboard Patterns” for each mode. For 31Et, the limited notational system of the Fokker organ, which is the same as the simplified system of
microtonal inflections found earlier, is compared to a complete polychromatic notation of only three colours or inflections (!,o,"). The original
simplified notation for the Fokker organ is incomplete for all possible scale degrees.
In showing the Polychromatic layouts we can see that different scale degrees duplicate in higher and lower levels. These pitch duplications
are necessary to maintain consistently and without confusion the layout of the different polychromatic levels. The layouts given then are the
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minimum complete systems where every mode can be created on every key with as few pitch duplications as possible. Pitch extensions are
also given if we need to duplicate more pitches for ergonomic or layout reasons. If we had a physical keyboard wrapped like a spiral around a
barbershop pole we could avoid any pitch duplication whatsoever. The spirals of each different temperament however would have to have a
slightly different note spacing to take into consideration the different note relations of each equal temperament.
Done! the Polychromatic Layouts and Notations are complete for Extended Tonality!
Partch 43 Tone Scale
Partch created great structure in his 43 tone scale, yet by his own admission he did not create a universal notation to compose in it. I have
now done so. Partch’s system of 43 tones is comprehensive in itself but for his compositions he created a different notation particular to each
of his instruments. To compose abstractly in it, it is necessary to have a universal notational system for the Partch 43 Tone Scale in the same
way we have a universal notational system when composing for Western instruments and voices with 12 notes to the octave. It is necessary
also that by this universal notation that we can as well understand the theoretical and acoustic foundations of the Partch 43 Tone Scale.
Partch chose to base his scale in Just Intonation and to expand his tonal and harmonic resources beyond our twelve notes to the octave by
the introduction of 7 and 11 Limit ratios. As mentioned above, 12Et is the worst of all temperaments for 7, 11 and 13 Limit ratios. 12Et does
not even come close to approximating the intervals of these Limit ratios. Partch’s 2, 3 and 5 Limit ratios are the same diatonic pitches we use
except with the perfection of Just Intonation. These represent no problems or demands to our notational system except we also now have
pitches that have the same letter name but that are separated by the Syntonic Comma of 81/80 (21.51 cents).
Briefly then, the simple 2, 3 and 5 Limit ratios are given by nothing more than naturals, sharps and flats. For the 2, 3 and 5 Limit ratios that
are a Syntonic Comma sharp or flat we simply add ! or " to the above notes to show the difference of the 81/80 Syntonic Comma, between
say G and G! or G". 7 Limit ratios for the most part are around a third of a semitone sharp or flat. Hence the 7 Limit ratios are all grouped
together by #!, #" and also by 1/6!, 1/6" symbols in front of the written pitch. The 11 Limit ratios for the most part are around a half of a
semitone sharp or flat. Hence the 11 Limit ratios are grouped together by ½!, ½" and also by 2/3!, 2/3" 5/6!, 5/6" symbols in front of the written
pitch. And so we can comprehensively and logically with full understanding of the 43 tone scale structure compose using our regular notation
with just the addition of a few fractional symbols. The fractional symbols are of course approximate and for the most part, also consistent with
one another. If we decided to write the pitches using polychromatic notation for the different Limits we wouldn’t need the fractional symbols at
all, instead creating a very colourful, comprehensive and clear score with complete understanding of the relationship between the different
Limits and Tonalities!
And so simply we have a universal notation for the Partch 43 Tone Scale and with this notation we can also have an immediate
understanding of the musical system we are in. The notation reveals to us the system, that is, when we have learned to understand how
Partch organized his scale with Otonalities, Utonalities, Identities and so forth! And so for a greater understanding of the depth of this scale the
reader is directed to the chapter on the Partch 43 Tone Scale!
10
Polychromatic Notation and Extended Tonality © Copyright Juhan Puhm 2016
Compendium Musica
11
Polychromatic Notation and Extended Tonality © Copyright Juhan Puhm 2016
Compendium Musica
12 Tone Equal Temperament
12ET
Ratio
Major Scale Intervals -> 2 2 1 2 2 2 1
Cents
+/- from 12ET
5 Limit
+/- from Just
Avg.->
2^(12/12)
2^(11/12)
2^(10/12)
2^(9/12)
2^(8/12)
2^(7/12)
2^(6/12)
2^(5/12)
2^(4/12)
2^(3/12)
2^(2/12)
2^(1/12)
2^(0/12)
2
1.887749
1.781797
1.681793
1.587401
1.498307
1.414214
1.334840
1.259921
1.189207
1.122462
1.059463
1
1200
1100
1000
900
800
700
600
500
400
300
200
100
0
0
0
0
0
0
0
0
0
0
0
0
0
0
31 Tone Equal Temperament
31ET
Ratio
2/1
15/8
16/9
5/3
8/5
3/2
45/32
4/3
5/4
6/5
9/8
16/15
1/1
1200
1088.27
996.09
9/5 1017.60
884.36
813.69
701.96
590.22 64/45 609.78
498.04
386.31
315.64
203.91
10/9 182.40
111.73
0
0
11.73
3.91
15.64
-13.69
-1.96
9.78
1.96
13.69
-15.64
-3.91
-11.73
0
12
2
1.955777
1.912532
1.870243
1.828889
1.788450
1.748905
1.710234
1.672418
1.635438
1.599276
1.563914
1.529334
1.495518
1.462450
1.430113
1.398491
1.367568
1.337329
1.307759
1.278843
1.250566
1.222914
1.195873
1.169431
1.143573
1.118287
1.093560
1.069380
1.045734
1.022611
1
+/- from Just
Avg.->
-17.60
7/4
12/7
14/9
968.83
933.13
764.92
-9.78
7/5
582.51
17.60
9/7
7/6
8/7
435.08
266.87
231.17
29.22
31.17
-33.13
35.08
10/7 617.49
17.49
-17.49
-35.08
33.13
-31.17
Major Scale Intervals -> 5 5 3 5 5 5 3
Cents
+/- from 12ET
5 Limit
+/- from Just
Avg.->
2^(31/31)
2^(30/31)
2^(29/31)
2^(28/31)
2^(27/31)
2^(26/31)
2^(25/31)
2^(24/31)
2^(23/31)
2^(22/31)
2^(21/31)
2^(20/31)
2^(19/31)
2^(18/31)
2^(17/31)
2^(16/31)
2^(15/31)
2^(14/31)
2^(13/31)
2^(12/31)
2^(11/31)
2^(10/31)
2^(9/31)
2^(8/31)
2^(7/31)
2^(6/31)
2^(5/31)
2^(4/31)
2^(3/31)
2^(2/31)
2^(1/31)
2^(0/31)
7 Limit
10.61
1200
1161.29
1122.58
1083.87
1045.16
1006.45
967.74
929.03
890.32
851.61
812.90
774.19
735.48
696.77
658.06
619.35
580.65
541.94
503.23
464.52
425.81
387.10
348.39
309.68
270.97
232.26
193.55
154.84
116.13
77.42
38.71
0
0
-38.71
22.58
-16.13
45.16
6.45
-32.26
29.03
-9.68
-48.39
12.90
-25.81
35.48
-3.23
-41.94
19.35
-19.35
41.94
3.23
-35.48
25.81
-12.90
48.39
9.68
-29.03
32.26
-6.45
-45.16
16.13
-22.58
38.71
0
2/1
1200
0
15/8
1088.27
-4.40
16/9
996.09
9/5
1017.60
10.36
5/3
884.36
5.96
8/5
813.69
-0.78
3/2
701.96
-5.18
64/45
45/32
609.78
590.22
9.58
-9.58
4/3
498.04
5.18
5/4
386.31
0.78
6/5
315.64
-5.96
9/8
203.91
10/9
182.40
-10.36
16/15
111.73
4.40
1/1
0
0
7 Limit
+/- from Just
6.77
Avg.->
-11.14
11.14
Polychromatic Notation and Extended Tonality © Copyright Juhan Puhm 2016
7/4
12/7
968.83
933.13
-1.08
-4.10
14/9
764.92
9.28
10/7
7/5
617.49
582.51
1.87
-1.87
9/7
435.08
-9.28
7/6
8/7
266.87
231.17
4.10
1.08
4.08
Compendium Musica
12 ET
"keyboard mapping"
12ET
Ratio
Cents
11 Limit
+/- from Just
Avg.->
0,12 8va
0,11 +7
0,10
-7
0,9
+6
0,8
-6
0,7
P5
0,6
x4
0,5
P4
0,4
+3
0,3
-3
0,2
+2
0,1
-2
0,0 Unis.
Unis.
-2
+2
-3
+3
P4
x4
P5
-6
+6
-7
+7
8va
0,0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
0,10
0,11
0,12
2^(12/12)
2^(11/12)
2^(10/12)
2^(9/12)
2^(8/12)
2^(7/12)
2^(6/12)
2^(5/12)
2^(4/12)
2^(3/12)
2^(2/12)
2^(1/12)
2^(0/12)
2
1.887749
1.781797
1.681793
1.587401
1.498307
1.414214
1.334840
1.259921
1.189207
1.122462
1.059463
1
1200
1100
1000
900
800
700
600
500
400
300
200
100
0
31ET
Ratio
Cents
21/11
20/11
1119.46
1035.00
-19.46
-35.00
11/7
22/15
782.49
663.05
17.51
36.95
15/11
14/11
536.95
417.51
-36.95
-17.51
11/10
22/21
165.00
80.54
13 Limit
+/- from Just
27.23
25/13
1132.10
22/13
910.79
13/7 1071.70
Avg.->
27.61
-32.10
28.30
-10.79
21/13 830.25
13/9
636.62
13/11
289.21
26/25
67.90
18/13 563.38
-30.25
-36.62
26/21 369.75
35.00
19.46
36.62
30.25
10.79
14/13 128.30
32.10
-28.30
31 ET
"keyboard mapping"
11 Limit
+/- from Just
Avg.->
0,31
0,30
0,29
0,28
0,27
0,26
0,25
0,24
0,23
0,22
0,21
0,20
0,19
0,18
0,17
0,16
0,15
0,14
0,13
0,12
0,11
0,10
0,9
0,8
0,7
0,6
0,5
0,4
0,3
0,2
0,1
0,0
P5
x4
P4
+3
-3
+2
-2
2-8va
+7
-7
+6
-6
P5
x4
P4
+3
-3
+2
-2
8va
+7
-7
+6
-6
P5
x4
P4
+3
-3
+2
-2
Unis.
13
Unis.
-2
+2
-3
+3
P4
x4
P5
-6
+6
-7
+7
8va
-2
+2
-3
+3
P4
x4
P5
-6
+6
-7
+7
2-8va
-2
+2
-3
+3
P4
x4
P5
0,0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
0,10
0,11
0,12
0,13
0,14
0,15
0,16
0,17
0,18
0,19
0,20
0,21
0,22
0,23
0,24
0,25
0,26
0,27
0,28
0,29
0,30
0,31
2^(31/31)
2^(30/31)
2^(29/31)
2^(28/31)
2^(27/31)
2^(26/31)
2^(25/31)
2^(24/31)
2^(23/31)
2^(22/31)
2^(21/31)
2^(20/31)
2^(19/31)
2^(18/31)
2^(17/31)
2^(16/31)
2^(15/31)
2^(14/31)
2^(13/31)
2^(12/31)
2^(11/31)
2^(10/31)
2^(9/31)
2^(8/31)
2^(7/31)
2^(6/31)
2^(5/31)
2^(4/31)
2^(3/31)
2^(2/31)
2^(1/31)
2^(0/31)
2
1.955777
1.912532
1.870243
1.828889
1.788450
1.748905
1.710234
1.672418
1.635438
1.599276
1.563914
1.529334
1.495518
1.462450
1.430113
1.398491
1.367568
1.337329
1.307759
1.278843
1.250566
1.222914
1.195873
1.169431
1.143573
1.118287
1.093560
1.069380
1.045734
1.022611
1
1200
1161.29
1122.58
1083.87
1045.16
1006.45
967.74
929.03
890.32
851.61
812.90
774.19
735.48
696.77
658.06
619.35
580.65
541.94
503.23
464.52
425.81
387.10
348.39
309.68
270.97
232.26
193.55
154.84
116.13
77.42
38.71
0
21/11
1119.46
3.12
11/6
1049.36 20/11 1035.00
-4.20
18/11
852.59
-0.98
11/7
782.49
-8.30
16/11
648.68
22/15
551.32
14/11
417.51
8.30
11/9
347.41
0.98
150.64
22/21
80.54
11/10
536.95
9.38
11/8
12/11
15/11
663.05
165.00
-9.38
4.20
-3.12
Polychromatic Notation and Extended Tonality © Copyright Juhan Puhm 2016
13 Limit
+/- from Just
5.88
10.17
Avg.->
25/13
13/7
24/13
1132.10
1071.70
1061.43
-9.52
12.17
-16.27
26/15
22/13
952.26
910.79
15.48
18.24
13/8
21/13
840.53
830.25
11.09
-17.35
20/13
745.79
-10.30
13/9
18/13
636.62
563.38
-17.26
17.26
13/10
454.21
10.30
26/21
16/13
369.75
359.47
17.35
-11.09
13/11
15/13
289.21
247.74
-18.24
-15.48
13/12
14/13
26/25
138.57
128.30
67.90
16.27
-12.17
9.52
-4.98
4.98
-10.17
14.19
Compendium Musica
43 Tone Equal Temperament
43ET
Ratio
Major Scale Intervals -> 7 7 4 7 7 7 4
Cents
+/- from 12ET
5 Limit
+/- from Just
Avg.->
2^(43/43)
2^(42/43)
2^(41/43)
2^(40/43)
2^(39/43)
2^(38/43)
2^(37/43)
2^(36/43)
2^(35/43)
2^(34/43)
2^(33/43)
2^(32/43)
2^(31/43)
2^(30/43)
2^(29/43)
2^(28/43)
2^(27/43)
2^(26/43)
2^(25/43)
2^(24/43)
2^(23/43)
2^(22/43)
2^(21/43)
2^(20/43)
2^(19/43)
2^(18/43)
2^(17/43)
2^(16/43)
2^(15/43)
2^(14/43)
2^(13/43)
2^(12/43)
2^(11/43)
2^(10/43)
2^(9/43)
2^(8/43)
2^(7/43)
2^(6/43)
2^(5/43)
2^(4/43)
2^(3/43)
2^(2/43)
2^(1/43)
2^(0/43)
14
2
1.968019
1.936549
1.905583
1.875112
1.845128
1.815624
1.786591
1.758022
1.729911
1.702249
1.675029
1.648244
1.621888
1.595953
1.570433
1.545321
1.520611
1.496296
1.472369
1.448825
1.425658
1.402861
1.380429
1.358355
1.336634
1.315261
1.294229
1.273534
1.253169
1.233131
1.213412
1.194009
1.174916
1.156129
1.137642
1.119450
1.101550
1.083936
1.066603
1.049547
1.032765
1.016250
1
1200
1172.09
1144.19
1116.28
1088.37
1060.47
1032.56
1004.65
976.74
948.84
920.93
893.02
865.12
837.21
809.30
781.40
753.49
725.58
697.67
669.77
641.86
613.95
586.05
558.14
530.23
502.33
474.42
446.51
418.60
390.70
362.79
334.88
306.98
279.07
251.16
223.26
195.35
167.44
139.53
111.63
83.72
55.81
27.91
0
0
-27.91
44.19
16.28
-11.63
-39.53
32.56
4.65
-23.26
48.84
20.93
-6.98
-34.88
37.21
9.30
-18.60
-46.51
25.58
-2.33
-30.23
41.86
13.95
-13.95
-41.86
30.23
2.33
-25.58
46.51
18.60
-9.30
-37.21
34.88
6.98
-20.93
-48.84
23.26
-4.65
-32.56
39.53
11.63
-16.28
-44.19
27.91
0
2/1
1200
0
15/8
1088.27
0.10
16/9
996.09
9/5
1017.60
8.56
5/3
884.36
8.66
8/5
813.69
-4.38
3/2
701.96
-4.28
64/45
45/32
609.78
590.22
4.18
-4.18
4/3
498.04
4.28
5/4
386.31
4.38
6/5
315.64
-8.66
9/8
203.91
10/9
182.40
-8.56
16/15
111.73
-0.10
1/1
0
0
Polychromatic Notation and Extended Tonality © Copyright Juhan Puhm 2016
7 Limit
+/- from Just
6.16
Avg.->
-12.95
12.95
7/4
968.83
7.92
12/7
933.13
-12.20
14/9
764.92
-11.43
10/7
7/5
617.49
582.51
-3.53
3.53
9/7
435.08
11.43
7/6
266.87
12.20
8/7
231.17
-7.92
8.77
Compendium Musica
43 ET
"keyboard mapping"
43ET
Ratio
Cents
11 Limit
+/- from Just
Avg.->
0,43
P5
Unis.
0,0
0,42
x4
-2
0,1
0,41
P4
+2
0,2
0,40
+3
-3
0,3
0,39
-3
+3
0,4
0,38
+2
P4
0,5
0,37
-2
x4
0,6
0,36 3-8va
0,35 +7
P5
0,7
0,8
-6
0,34
-7
+6
0,9
0,33
0,32
+6
-7
-6
+7
0,10
0,11
0,31
P5
8va
0,12
0,30
x4
-2
0,13
0,29
P4
+2
0,14
0,28
+3
-3
0,15
0,27
-3
+3
0,16
0,26
+2
P4
0,17
0,25
-2
x4
0,18
0,24
2-8va
P5
0,19
0,23
+7
-6
0,20
0,22
0,21
-7
+6
+6
-7
0,21
0,22
0,20
-6
+7
0,23
0,19
P5
2-8va
0,24
0,25
0,18
x4
-2
0,17
P4
+2
0,26
0,16
+3
-3
0,27
0,15
-3
+3
0,28
0,14
+2
P4
0,29
0,13
-2
x4
0,30
0,12
8va
P5
0,31
0,11
0,10
+7
-6
-7
+6
0,32
0,33
0,9
+6
-7
0,34
0,8
0,7
-6
+7
P5
3-8va
0,35
0,36
0,6
x4
-2
0,37
0,5
P4
+2
0,38
0,4
+3
-3
0,39
0,3
-3
+3
0,40
0,2
+2
P4
0,41
0,1
-2
x4
0,42
0,0
Unis.
P5
0,43
15
2^(43/43)
2^(42/43)
2^(41/43)
2^(40/43)
2^(39/43)
2^(38/43)
2^(37/43)
2^(36/43)
2^(35/43)
2^(34/43)
2^(33/43)
2^(32/43)
2^(31/43)
2^(30/43)
2^(29/43)
2^(28/43)
2^(27/43)
2^(26/43)
2^(25/43)
2^(24/43)
2^(23/43)
2^(22/43)
2^(21/43)
2^(20/43)
2^(19/43)
2^(18/43)
2^(17/43)
2^(16/43)
2^(15/43)
2^(14/43)
2^(13/43)
2^(12/43)
2^(11/43)
2^(10/43)
2^(9/43)
2^(8/43)
2^(7/43)
2^(6/43)
2^(5/43)
2^(4/43)
2^(3/43)
2^(2/43)
2^(1/43)
2^(0/43)
2
1.968019
1.936549
1.905583
1.875112
1.845128
1.815624
1.786591
1.758022
1.729911
1.702249
1.675029
1.648244
1.621888
1.595953
1.570433
1.545321
1.520611
1.496296
1.472369
1.448825
1.425658
1.402861
1.380429
1.358355
1.336634
1.315261
1.294229
1.273534
1.253169
1.233131
1.213412
1.194009
1.174916
1.156129
1.137642
1.119450
1.101550
1.083936
1.066603
1.049547
1.032765
1.016250
1
1200
1172.09
1144.19
1116.28
1088.37
1060.47
1032.56
1004.65
976.74
948.84
920.93
893.02
865.12
837.21
809.30
781.40
753.49
725.58
697.67
669.77
641.86
613.95
586.05
558.14
530.23
502.33
474.42
446.51
418.60
390.70
362.79
334.88
306.98
279.07
251.16
223.26
195.35
167.44
139.53
111.63
83.72
55.81
27.91
0
21/11
1119.46
-3.18
11/6
20/11
1049.36
1035.00
11.10
-2.44
18/11
11/7
852.59
782.49
13 Limit
+/- from Just
6.27
Avg.->
25/13
1132.10
24/13
1061.43
26/15
22/13
952.26
910.79
13/8
840.53
20/13
745.79
7.70
13/9
636.62
5.24
18/13
563.38
-5.24
13/10
454.21
-7.70
16/13
359.47
13/11
15/13
289.21
247.74
13/12
138.57
26/25
67.90
6.79
12.09
13/7 1071.70
-0.96
-11.24
-3.42
10.14
12.52
21/13 830.25
-3.32
6.96
-1.10
22/15
16/11
663.05
648.68
6.72
-6.82
11/8
15/11
551.32
536.95
6.82
-6.72
14/11
417.51
1.10
11/9
347.41
-12.52
11/10
12/11
165.00
150.64
2.44
-11.10
22/21
80.54
3.18
Polychromatic Notation and Extended Tonality © Copyright Juhan Puhm 2016
26/21 369.75
3.32
-6.96
-10.14
3.42
14/13 128.30
0.96
-12.09
11.24
Compendium Musica
53 Tone Equal Temperament
53ET
Ratio
Major Scale Intervals -> 8 1 8 5 9 8 9 5
Cents
+/- from 12ET
5 Limit
+/- from Just
Avg.->
2^(53/53)
2^(52/53)
2^(51/53)
2^(50/53)
2^(49/53)
2^(48/53)
2^(47/53)
2^(46/53)
2^(45/53)
2^(44/53)
2^(43/53)
2^(42/53)
2^(41/53)
2^(40/53)
2^(39/53)
2^(38/53)
2^(37/53)
2^(36/53)
2^(35/53)
2^(34/53)
2^(33/53)
2^(32/53)
2^(31/53)
2^(30/53)
2^(29/53)
2^(28/53)
2^(27/53)
2^(26/53)
2^(25/53)
2^(24/53)
2^(23/53)
2^(22/53)
2^(21/53)
2^(20/53)
2^(19/53)
2^(18/53)
2^(17/53)
2^(16/53)
2^(15/53)
2^(14/53)
2^(13/53)
2^(12/53)
2^(11/53)
2^(10/53)
2^(9/53)
2^(8/53)
2^(7/53)
2^(6/53)
2^(5/53)
2^(4/53)
2^(3/53)
2^(2/53)
2^(1/53)
2^(0/53)
16
2
1.974014
1.948365
1.923050
1.898064
1.873402
1.849061
1.825036
1.801323
1.777918
1.754817
1.732017
1.709512
1.687301
1.665377
1.643739
1.622382
1.601302
1.580496
1.559960
1.539692
1.519686
1.499941
1.480452
1.461216
1.442231
1.423492
1.404996
1.386741
1.368723
1.350939
1.333386
1.316061
1.298961
1.282084
1.265426
1.248984
1.232756
1.216738
1.200929
1.185325
1.169924
1.154723
1.139720
1.124911
1.110295
1.095869
1.081630
1.067577
1.053705
1.040015
1.026502
1.013164
1
1200
1177.36
1154.72
1132.08
1109.43
1086.79
1064.15
1041.51
1018.87
996.23
973.58
950.94
928.30
905.66
883.02
860.38
837.74
815.09
792.45
769.81
747.17
724.53
701.89
679.25
656.60
633.96
611.32
588.68
566.04
543.40
520.75
498.11
475.47
452.83
430.19
407.55
384.91
362.26
339.62
316.98
294.34
271.70
249.06
226.42
203.77
181.13
158.49
135.85
113.21
90.57
67.92
45.28
22.64
0
0
-22.64
-45.28
32.08
9.43
-13.21
-35.85
41.51
18.87
-3.77
-26.42
-49.06
28.30
5.66
-16.98
-39.62
37.74
15.09
-7.55
-30.19
47.17
24.53
1.89
-20.75
-43.40
33.96
11.32
-11.32
-33.96
43.40
20.75
-1.89
-24.53
-47.17
30.19
7.55
-15.09
-37.74
39.62
16.98
-5.66
-28.30
49.06
26.42
3.77
-18.87
-41.51
35.85
13.21
-9.43
-32.08
45.28
22.64
0
2/1
1200
0
15/8
1088.27
-1.48
9/5
16/9
1017.60
996.09
1.27
0.14
5/3
884.36
-1.34
8/5
813.69
1.41
3/2
701.96
-0.07
64/45
45/32
609.78
590.22
1.54
-1.54
4/3
498.04
0.07
5/4
386.31
-1.41
6/5
315.64
1.34
9/8
10/9
203.91
182.40
-0.14
-1.27
16/15
111.73
1.48
1/1
0
0
7 Limit
+/- from Just
0.95
Polychromatic Notation and Extended Tonality © Copyright Juhan Puhm 2016
Avg.->
7/4
968.83
4.76
12/7
933.13
-4.83
14/9
764.92
4.90
10/7
7/5
617.49
582.51
-6.17
6.17
9/7
435.08
-4.90
7/6
266.87
4.83
8/7
231.17
-4.76
5.16
Compendium Musica
53 ET
"keyboard mapping"
53ET
Ratio
Cents
11 Limit
+/- from Just
Avg.->
0,53
0,52
0,51
0,50
0,49
0,48
0,47
0,46
0,45
0,44
0,43
0,42
0,41
0,40
0,39
0,38
0,37
0,36
0,35
0,34
0,33
0,32
0,31
0,30
0,29
0,28
0,27
0,26
0,25
0,24
0,23
0,22
0,21
0,20
0,19
0,18
0,17
0,16
0,15
0,14
0,13
0,12
0,11
0,10
0,9
0,8
0,7
0,6
0,5
0,4
0,3
0,2
0,1
0,0
17
P4
+3
-3
+2
-2
4-8va
+7
-7
+6
-6
P5
x4
P4
+3
-3
+2
-2
3-8va
+7
-7
+6
-6
P5
x4
P4
+3
-3
+2
-2
2-8va
+7
-7
+6
-6
P5
x4
P4
+3
-3
+2
-2
8va
+7
-7
+6
-6
P5
x4
P4
+3
-3
+2
-2
Unis.
Unis.
-2
+2
-3
+3
P4
x4
P5
-6
+6
-7
+7
8va
-2
+2
-3
+3
P4
x4
P5
-6
+6
-7
+7
2-8va
-2
+2
-3
+3
P4
x4
P5
-6
+6
-7
+7
3-8va
-2
+2
-3
+3
P4
x4
P5
-6
+6
-7
+7
4-8va
-2
+2
-3
+3
P4
0,0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
0,10
0,11
0,12
0,13
0,14
0,15
0,16
0,17
0,18
0,19
0,20
0,21
0,22
0,23
0,24
0,25
0,26
0,27
0,28
0,29
0,30
0,31
0,32
0,33
0,34
0,35
0,36
0,37
0,38
0,39
0,40
0,41
0,42
0,43
0,44
0,45
0,46
0,47
0,48
0,49
0,50
0,51
0,52
0,53
2^(53/53)
2^(52/53)
2^(51/53)
2^(50/53)
2^(49/53)
2^(48/53)
2^(47/53)
2^(46/53)
2^(45/53)
2^(44/53)
2^(43/53)
2^(42/53)
2^(41/53)
2^(40/53)
2^(39/53)
2^(38/53)
2^(37/53)
2^(36/53)
2^(35/53)
2^(34/53)
2^(33/53)
2^(32/53)
2^(31/53)
2^(30/53)
2^(29/53)
2^(28/53)
2^(27/53)
2^(26/53)
2^(25/53)
2^(24/53)
2^(23/53)
2^(22/53)
2^(21/53)
2^(20/53)
2^(19/53)
2^(18/53)
2^(17/53)
2^(16/53)
2^(15/53)
2^(14/53)
2^(13/53)
2^(12/53)
2^(11/53)
2^(10/53)
2^(9/53)
2^(8/53)
2^(7/53)
2^(6/53)
2^(5/53)
2^(4/53)
2^(3/53)
2^(2/53)
2^(1/53)
2^(0/53)
2
1.974014
1.948365
1.923050
1.898064
1.873402
1.849061
1.825036
1.801323
1.777918
1.754817
1.732017
1.709512
1.687301
1.665377
1.643739
1.622382
1.601302
1.580496
1.559960
1.539692
1.519686
1.499941
1.480452
1.461216
1.442231
1.423492
1.404996
1.386741
1.368723
1.350939
1.333386
1.316061
1.298961
1.282084
1.265426
1.248984
1.232756
1.216738
1.200929
1.185325
1.169924
1.154723
1.139720
1.124911
1.110295
1.095869
1.081630
1.067577
1.053705
1.040015
1.026502
1.013164
1
1200
1177.36
1154.72
1132.08
1109.43
1086.79
1064.15
1041.51
1018.87
996.23
973.58
950.94
928.30
905.66
883.02
860.38
837.74
815.09
792.45
769.81
747.17
724.53
701.89
679.25
656.60
633.96
611.32
588.68
566.04
543.40
520.75
498.11
475.47
452.83
430.19
407.55
384.91
362.26
339.62
316.98
294.34
271.70
249.06
226.42
203.77
181.13
158.49
135.85
113.21
90.57
67.92
45.28
22.64
0
21/11
1119.46
-10.03
11/6
1049.36 20/11 1035.00
-7.85
18/11
11/7
16/11
11/8
852.59
782.49
648.68 22/15 663.05
551.32 15/11 536.95
7.92
-7.92
-9.96
11/9
347.41
-7.79
22/21
80.54
Avg.->
25/13
1132.10
-0.02
24/13
1061.43 13/7 1071.70
2.72
26/15
952.26
-1.32
22/13
910.79
-5.13
13/8
840.53 21/13 830.25
-2.79
20/13
745.79
1.38
13/9
636.62
-2.66
18/13
563.38
2.66
13/10
454.21
-1.38
16/13
359.47 26/21 369.75
2.79
13/11
289.21
5.13
15/13
247.74
1.32
13/12
138.57 14/13 128.30
-2.72
26/25
67.90
0.02
3.45
-7.55
6.51
7.48
9.96
417.51
150.64 11/10 165.00
+/- from Just
7.79
14/11
12/11
13 Limit
8.07
7.85
-6.45
6.45
-6.51
10.03
Polychromatic Notation and Extended Tonality © Copyright Juhan Puhm 2016
-7.48
7.55
Compendium Musica
Notation for 12 Notes Per Octave
Scale degrees by the Cycle of Fifths (7n mod 12)
0
B#
C
Cycle
of
Fifths
1
2
Bx
C#
Cx
3
Db
Ebb
Eb
Fbb
E
Fb
E#
F
6
7
Ex
F#
Fx
8
9
Gb
10
Gx
G#
G
Gbb
Ab
Bbb
12
Ax
A#
A
Abb
11
Bb
Cbb
B
Cb
B#
C
7 Note Diatonic Scales by the Cycle of Fifths (monochromatic notation)
Bx
Cx
Dx
Ex
B#
Cx
Dx
Ex
B#
Cx
Dx
E#
B#
Cx
Dx
E#
B#
Cx
D#
E#
B#
Cx
D#
E#
B#
C#
D#
E#
Cx
Fx
B#
E#
A#
D#
G#
1
6
11
4
9
2
7
C#
F#
B
E
A
D
G
B#
C
D
E
E
E
0
C
C
D
E
5
10
3
8
1
6
11
F
Bb
Eb
Ab
Db
Gb
Cb
C
C
C
C
C
D
D
D
E
4
9
2
7
0
5
10
Fb
Bbb
Ebb
Abb
Dbb
Gbb
Cbb
C#
C#
C#
C#
C#
C#
D#
D#
D#
D#
D
D
Db
Db
Db
Db
Db
Db
Db
Dbb
Dbb
Dbb
Dbb
E
B
F
G
G
G
G
A
A
Eb
Fb
Gb
Gb
Gb
Gb
Gbb
Gbb
Gbb
Gb
Ab
Ab
Ab
Ab
Gb
Ab
Ab
Abb
Abb
Abb
Abb
Abb
Polychromatic Notation and Extended Tonality © Copyright Juhan Puhm 2016
Bb
Bb
Bb
Bb
Bb
Bb
Bb
Ab
Bbb
Bbb
Bbb
Bbb
Bbb
Bbb
Cbb
Cbb
2
7
0
5
10
3
8
1
6
11
4
9
2
7
C
C
0
C
C
C
C
C
F
Bb
Eb
Ab
Db
Gb
Cb
5
10
3
8
1
6
11
Fb
Bbb
Ebb
Abb
Dbb
Gbb
Cbb
4
9
2
7
0
5
10
Cb
Cb
Cb
Cb
Cb
Cb
Cb
Bbb
Cx
Fx
B#
E#
A#
D#
G#
C
B
A
A
Keys
C#
F#
B
E
A
D
G
B#
G
G
A
B#
B
F
Gb
Fbb
A#
A#
A#
A#
A#
B#
B#
B#
B#
B#
A#
A
A
Fb
Ebb
A#
G
Eb
Eb
Eb
Eb
Fb
Fb
Fb
Fb
Fb
G#
G#
G#
G#
G#
Ax
Ax
Ax
B
B
B
B
F
F
F
F
F
Eb
F#
F#
F#
F#
F#
F#
F#
E
Eb
Ebb
Ebb
Ebb
Ebb
Ebb
E#
E#
Major Scale Intervals -> 2 2 1 2 2 2 1
Fx
Gx
Fx
Gx
Fx
Gx
Fx
Gx
Fx
Gx
Fx
G#
Fx
G#
Cycle
of
Fifths
Dbb
Keys
2
7
0
5
10
3
8
18
5
Dx
D#
D
Dbb
4
Dbb
Dbb
Dbb
Dbb
Compendium Musica
Notations for 31 Notes Per Octave
Scale degrees by the Cycle of Fifths (18n mod 31)
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
Fx#
Bx
Cx
Dx
C#
Ex
D#
C
E#
D
E
Db
Eb
Dbb
Ebb
Fx
Gx
F#
F
B
Ab
Gbb
B#
A
Gb
Fbb
A#
G
Fb
Ax
G#
Bb
Abb
Bbb
C
Cb
Cbb
Bbbb
Incomplete Notation by Microtonal Inflection
Cycle
of
Fifths
C
C!
C#
Db
D"
D
D!
D#
Eb
E"
E
F"
E#
F
F!
F#
Gb
G"
G
G!
G#
Ab
A"
A
A!
A#
Bb
B"
B
B!
C"
Cb
B#
C
Keys
7 Note Diatonic Scales by the Cycle of Fifths (monochromatic notation)
Bx
Cx
Dx
22
Gx
4
17
30
12
25
7
20
Cx
Fx
B#
E#
A#
D#
G#
2
15
28
10
23
5
18
C#
F#
B
E
A
D
G
C
D
E
E
E
0
C
C
D
E
13
26
8
21
3
16
29
F
Bb
Eb
Ab
Db
Gb
Cb
C
C
C
C
C
D
D
D
E
11
24
6
19
1
14
27
Fb
Bbb
Ebb
Abb
Dbb
Gbb
Cbb
9
Fbb
19
E!
Fb
Cx
Bx
Dx
Dx
Dx
Dx
Cx
Cx
Cx
Cx
Cx
C#
D#
D#
D#
D#
Dbb
B#
B#
A#
B#
B#
B#
B#
A#
A#
A#
F
Bb
Eb
Ab
Db
Gb
Cb
13
26
8
21
3
16
29
Cbb
Fb
Bbb
Ebb
Abb
Dbb
Gbb
Cbb
11
24
6
19
1
14
27
Cbb
Fbb
9
A
B
F
G
G
G
G
A
A
Fb
Gb
Eb
Fb
Gb
Gb
Gb
Gb
Bbb
Abb
Fbb
Gbb
Dbb
Ebb
Fbb
Gbb
Abb
Polychromatic Notation and Extended Tonality © Copyright Juhan Puhm 2016
Cb
Cb
Cb
Cb
Cb
Cb
Cb
Bbb
Bbb
Bbb
Bbb
Bbb
Bbb
Abb
Abb
Abb
Abb
Gbb
Bb
Bb
Bb
Bb
Bb
Ab
Ab
Dbb
Dbb
Gbb
Bb
Bb
Ab
Ab
Ab
Bbbb
2
15
28
10
23
5
18
C
C
C
C
C
G
Gb
4
17
30
12
25
7
20
0
F
Gb
Cx
Fx
B#
E#
A#
D#
G#
C
A
A
Ab
22
C
B
A
Ab
Gx
C
B#
B
A
G
Keys
C#
F#
B
E
A
D
G
A#
A#
A#
G
Ebb
Ebb
Ebb
Ebb
Dbb
Ax
Ax
Ax
B
B
B
B
Eb
Eb
Eb
Eb
Ebb
Gx
Gx
Gx
Gx
Gx
G#
G#
G#
G#
G#
F#
F#
F#
F#
F#
Fb
Fb
Fb
Fb
Fb
Ax
G#
F#
F
F
F
F
F
Ebb
Gx
G#
F#
E
Eb
Db
Db
Fx
Fx
Fx
Fx
Fx
E
Eb
Db
Db
Db
Db
E#
E#
E#
D
Db
Fx
E#
E#
E#
D
Fx#
Fx
E#
D#
D#
C#
C#
C#
C#
C#
Ex
Ex
D#
C#
Major Scale Intervals -> 5 5 3 5 5 5 3
Ex
Cycle
of
Fifths
Cbb
Compendium Musica
Notations for 43 Notes Per Octave
Scale degrees by the Cycle of Fifths (25n mod 43)
0
1
2
3
4
5
6
7
8
Bx#
Bx
9
10
11
12
13
14
15
Cx#
16
18
19
E#
E
Db
Eb
Ebb
Ebbb
22
23
Ex
D#
D
Dbb
21
24
25
27
Fx#
Fx
F#
F
G
Fb
Gb
Fbb
Gbb
Fbbb
26
Ex#
Dx
C#
20
Dx#
Cx
C
17
Abb
Gbbb
Abbb
Incomplete Notation by Microtonal Inflection
C
Cycle
of
Fifths
Keys
9
27
2
20
38
13
31
Cx#
Fx#
Bx
Ex
Ax
Dx
Gx
6
24
42
17
35
10
28
Cx
Fx
B#
E#
A#
D#
G#
3
21
39
14
32
7
25
C#
F#
B
E
A
D
G
0
18
36
11
29
4
22
40
20
C!
C#"
C#
Db
Db!
D"
D
D!
D#"
D#
Eb
Eb!
E"
E
E!
F"
F
F!
F#"
F#
Gb
Gb!
G"
G
G!
G#"
Fb Fb!/E#" E#
7 Note Diatonic Scales by the Cycle of Fifths (monochromatic notation)
Bx#
Cx#
Bx
Cx#
Bx
Cx#
Bx
Cx#
Bx
Cx#
Bx
Cx#
Bx
Cx
Major Scale Intervals -> 7 7 4 7 7 7 4
Dx#
Dx#
Dx#
Dx#
Dx
C#
D#
D#
D#
D#
C#
C#
C#
C#
C#
Ex
Ex
Ex
Fx
Fx
E#
Fx
Fx
Fx
Fx
Fx
E#
D#
D#
Fx#
Fx#
Fx#
Fx#
Fx#
Ex
Ex
E#
E#
E#
D#
C#
Ex
Dx
Dx
Dx
Dx
Dx
Cx
Cx
Cx
Cx
Cx
Fx#
Fx#
Ex
Dx
Cx
Bx
Ex#
Ex#
E#
E#
F#
F#
E
D
E
E
F#
F#
F#
F#
C
D
E
F#
C
C
D
E
F
G
F
Bb
Eb
Ab
Db
Gb
Cb
C
C
C
C
C
D
D
D
E
F
G
G
G
G
E
D
Db
Db
Db
Db
Eb
F
F
F
F
F
Eb
Eb
Eb
Eb
Eb
Fb
Polychromatic Notation and Extended Tonality © Copyright Juhan Puhm 2016
Gb
Gb
Gb
G
G
Compendium Musica
15
33
8
26
1
19
37
Fb
Bbb
Ebb
Abb
Dbb
Gbb
Cbb
12
30
5
23
41
16
34
Fbb
Bbbb
Ebbb
Abbb
Dbbb
Gbbb
Cbbb
Db
Db
Db
Dbb
Dbb
Dbb
Ebb
Ebb
Ebb
Ebb
Fbb
Gbb
Ebb
Fbb
Gbb
Gbb
Gbb
Gbb
Ebbb
Fbb
Fbb
Fbb
Fbb
Fbb
Ebbb
Ebbb
Ebbb
Ebbb
Ebbb
Gb
Gb
Gb
Gb
Fb
Fb
Fb
Fb
Fb
Ebb
Dbb
Dbb
Dbb
Dbb
Fb
Eb
Ebb
Fbbb
Gbb
Abb
Abb
Abb
Gbb
Gbbb
Gbbb
Gbbb
(foldout)
21
Abb
Abb
Polychromatic Notation and Extended Tonality © Copyright Juhan Puhm 2016
Abb
Abb
Abbb
Abbb
Abbb
Abbb
Abbb
Compendium Musica
28
29
30
31
32
33
34
35
36
37
38
39
40
Gx#
A#
B#
A
B
Ab
Bb
Bbb
Bbbb
Ab!
C
Cb
Cbb
Cbbb
A"
43
Ax
G#
Ab
42
Ax#
Gx
G#
41
A
A!
A#"
Dbbb
A#
Bb
Bb!
B"
B
B!
C"
C
Keys
Cycle
of
Fifths
Cx#
Fx#
Bx
Ex
Ax
Dx
Gx
9
27
2
20
38
13
31
Cx
Fx
B#
E#
A#
D#
G#
6
24
42
17
35
10
28
3
21
39
14
32
7
Cb Cb!/B#" B#
Gx#
Gx#
Gx#
Gx#
Gx#
Ax#
Ax#
Ax#
Ax
Ax
Gx
Ax
Ax
Gx
Gx
Gx
Gx
Gx
Gx
Ax
Ax
Ax
B#
B#
A#
B#
B#
B#
B#
A#
G#
A#
A#
G#
G#
G#
G#
G#
G#
A
B
B
B
A
B
C
C#
F#
B
E
A
D
G
A
B
C
C
0
C
C
C
C
C
F
Bb
Eb
Ab
Db
Gb
Cb
18
36
11
29
4
22
40
A#
A#
A#
B#
B
B
A
A
A
A
Ab
Ab
Ab
Ab
Ab
22
Bb
Bb
Bb
Bb
Bb
Bb
Bb
Cb
Cb
25
Polychromatic Notation and Extended Tonality © Copyright Juhan Puhm 2016
Compendium Musica
Ab
Ab
Bbb
Cb
Cb
Cb
Cb
Cb
Bbb
Bbb
Bbb
Bbb
Bbb
Bbb
Cbb
Cbb
Bbbb
Cbb
Cbb
Cbb
Cbb
Cbb
Bbbb
Bbbb
Bbbb
Bbbb
Bbbb
Bbbb
Cbbb
Cbbb
Dbbb
Dbbb
Dbbb
Dbbb
Fb
Bbb
Ebb
Abb
Dbb
Gbb
Cbb
15
33
8
26
1
19
37
Fbb
Bbbb
Ebbb
Abbb
Dbbb
Gbbb
Cbbb
12
30
5
23
41
16
34
(foldout)
23
Polychromatic Notation and Extended Tonality © Copyright Juhan Puhm 2016
Compendium Musica
Notations for 53 Notes Per Octave
Scale degrees by the Cycle of Fifths (31n mod 53)
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Axx
17
18
19
Bx#
Bx
20
21
22
Cx#
D#
26
E#
D
E
Db
Eb
Ebb
27
F#
F
Fb
Gb
Fbb
Gbb
Fbbb
Gbbb
Abbb
Gbbbb
(Fbbbb)
25
Dx
C#
Ebbb
24
Dx#
Cx
C
23
Cxx
(Bxx)
Ax#
B#
16
Abbbb
Incomplete Notation by Microtonal Inflection
C
Cycle
of
Fifths
Keys
11
33
2
24
46
Bx#
Ex#
Ax#
Dx#
Gx#
15
37
6
28
50
19
41
Cx#
Fx#
Bx
Ex
Ax
Dx
Gx
10
32
1
23
45
14
36
Cx
Fx
B#
E#
A#
D#
G#
5
27
49
18
40
9
B#"
31
C#
F#
B
E
A
D
G
0
C
C
24
C!
C!!
C#""
C#"
Db!
Db!!
D""
D"
D
D!
8 Note Diatonic Scales by the Cycle of Fifths (duochromatic notation)
Axx"
Ax#
Ax#
Bx#"
Ax#
Bx#"
Ax#" Ax#
Bx#"
Ax#"
Ax#"
Ax#"
Eb"
Eb
Eb!
E""
E"
E
Bx#
Bx#
Cx#"
Cx#"
Cx#"
Cx#"
Cx"
Cx"
Cx"
Cx"
Gb!
Dx#
Dx
Dx"
Dx"
Dx"
Dx"
Ex"
Ex"
Dx
Ex"
Ex"
E#
E#
D#
E#"
E#"
D#
F#
F#
E
F#"
F#"
F#"
E
D
E"
D
E"
D
E"
Polychromatic Notation and Extended Tonality © Copyright Juhan Puhm 2016
E#
E#
E#"
E#"
D#
D
D"
F#"
Dx
Cx
Cx
D#"
D#"
D#"
D#"
C
F#""
Dx#
Dx#"
Dx#"
Dx#"
Dx#"
D#
C#
C#
F!!
Dx
C#
C#
F!
Dx#
Cx#
Cx#
Cx
B#
B#
Cxx
F
Dx#
Cx
B#
Cxx"
Cxx"
Cxx"
Cxx"
F"
Cx#
Bx
Bx
Bx"
F""
Cx#
Bx
B#
E!
Major Scale Intervals -> 8 1 8 5 9 8 9 5
Bx#"
C#"
C#"
C#"
C#"
Eb!!
Bx#
Bx
Bx"
Bx"
Bx"
B#"
B#"
B#"
D!!
E
E
F#"
F
F#
F#
Compendium Musica
22
44
13
35
4
26
48
F
Bb
Eb
Ab
Db
Gb
Cb
17
39
8
30
52
21
43
Fb
Bbb
Ebb
Abb
Dbb
Gbb
Cbb
12
34
3
25
47
16
38
Fbb
Bbbb
Ebbb
Abbb
Dbbb
Gbbb
Cbbb
7
29
51
20
42
Fbbb
Bbbbb
Ebbbb
Abbbb
Dbbbb
C
C
D!
D!
D!
Eb
Db
Db
Db!
Eb!
Eb!
Eb!
Db
Db
Db!
Db!
Db!
F!
F!
F!
F!
Eb
Eb
Fb!
Fb!
Fb!
Fb!
Ebb
Ebb
Fbb!
Fbb!
Fbb!
Fbb!
Fbbb
Gbbb
Ebbb!
Fbbb
Gbbb! Gbbb
Gbbb!
Gbbb!
Gbbb!
Gbbb
Gbbbb
Polychromatic Notation and Extended Tonality © Copyright Juhan Puhm 2016
Gbb
Abbb
Abbb
Abbb
Abbb! Abbb
Abbb!
Gbbb
(foldout)
25
Gbb
Gbb
Gbb!
Gbb!
Gbb!
Gbb!
Fbb
Fbb
Ebbb
Ebbb! Ebbb
Ebbb!
Ebbb!
Fbbb
Fbbb! Fbbb
Fbbb!
Fbbb!
Gb!
Gb!
Gb!
Gb!
Fb
Fb
Gbb
Fbb
Ebb!
Ebbb
Gb
Fbb
Ebbb
Gb
Gb
Fb
Eb!
Ebb
F
F
Fb
Ebb
Ebb!
Ebb!
Ebb!
F
E!
Eb
Abbb!
Abbb!
Abbbb
Abbbb
Abbbb
Gb
Compendium Musica
28
29
30
31
32
33
34
35
36
37
Dxx
38
39
40
41
Ex
43
44
Bbb
Bbbb
Dxx"
Dxx"
Ab
Ab!
Cbb
Ab!!
A""
A"
A
A!
Ex#
Fxx"
Fxx"
Fxx"
Fxx"
Ex#
Ex#
Ex#"
Ex#"
Bb"
Bb
Bb!
Fx#"
Fx#"
Fx#"
Fx#"
Ex
Ex
B""
B"
Cb!
Fx
Fx
Ax
Ax
Ax"
Gx
Gx
Ax"
Ax"
Ax"
A#"
G#
G#
A#"
A#"
A#"
B#"
B#"
B#"
A#
A#
B#"
A"
Bx#
Ex#
Ax#
Dx#
Gx#
11
33
2
24
46
Cx#
Fx#
Bx
Ex
Ax
Dx
Gx
15
37
6
28
50
19
41
Cx
Fx
B#
E#
A#
D#
G#
10
32
1
23
45
14
36
5
27
49
18
40
9
0
A
B"
B"
A
B"
C
B"
C
C
A
G
Keys
Cycle
of
Fifths
C#
F#
B
E
A
D
G
B
B
A
A"
Ax
Ax
A#
A#
G#
G
C
Gx#
Gx#
Gx
G#
G
C"
Gx#
Gx
Gx"
Gx"
Gx"
Gx"
C""
Gx#
Fx#
Fx#
Fx
Cb!!
Gxx"
Gxx"
Gxx"
Gx#"
Gx#"
Gx#"
Gx#"
Fx
G#"
G#"
G#"
G#"
Bb!!
Fxx
Fxx
Fx#
Ex
26
Bb""
Ebbbb
Fx#
Ex
Fx"
Fx"
Fx"
Fx"
Dbb
Dbbb
Ex#
Ex#"
Ex#"
53
C
Dbbbb
Ab"
52
Cb
Cbbb
(Cbbbb)
G!!
51
B
Bb
Abb
G!
50
Ax
A
Bbbbb
49
A#
Ab
G
48
Gx
G
G"
47
Gx#
Fx
G""
46
Gxx
Fx#
G#
Gb!!
45
Fxx
(Exx)
Ex#
42
Polychromatic Notation and Extended Tonality © Copyright Juhan Puhm 2016
B
B
31
Compendium Musica
G!
G!
G!
G!
G
A!
A!
Ab
Ab
Ab!
Ab!
Bb!
Bb!
Bb!
Bb!
Ab
Ab
Ab!
Ab!
Abb!
Abb!
Bbbbb
Bbbbb
Bbbbb
Bbbbb! Bbbbb
Bbbbb!
Cb
Cb
Cb!
Cb!
Cb!
Cb!
Bbb
Bbb
Cbb!
Cbb!
Cbb!
Cbb!
Bbbb
Bbbb
Cbbb
Cbb
Dbb!
Cbb
Cbb
Dbb!
Dbb!
Dbb!
Dbbb
Dbbb
Dbbb!
Dbbb!
Dbbb!
Dbbbb
Dbbbb
(foldout)
27
Dbb
Dbb
Dbbb
Dbbb! Dbbb
Cbbb
Cbbb
Cbbb! Cbbb
Cbbb!
Cbbb!
Cbbb!
Cb
Cb
Cbb
Bbbb
Bbbb
Bbbb!
Bbbb!
Bbbb!
Bbbb!
Bb
Bb
Bbb
Bbb!
Bbb!
Bbb!
Bbb!
Abb
Abb
C!
C!
C!
C!
Bbb
Abb
Abb
Abb!
Abb!
Bb
Bb
Polychromatic Notation and Extended Tonality © Copyright Juhan Puhm 2016
Ebbbb
Ebbbb
Ebbbb
Ebbbb! Ebbbb
Dbb
Dbb
C
C
F
Bb
Eb
Ab
Db
Gb
Cb
22
44
13
35
4
26
48
Fb
Bbb
Ebb
Abb
Dbb
Gbb
Cbb
17
39
8
30
52
21
43
Fbb
Bbbb
Ebbb
Abbb
Dbbb
Gbbb
Cbbb
12
34
3
25
47
16
38
Fbbb
Bbbbb
Ebbbb
Abbbb
Dbbbb
7
29
51
20
42
Compendium Musica
Polychromatic Layout for 31Et
Minimum Complete 31Et System
Third Extension
!
(optional)
!
(31 notes + (2 x 7) duplicate notes = 45 Notes)
C#!!! 5
Db!! 5
F#!!! 18
Gb!! 18
B!!! 0
Cb!! 0
!
D#!!!
G#!!!
A#!!!
Second Extension
!
Eb!! 10
Ab!! 23
Bb!! 28
(optional)
!
Fbb!
Bbbb!
Cbb!
!
B#!!!
E#!!!
Fx!!!
B#!!!
First Extension
!
C!! 2
F!! 15
G!! 20
C!! 2
(optional)
!
Dbb!
Gbb!
Abb!
(incomplete Upper
(incomplete Upper
modes on 7)
modes on 12)
modes on 25)
Cx!!!
Dx!!!
Gx!!!
D!! 7
E!! 12
A!! 25
Complete Lydian, Ionian,
Mixolydian Upper modes and
Dorian Middle mode on each key.
Complete Aeolian mode except 30
!!
!!
!
!
!
Complete Phrygian mode except 17, 30
!
!
Complete Locrian mode except 4, 17, 30
o
!
(20, 2, 15 required for complete Lower modes)
Complete Locrian, Phrygian,
Aeolian Lower modes and
Dorian Middle mode on each key.
o
!
"
""
Complete Mixolydian mode except 1
!
"
!
""
!
B#!!
C#"" 4
Db" 4
Ebb!
Cx!!
C" 1 !!!!!!!!!!!!!!!!!!!!!!!!!!! D" 6
C#" 3
Dbb
Ebb
Db 3
B#!
Cx!
C 0 !!!!!!!!!!!!!!!!!!!!!!!!!!! D 5
C# 2
Dbb"
Ebb$
Db$ 2
B#
Cx
C$ 30!!!!!!!!!!!!!!!!!!!!!!!!!!! D$ 4
C#$ 1
Dbb""
Ebb$$
Db$$ 1
Complete Ionian mode except 14, 1
Complete Lydian mode except 27, 14, 1
(11,29,16 required for complete Upper modes)
Fb!
D#""
Dx!!
Eb" 9
E" 11
Fbb
Fb
D#"
Dx!
Eb 8
E 10
Fbb"
Fb$
D#
Dx
Eb$ 7
E$ 9
Fbb""
Fb$$
D#"
E#!!
F#"" 17
Gb" 17
B"" 30
Cb" 30
Bbb!
Fx!!
F" 14 !!!!!!!!!!!!!!!!!!!!!!!!!!! G" 19
F#" 16
Gbb
Abb
Gb 16
E#!
Fx!
F 13 !!!!!!!!!!!!!!!!!!!!!!!!!!! G 18
F# 15
Gbb"
Abb"
Gb$ 15
E#
Fx
F$ 12 !!!!!!!!!!!!!!!!!!!!!!!!!!! G$ 17
F#$ 14
Gbb""
Abb""
Gb$$ 14
G#""
Gx!!
Ab" 22
A" 24
Bbbb
Bbb
G#"
Gx!
Ab 21
A 23
Bbbb"
Bbb$
G#
Gx
Ab$ 20
A$ 22
Bbbb""
Bbb$$
G#"
A#""
B#!!
Bb" 27!!!!!!!!!!!!!!!!!!!!!!!!!!! C" 1
B" 29
Cbb
Dbb
Cb 29
A#"
B#!
Bb 26!!!!!!!!!!!!!!!!!!!!!!!!!!! C 0
B 28
Cbb"
Dbb"
Cb$ 28
A#
B#
Bb$ 25!!!!!!!!!!!!!!!!!!!!!!!!!!! C$ 30
B$ 27
Cbb""
Dbb""
Cb$$ 27
A#"
Eb"" 6
Ab"" 19
Fbb"""
Bbbb"""
Cbb"""
(incomplete Lower
(incomplete Lower
(incomplete Lower
modes on 19)
modes on 24)
modes on 6)
28
Dbb!
(incomplete Upper
Bb"" 24
!
B#"
E#"
Fx"
B#"
First Extension
!
C"" 29
F"" 11
G"" 16
C"" 29
(optional)
!
Dbb"""
Gbb"""
Abb"""
Dbb"""
!
Cx"
Dx"
Gx"
Second Extension
!
D"" 3
E"" 8
A"" 21
(optional)
!
Ebb"""
Fb"""
Bbb"""
Third Extension
!
(optional)
!
C#"" 0
Db""" 0
F#"" 13
Gb""" 13
Polychromatic Notation and Extended Tonality © Copyright Juhan Puhm 2016
B"" 26
Cb""" 26
#
!
#
o
!
Complete Upper,
#!!!!!!!!!!!!!!!!! o
Middle and Lower
"
modes on all keys
#
o
#
"
Compendium Musica
Modes for 31Et
Complete Locrian to Lydian modes from Ebb to A#
Lower Modes
Upper Modes
(retrograde inversion around C)
! "
C Locrian (Db+)
Gb - 16
Eb - 8
C Phrygian (Ab+)
Db - 3
Bb - 26
Db - 3
Bb - 26
C Aeolian (Eb+)
Ab - 21
Ab - 21
Eb - 8
Eb - 8
C-0
Eb - 8
A - 23
Bb - 26
C Ionian (C+)
G - 18
D-5
C Mixolydian (F+)
C Dorian (Bb+)
29
Eb - 8
C-0
Bb - 26
G - 18
C-0
F - 13
D-5
A - 23
Bb - 26
G - 18
A - 23
Polychromatic Notation and Extended Tonality © Copyright Juhan Puhm 2016
D-5
B - 28
C-0
F - 13
Middle Mode
(self-symmetrical around C)
G - 18
E - 10
D-5
F - 13
Bb - 26
G - 18
C-0
C Lydian (G+)
C-0
C-0
F - 13
F - 13
Ab - 21
F - 13
G - 18
E - 10
D-5
B - 28
C-0
F - 13
D-5
A - 23
F# - 15
A - 23
G - 18
E - 10
Compendium Musica
Polychromatic Layout for 31Et
Generalised Isomorphic Keyboard
B!!! 0
Cb!! 0
0
Third Extension
(optional)
F#!!! 18
Gb!! 18
18
C#!!! 5
Db!! 5
5
23
Second Extension
10
Eb!! 10
(optional)
28
Fbb!
First Extension
Bbbb!
A#!!!
Bb!! 28
E#!!!
15
B#!!!
F!! 15
2
20
C!! 2
Gbb!
Cbb!
Cx!!!
G!! 20
D!! 7
Abb!
complete Upper and Middle
25
Ebb!
Dx!!!
A!! 25
modes on top row keys
12
E!! 12
Bbb!
30
Fb!
Enharmonic dual keys
17
Start of 31Et note duplication
22
Dbb!
9
Eb! 9
Fbb
Bbbb
14
B#!!
F! 14
1
C! 1
Gbb
19
Dbb
Cx!!
G! 19
D! 6
Abb
24
Ebb
Dbb
Gx!!
Dx!!
A! 24
E! 11
Bbb
29
Fb
B! 29
Cb 29
F#! 16
Gb 16
C#! 3
Db 3
G#!
Ab 21
D#!
8
Eb 8
26
Fbb"
Bbbb"
A#!
Bb 26
E#!
13
B#!
F 13
0
C0
Gbb"
18
Dbb"
Cbb"
Cx!
G 18
D5
Abb"
23
Ebb"
Dbb"
Gx!
Dx!
A 23
10
E 10
Bbb"
28
Fb"
C# 2
Db" 2
2
B 28
Cb" 28
F# 15
Gb" 15
15
G#
Ab" 20
D#
7
Eb" 7
25
Fbb""
Bbbb""
A#
Bb" 25
E#
12
B#
F" 12
30
C" 30
Gbb""
17
Dbb""
Cbb""
B#
C" 30
Fx
Cx
G" 17
4
D" 4
Abb""
22
Ebb""
Dbb""
Gx
Dx
A" 22
9
E" 9
Bbb""
27
Fb""
14
B#!
C0
Fx!
5
20
B#!!
C! 1
11
3
Enharmonic dual keys
Cbb
Fx!!
6
21
Start of 31Et note duplication
A#!!
Bb! 27
E#!!
16
31 ET
G#!!
Ab! 22
D#!!
27
Dbb!
Gx!!!
B!! 30
Cb! 30
F#!! 17
Gb! 17
C#!! 4
Db! 4
4
B#!!!
C!! 2
Fx!!!
7
(optional)
Necessary notes for
30
G#!!!
Ab!! 23
D#!!!
F#"
14
Polychromatic Notation and Extended Tonality © Copyright Juhan Puhm 2016
B" 27
Cb"" 27
Compendium Musica
C#! 1
Db!! 1
1
Necessary notes for
19
Gb!! 14
complete Lower and Middle
6
Eb!! 6
modes on bottom row keys
24
Fbb!!!
First Extension
(optional)
Second Extension
(optional)
B#!
F!! 11
29
16
C!! 29
Gbb!!!
Dbb!!!
(optional)
0
Cbb!!!
C#!! 0
Db!!! 0
Cx!
G!! 16
Abb!!!
Dbb!!!
Gx!
Dx!
A!! 21
E!! 8
Bbb!!!
F#!! 13
Gb!!! 13
(foldout)
31
B#!
C!! 29
Fx!
Fb!!!
13
A#!
Bb!! 24
D!! 3
Ebb!!!
26
Third Extension
Bbbb!!!
E#!
11
3
21
8
G#!
Ab!! 19
D#!
Polychromatic Notation and Extended Tonality © Copyright Juhan Puhm 2016
B!! 26
Cb!!! 26
Compendium Musica
Isomorphic Keyboard Patterns for 31Et
C Lydian (G+)
C-0
C-0
C Ionian (C+)
F - 13
C-0
G - 18
C-0
D-5
G - 18
A - 23
D-5
E - 10
A - 23
B - 28
!
E - 10
!
F# - 15
(retrograde inversion around C)
(retrograde inversion around C)
!
!
C Locrian (Db+)
Gb - 16
B - 28
C Phrygian (Ab+)
Db - 3
Db - 3
Ab - 21
Ab - 21
Eb - 8
Eb - 8
Bb - 26
Bb - 26
F - 13
C-0
F - 13
C-0
C-0
C-0
C Mixolydian (F+)
Bb - 26
G - 18
C Dorian (Bb+)
F - 13
Eb - 8
Bb - 26
(self-symmetrical around C)
C-0
C-0
F - 13
C-0
G - 18
C-0
D-5
G - 18
A - 23
!
D-5
E - 10
A - 23
(retrograde inversion around C)
!
C Aeolian (Eb+)
Ab - 21
Eb - 8
Bb - 26
F - 13
C-0
C-0
G - 18
D-5
32
Polychromatic Notation and Extended Tonality © Copyright Juhan Puhm 2016
Compendium Musica
(foldout)
33
Polychromatic Notation and Extended Tonality © Copyright Juhan Puhm 2016
Compendium Musica
Fokker Organ Layout for 31Et
Polychromatic Layout
Complete Locrian to Lydian modes from Ebb! to A#"
-first octave only
-67+1 keys to play an octave of 31+1 notes
Dx!
Cx!
E 10
A#
B#
Cx
Fb!
F# 15
Gb! 15
G#
D5
Ab! 20
Bb! 25
Ebb!
Fbb!
Gbb!
Abb!
Bbb"
C! 30
Cb" 30
D! 4
Db" 4
C# 2
Db! 2
D#
E#
Fx
Gx
Eb! 7
F! 12
Dbb!
Ebb"
G! 17
Gb" 17
A! 22
Ab" 22
Cx
D! 4
Db" 4
Fb"
C#! 1
C" 1
Cbb
Dbb
E! 9
Eb" 9
F#! 14
F" 14
A#!
B#"
Cx"
G" 19
A" 24
Fbb
Gbb
Abb
Bbb
B" 29
Cb 29
C#" 3
Db 3
G#!
Bbbb
C#! 1
C" 1
D#!
E#"
Fx"
Gx"
A#"
B#!
D" 6
E" 11
G#"
Bb 26
C0
Dbb
Ebb
F#" 16
Gb 16
Ab 21
Cbb"
Dbb"
Dx"
Fb
Cx"
D#"
E#!
Fx!
28
C# 2
C#" 3
Db 3
Eb 8
F 13
G 18
A 23
Cb! 28
Db! 2
Gbb"
Abb"
Bbb!
Cbb!
Dbb!
Fbb"
Dx!
B#!
Cx!
E 10
C0
D5
Fb!
Dbb"
34
Dx
B! 27
Bb" 27
Ebb!
Fbb!
C# 2
Db! 2
Eb! 7
Dbb!
Ebb"
D#
B#
Cx
C! 30
Cb" 30
D! 4
Db" 4
F# 15
Gb! 15
Dx
Bbbb"
G#
Bbbb!
Ab! 20
Gbb!
Abb!
Fx
Gx
F! 12
G! 17
Gb" 17
A! 22
Ab" 22
E! 9
Eb" 9
F#! 14
F" 14
Fbb
Gbb
G#!
A#
B#
Bb! 25
C! 30
Cb" 30
Bbb"
E#
Fb"
B
Gx!
Bbbb
B! 27
Bb" 27
C#! 1
C" 1
Cbb
Dbb
A#!
B#"
G" 19
A" 24
Abb
Bbb
B" 29
Cb 29
C#! 1
C" 1
D#!
E#"
Fx"
Gx"
A#"
B#!
D" 6
E" 11
G#"
Bb 26
C0
Dbb
Ebb
F#" 16
Gb 16
Ab 21
Cbb"
Dbb"
Dx"
Fb
D#"
E#!
Fx!
Eb 8
F 13
G 18
A 23
Cb! 28
Gbb"
Abb"
Bbb!
Cbb!
Dx!
Gx!
28
Cx"
C#" 3
Db 3
Fbb"
Bbbb"
B
B#"
B" 29
Cb 29
B#!
Cx!
E 10
B#
D5
Fb!
F# 15
Gb! 15
A#
C0
Ab! 20
Bb! 25
Dbb"
Ebb!
Fbb!
Gbb!
Abb!
Bbb"
C! 30
Cb" 30
G#
Polychromatic Notation and Extended Tonality © Copyright Juhan Puhm 2016
Bbbb!
Compendium Musica
Original Colouration
-first octave only
-67+1 keys to play an octave of 31+1 notes
(Archiphone Keyboard)
D5
C# 2
E 10
D# 7
D! 4
C" 1
E! 9
Db 3
C! 30
D! 4
B" 29
E! 9
Db 3
C0
F" 14
Eb 8
D5
F 13
E 10
B! 27
Ab 21
Polychromatic Notation and Extended Tonality © Copyright Juhan Puhm 2016
C" 1
B" 29
Bb 26
A 23
G# 20
C# 2
C! 30
A" 24
G 18
F# 15
C0
A# 25
G" 19
Db 3
B 28
A! 22
Gb 16
C" 1
Bb 26
G# 20
D! 4
B" 29
A 23
G! 17
E" 11
B! 27
Ab 21
F# 15
C! 30
A" 24
G 18
F! 12
D" 6
A! 22
Gb 16
E 10
A# 25
G" 19
F 13
D# 7
C" 1
G! 17
E" 11
D5
G# 20
F" 14
Eb 8
C# 2
35
F! 12
D" 6
C0
F# 15
C0
B 28
A# 25
C! 30
Compendium Musica
Isomorphic Keyboard Patterns for the Fokker Organ Keyboard in 31Et
Complete Locrian to Lydian modes from Ebb! to A#"
Polychromatic Layout
C-0
C Lydian (G+)
"
C-0
D-5
E - 10
F# - 15
Db - 3
Eb - 8
F - 13
G - 18
A - 23
B - 28
Gb - 16
Ab - 21
Bb - 26
G - 18
A - 23
B - 28
Ab - 21
Bb - 26
C-0
Bb - 26
C-0
Bb - 26
C-0
Bb - 26
C-0
(retrograde inversion around C)
!
C Locrian (Db+)
C-0
C-0
C-0
C Ionian (C+)
F - 13
"
C-0
D-5
E - 10
(retrograde inversion around C)
!
C Phrygian (Ab+)
Db - 3
Eb - 8
F - 13
G - 18
F - 13
G - 18
F - 13
G - 18
C-0
C Mixolydian (F+)
"
C-0
D-5
A - 23
E - 10
(retrograde inversion around C)
!
C Aeolian (Eb+)
Ab - 21
Eb - 8
C-0
D-5
C Dorian (Bb+)
Eb - 8
(self-symmetrical around C)
C-0
F - 13
G - 18
A - 23
D-5
(foldout)
36
Polychromatic Notation and Extended Tonality © Copyright Juhan Puhm 2016
Compendium Musica
Polychromatic Layout for 43Et
Minimum Complete 43Et System
!
D#!!!!
G#!!!!
A#!!!!
Third Extension
!
Eb!!! 14
Ab!!! 32
Bb!!! 39
(optional)
!
Fbb!!
Bbbb!!
Cbb!!
!
B#!!!!
E#!!!!
Fx!!!!
B#!!!!
Second Extension
!
C!!! 3
F!!! 21
G!!! 28
C!!! 3
(optional)
!
Dbb!!
Gbb!!
Abb!!
Dbb!!
!
Cx!!!!
Dx!!!!
Gx!!!!
First Extension
!
D!!! 10
E!!! 17
A!!! 35
(optional)
!
Ebb!!
Fb!!
Complete Lydian, Ionian,
Mixolydian Upper modes and
Dorian Middle mode on each key.
!!!
!!
!
!
!
!
Complete Aeolian mode except 31
Complete Phrygian mode except 13, 31
Complete Locrian mode except 38, 13, 31
(17, 35, 10 required for complete Lower modes)
Complete Locrian, Phrygian,
Aeolian Lower modes and
Dorian Middle mode on each key.
Complete Mixolydian mode except 12
"
!
Complete Ionian mode except 30, 12
""
!
"""
!
Complete Lydian mode except 5, 30, 12
(26, 8, 33 required for complete Upper modes)
37
(43 notes + (2 x 10) duplicate notes = 63 Notes)
B#!!!
Bbb!!
(incomplete Upper
(incomplete Upper
(incomplete Upper
modes on 6)
modes on 24)
modes on 42)
C#!!! 6
Db!! 6
F#!!! 24
Gb!! 24
C"" 2!!!!!!!!!!!!!!!!!!!!!!!!!!!
C#"" 5
Dbb!
Db" 5
B#!!
C" 1 !!!!!!!!!!!!!!!!!!!!!!!!!!!
C#" 4
Dbb
Db 4
B#!
C 0 !!!!!!!!!!!!!!!!!!!!!!!!!!!
C# 3
Dbb"
Db$ 3
B#
C$ 42!!!!!!!!!!!!!!!!!!!!!!!!!!!
C#$ 2
Dbb""
Db$$ 2
B#"
C$$ 41!!!!!!!!!!!!!!!!!!!!!!!!!!!
C#"" 1
Dbb"""
Db""" 1
Cx!!!
D#"""
Dx!!!
D"" 9
Eb"" 13
E"" 16
Ebb"
Fbb!
Fb"
Cx!!
D#""
Dx!!
D" 8
Eb" 12
E" 15
Ebb
Fbb
Fb
Cx!
D#"
Dx!
D7
Eb 11
E 14
Ebb$
Fbb"
Fb$
Cx
D#
Dx
D$ 6
Eb$ 10
E$ 13
Ebb$$
Fbb""
Fb$$
Cx"
D#$
Dx"
D$$ 5
Eb$$ 9
E$$ 12
Ebb$$$
Fbb"""
Fb$$$
E#!!!
F"" 20 !!!!!!!!!!!!!!!!!!!!!!!!!!!
F#"" 23
Gbb!
Gb" 23
E#!!
F" 19 !!!!!!!!!!!!!!!!!!!!!!!!!!!
F#" 22
Gbb
Gb 22
E#!
F 18 !!!!!!!!!!!!!!!!!!!!!!!!!!!
F# 21
Gbb"
Gb$ 21
E#
F$ 17 !!!!!!!!!!!!!!!!!!!!!!!!!!!
F#$ 20
Gbb""
Gb$$ 20
E#"
F$$ 16 !!!!!!!!!!!!!!!!!!!!!!!!!!!
F#"" 19
Gbb"""
Gb""" 19
(incomplete Lower
(incomplete Lower
modes on 1)
modes on 19)
Fx!!!
G#"""
Gx!!!
G"" 27
Ab"" 31
A"" 34
Abb!
Bbbb!
Bbb"
Fx!!
G#""
Gx!!
G" 26
Ab" 30
A" 33
Abb
Bbbb
Bbb
Fx!
G#"
Gx!
G 25
Ab 29
A 32
Abb"
Bbbb"
Bbb$
Fx
G#
Gx
G$ 24
Ab$ 28
A$ 31
Abb""
Bbbb""
Bbb$$
Fx"
G#$
Gx"
G$$ 23
Ab$$ 27
A$$ 30
Abb"""
Bbbb"""
Bbb$$$
A#"""
B!!! 42
Cb!! 42
Bb"" 38!!!!!!!!!!!!!!!!!!!!!!!!!!!
B"" 41
Cbb!
Cb" 41
A#""
Bb" 37!!!!!!!!!!!!!!!!!!!!!!!!!!!
B" 40
Cbb
Cb 40
A#"
Bb 36 !!!!!!!!!!!!!!!!!!!!!!!!!!!
B 39
Cbb"
Cb$ 39
A#
Bb$ 35!!!!!!!!!!!!!!!!!!!!!!!!!!!
B$ 38
Cbb""
Cb$$ 38
A#$
Bb$$ 34!!!!!!!!!!!!!!!!!!!!!!!!!!!
B"" 37
Cbb"""
Cb""" 37
B#!!!
C"" 2
#
B#!!
#
!
C" 1
#
!!!!!!!!!!!
!
Dbb
#
!
o
o
!
B#!
#
#!!!!!!!!!!!!!!!!! o
Middle and Lower
Dbb"
#
o
"
modes on all keys
B#
#
"
o
C$ 42
G#""
A#""
First Extension
!
Eb""" 8
Ab""" 26
Bb""" 33
(optional)
!
Fbb""""
Bbbb""""
Cbb""""
#
"
B#"
#
""
C$$ 41
Dbb"""
!
B#""
E#""
Fx""
B#""
Second Extension
!
C""" 40
F""" 15
G""" 22
C""" 40
(optional)
!
Dbb""""
Gbb""""
Abb""""
Dbb""""
Third Extension
!
Cx""
Dx""
Gx""
(optional)
!
D""" 4
E""" 11
A""" 29
!
Ebb""""
Fb""""
Bbb""""
Polychromatic Notation and Extended Tonality © Copyright Juhan Puhm 2016
# !!!!!!!!!!!
Dbb""
modes on 37)
D#""
!!
C0
(incomplete Lower
!
!!
Dbb!
"
""
Complete Upper,
Compendium Musica
Modes for 43Et
Complete Locrian to Lydian modes from Ebb to A#
Lower Modes
Upper Modes
(retrograde inversion around C)
! "
C Locrian (Db+)
Gb - 22
Eb - 11
C Phrygian (Ab+)
Db - 4
Bb - 36
Db - 4
Bb - 36
C Aeolian (Eb+)
Ab - 29
Ab - 29
Eb - 11
Eb - 11
C-0
Eb - 11
A - 32
Bb - 36
C Ionian (C+)
G - 25
D-7
Mixolydian (F+)
C Dorian (Bb+)
38
Eb - 11
C-0
Bb - 36
G - 25
C-0
F - 18
D-7
A - 32
Bb - 36
G - 25
A - 32
Polychromatic Notation and Extended Tonality © Copyright Juhan Puhm 2016
D-7
B - 39
C-0
F - 18
Middle Mode
(self-symmetrical around C)
G - 25
E - 14
D-7
F - 18
Bb - 36
G - 25
C-0
C Lydian (G+)
C-0
C-0
F - 18
F - 18
Ab - 29
F - 18
G - 25
E - 14
D-7
B - 39
C-0
F - 18
D-7
A - 32
F# - 21
A - 32
G - 25
E - 14
Compendium Musica
Polychromatic Layout for 43Et
Generalised Isomorphic Keyboard
G#!!!!
32
D#!!!!
Third Extension
14
Eb!!! 14
(optional)
39
Fbb!!
Second Extension
Ab!!! 32
Bbbb!!
21
B#!!!!
F!!! 21
3
C!!! 3
Gbb!!
28
Dbb!!
Cbb!!
Cx!!!!
G!!! 28
D!!! 10
Abb!!
First Extension
35
Ebb!!
Dx!!!!
A!!! 35
(optional)
17
E!!! 17
Bbb!!
Necessary notes for
42
Fb!!
complete Upper and Middle
24
modes on top row keys
Start of 43Et
note duplication
Eb!! 13
38
Fbb!
Bbbb!
A#!!!
Bb!! 38
E#!!!
20
B#!!!
F!! 20
2
C!! 2
Gbb!
27
Dbb!
Cbb!
Cx!!!
G!! 27
D!! 9
Abb!
34
Ebb!
Dbb!
Gx!!!
Dx!!!
A!! 34
16
E!! 16
Bbb!
41
Fb!
B!! 41
Cb! 41
F#!! 23
Gb! 23
C#!! 5
Db! 5
5
30
G#!!
Ab! 30
D#!!
12
Eb! 12
37
Fbb
Bbbb
A#!!
Bb! 37
E#!!
19
B#!!
F! 19
1
C! 1
Gbb
26
Dbb
Cbb
Cx!!
G! 26
D! 8
Abb
33
Ebb
Dbb
Gx!!
Dx!!
A! 33
15
E! 15
Bbb
40
Fb
C#! 4
Db 4
4
B! 40
Cb 40
F#! 22
Gb 22
22
G#!
Ab 29
D#!
11
Eb 11
36
Fbb"
Bbbb"
A#!
Bb 36
E#!
18
B#!
F 18
0
C0
Gbb"
25
Dbb"
Cbb"
B#!
C0
Fx!
Cx!
G 25
7
D7
Abb"
32
Ebb"
Dbb"
Gx!
Dx!
A 32
14
E 14
Bbb"
39
Fb"
21
B#!!
C! 1
Fx!!
8
29
B#!!!
C!! 2
Fx!!!
9
23
43 ET
G#!!!
Ab!! 31
D#!!!
13
Dbb!!
Gx!!!!
B!!! 42
Cb!! 42
F#!!! 24
Gb!! 24
C#!!! 6
Db!! 6
6
31
B#!!!!
C!!! 3
Fx!!!!
10
(optional)
39
A#!!!!
Bb!!! 39
E#!!!!
F#
21
Polychromatic Notation and Extended Tonality © Copyright Juhan Puhm 2016
B 39
Cb" 39
Compendium Musica
C# 3
Db! 3
3
28
Gb! 21
G#
Ab! 28
D#
10
Eb! 10
35
Fbb!!
Bbbb!!
17
B#
F! 17
42
C! 42
Gbb!!
24
Dbb!!
Cbb!!
Cx
G! 24
D! 6
Abb!!
31
Ebb!!
Dbb!!
Gx
Dx
A! 31
13
E! 13
Bbb!!
38
Fb!!
F#!
C#! 2
Db!! 2
2
27
B! 38
Cb!! 38
20
Gb!! 20
G#!
Ab!! 27
D#!
9
Eb!! 9
Start of 43Et
34
Fbb!!!
note duplication
16
B#!
F!! 16
41
C!! 41
Gbb!!!
23
Dbb!!!
Bbbb!!!
A#!
Bb!! 34
E#!
Cbb!!!
Cx!
G!! 23
5
D!! 5
Abb!!!
30
Ebb!!!
Dbb!!!
Gx!
Dx!
A!! 30
E!! 12
Bbb!!!
Necessary notes for
37
Fb!!!
complete Lower and Middle
19
First Extension
(optional)
Second Extension
(optional)
Third Extension
(optional)
B!! 37
Cb!!! 37
F#!! 19
Gb!!! 19
C#!! 1
Db!!! 1
1
26
G#!!
Ab!!! 26
D#!!
8
Eb!!! 8
33
Fbb!!!!
Bbbb!!!!
A#!!
Bb!!! 33
E#!!
15
B#!!
F!!! 15
40
22
C!!! 40
Gbb!!!!
Cbb!!!!
Cx!!
G!!! 22
D!!! 4
Abb!!!!
29
11
Ebb!!!!
Dbb!!!!
Gx!!
Dx!!
A!!! 29
E!!! 11
Bbb!!!!
Fb!!!!
(foldout)
Polychromatic Notation and Extended Tonality © Copyright Juhan Puhm 2016
B#!!
C!!! 40
Fx!!
4
Dbb!!!!
B#!
C!! 41
Fx!
12
modes on bottom row keys
B#
C! 42
Fx
6
20
40
A#
Bb! 35
E#
Compendium Musica
Isomorphic Keyboard Patterns for 43Et
C Lydian (G+)
C-0
C-0
C Ionian (C+)
F - 18
C-0
G - 25
C-0
D-7
G - 25
A - 32
D-7
E - 14
A - 32
B - 39
!
E - 14
!
F# - 21
(retrograde inversion around C)
(retrograde inversion around C)
!
!
C Locrian (Db+)
Gb - 22
B - 39
C Phrygian (Ab+)
Db - 4
Db - 4
Ab - 29
Ab - 29
Eb - 11
Eb - 11
Bb - 36
Bb - 36
F - 18
C-0
F - 18
C-0
C-0
C-0
C Mixolydian (F+)
Bb - 36
G - 25
C Dorian (Bb+)
F - 18
Eb - 11
Bb - 36
(self-symmetrical around C)
C-0
C-0
F - 18
C-0
G - 25
C-0
D-7
G - 25
A - 32
!
D-7
E - 14
A - 32
(retrograde inversion around C)
!
C Aeolian (Eb+)
Ab - 29
Eb - 11
Bb - 36
F - 18
C-0
C-0
G - 25
D-7
41
Polychromatic Notation and Extended Tonality © Copyright Juhan Puhm 2016
Compendium Musica
(foldout)
42
Polychromatic Notation and Extended Tonality © Copyright Juhan Puhm 2016
Compendium Musica
Polychromatic Layout for 12nEt
(12, 24, 36, 48, 60, 72, 84, 96, 108 ET)
n
Range
12ET
o
!or"
24ET
36ET
!,"
!!or""
48ET
60ET
!!,""
!!!or"""
72ET
84ET
!!!,"""
96ET !!!!or""""
108ET !!!!,""""
1
2
3
4
5
6
7
8
9
43
B#!!!!
C#!!!!
Cx!!!!
D#!!!!
Dx!!!!
E#!!!!
F#!!!!
Fx!!!!
G#!!!!
Gx!!!!
A#!!!!
B!!!!
B#!!!!
= 1/4 tone, 1/2 semitone
C!!!!
(Bx!!!!)
D!!!!
Eb!!!!
E!!!!
F!!!!
(Ex!!!!)
G!!!!
Ab!!!!
A!!!!
Bb!!!!
(Ax!!!!)
C!!!!
= 1/6 tone, 1/3 semitone
Dbb!!!!
Fbb!!!!
Fb!!!!
Gbb!!!!
Bbbb!!!!
Bbb!!!!
Cbb!!!!
D#!!!
Dx!!!
E#!!!
Fx!!!
G#!!!
Gx!!!
A#!!!
Cb!!!!
B!!!
Dbb!!!!
Cx!!!
Gb!!!!
F#!!!
Abb!!!!
B#!!!
Db!!!!
C#!!!
Ebb!!!!
= 1/8 tone, 1/4 semitone
= 1/10 tone, 1/5 semitone
C!!!
(Bx!!!)
D!!!
Eb!!!
E!!!
F!!!
(Ex!!!)
G!!!
Ab!!!
A!!!
Bb!!!
(Ax!!!)
C!!!
= 1/12 tone, 1/6 semitone
Dbb!!!
Fbb!!!
Fb!!!
Gbb!!!
Bbbb!!!
Bbb!!!
Cbb!!!
D#!!
Dx!!
E#!!
Fx!!
G#!!
Gx!!
A#!!
Cb!!!
B!!
Dbb!!!
Cx!!
Gb!!!
F#!!
Abb!!!
B#!!
Db!!!
C#!!
Ebb!!!
= 1/14 tone, 1/7 semitone
= 1/16 tone, 1/8 semitone
C!!
(Bx!!)
D!!
Eb!!
E!!
F!!
(Ex!!)
G!!
Ab!!
A!!
Bb!!
(Ax!!)
C!!
= 1/18 tone, 1/9 semitone
Dbb!!
Db!!
C#!
Ebb!!
Fbb!!
Fb!!
Gbb!!
Bbbb!!
Bbb!!
Cbb!!
D#!
Dx!
E#!
Fx!
G#!
Gx!
A#!
Cb!!
B!
Dbb!!
Cx!
Gb!!
F#!
Abb!!
B#!
C!
(Bx!)
D!
Eb!
E!
F!
(Ex!)
G!
Ab!
A!
Bb!
(Ax!)
C!
Dbb!
Db!
C#
Ebb!
Fbb!
Fb!
Gbb!
Bbbb!
Bbb!
Cbb!
D#
Dx
E#
Fx
G#
Gx
A#
Cb!
B
Dbb!
Cx
Gb!
F#
Abb!
B#
C
(Bx)
D
Eb
E
F
(Ex)
G
Ab
A
Bb
(Ax)
C
Dbb
Db
C#"
Ebb
Fbb
Fb
Gbb
Bbbb
Bbb
Cbb
D#"
Dx"
E#"
Fx"
G#"
Gx"
A#"
Cb
B"
Dbb
Cx"
Gb
F#"
Abb
B#"
C"
(Bx")
D"
Eb"
E"
F"
(Ex")
G"
Ab"
A"
Bb"
(Ax")
C"
Dbb"
Db"
C#""
Ebb"
Fbb"
Fb"
Gbb"
Bbbb"
Bbb"
Cbb"
D#""
Dx""
E#""
Fx""
G#""
Gx""
A#""
Cb"
B""
Dbb"
Cx""
Gb"
F#""
Abb"
B#""
C""
(Bx"")
D""
Eb""
E""
F""
(Ex"")
G""
Ab""
A""
Bb""
(Ax"")
C""
Dbb""
Db""
C#"""
Ebb""
Fbb""
Fb""
Gbb""
Bbbb""
Bbb""
Cbb""
D#"""
Dx"""
E#"""
Fx"""
G#"""
Gx"""
A#"""
Cb""
B"""
Dbb""
Cx"""
Gb""
F#"""
Abb""
B#"""
C"""
(Bx""")
D"""
Eb"""
E"""
F"""
(Ex""")
G"""
Ab"""
A"""
Bb"""
(Ax""")
C"""
Dbb"""
Db"""
C#""""
Ebb"""
Fbb"""
Fb"""
Gbb"""
Bbbb"""
Bbb"""
Cbb"""
D#""""
Dx""""
E#""""
Fx""""
G#""""
Gx""""
A#""""
Cb"""
B""""
Dbb"""
Cx""""
Gb"""
F#""""
Abb"""
B#""""
C""""
(Bx"""")
D""""
Eb""""
E""""
F""""
(Ex"""")
G""""
Ab""""
A""""
Bb""""
(Ax"""")
C""""
Dbb""""
Db""""
Ebb""""
Fbb""""
Fb""""
Gbb""""
Gb""""
Abb""""
Bbbb""""
Bbb""""
Cbb""""
Cb""""
Dbb""""
= 1/2 tone, 1 semitone
Polychromatic Notation and Extended Tonality © Copyright Juhan Puhm 2016
B#!!!
B#!!
B#!
B#
B#"
B#""
B#"""
B#""""
Compendium Musica
44
Polychromatic Notation and Extended Tonality © Copyright Juhan Puhm 2016
Compendium Musica
Polychromatic Layout for 53Et
Minimum Complete 53Et System
Third Extension
!
(optional)
!
(53 notes + (2 x 8) duplicate notes = 69 Notes)
Db!!!! 8
C#!!! 8
Gb!!!! 30
F#!!! 30
Cb!!!! 52
B!!! 52
!
Ebb!!!!
Fb!!!!
Bbb!!!!
Second Extension
!
D!!! 12
E!!! 21
A!!! 43
(optional)
!
Cx!!
Dx!!
Gx!!
!
Dbb!!!!
Gbb!!!!
Abb!!!!
Dbb!!!!
First Extension
!
C!!! 3
F!!! 25
G!!! 34
C!!! 3
(optional)
!
B#!!
E#!!
Fx!!
(incomplete Upper
(incomplete Upper
modes on 16)
modes on 38)
modes on 47)
Fbb!!!!
Bbbb!!!!
Cbb!!!!
Eb!!! 16
Ab!!! 38
Bb!!! 47
Complete Lydian, Ionian,
!!!
!
!!
!
Dbb!!!
!!
!
C"" 2
!
!
Mixolydian, Dorian 1
Upper modes on each key,
plus Lower modes
Dorian 2 on 46, 15, 37
!!!
B#!
Aeolian on 24, 46, 15, 37
Dbb!!
Phrygian on 2, 24, 46, 15, 37
C" 1
B#
Locrian on 33, 2, 24, 46, 15, 37
(3 extensions required for
Dbb!
complete Lower modes)
C0
B#"
Dbb
C$ 52
Complete Locrian, Phrygian,
B#""
Aeolian, Dorian 2
Lower modes on each key,
plus Upper modes
Dorian 1 on 7, 38, 16
"
!
Dbb"
""
!
C$$ 51
""
!
B#"""
"""
!
"""
Db""" 7
C#"" 7
D#!!
!!!!!!!!!!!!!!!!!!!!!!!!!!!
Db"" 6
C#" 6
!!!!!!!!!!!!!!!!!!!!!!!!!!!
Db" 5
C# 5
!!!!!!!!!!!!!!!!!!!!!!!!!!!
Db 4
C#$ 4
!!!!!!!!!!!!!!!!!!!!!!!!!!!
Db$ 3
C#$$ 3
!!!!!!!!!!!!!!!!!!!!!!!!!!!
Db$$ 2
C#$$$ 2
B#!!
(incomplete Upper
Ebb"""
Fbb!!!
Fb"""
Gbb!!!
D"" 11
Eb"" 15
E"" 20
F"" 24
Cx!
D#"
Dx!
E#!
Ebb""
Fbb!!
Fb""
Gbb!!
D" 10
Eb" 14
E" 19
F" 23
Cx
D#
Dx
E#
Ebb"
Fbb!
Fb"
Gbb!
D9
Eb 13
E 18
F 22
Cx"
D#$
Dx"
E#"
Ebb
Fbb
Fb
Gbb
D$ 8
Eb$ 12
E$ 17
F$ 21
Cx""
D#$$
Dx""
E#""
Ebb$
Fbb"
Fb$
Gbb"
D$$ 7
Eb$$ 11
E$$ 16
F$$ 20
Cx"""
D#$$$
Dx"""
E#"""
Gb""" 29
F#"" 29
!!!!!!!!!!!!!!!!!!!!!!!!!!!
Gb"" 28
F#" 28
!!!!!!!!!!!!!!!!!!!!!!!!!!!
Gb" 27
F# 27
!!!!!!!!!!!!!!!!!!!!!!!!!!!
Gb 26
F#$ 26
!!!!!!!!!!!!!!!!!!!!!!!!!!!
Gb$ 25
F#$$ 25
!!!!!!!!!!!!!!!!!!!!!!!!!!!
Gb$$ 24
F#$$$ 24
G#!!
A#!!
Abb!!!
Bbbb!!!
Bbb"""
G"" 33
Ab"" 37
A"" 42
Fx!
G#"
Gx!
Abb!!
Bbbb!!
Bbb""
G" 32
Ab" 36
A" 41
Fx
G#
Gx
Abb!
Bbbb!
Bbb"
G 31
Ab 35
A 40
Fx"
G#$
Gx"
Abb
Bbbb
Bbb
G$ 30
Ab$ 34
A$ 39
Fx""
G#$$
Gx""
Abb"
Bbbb"
Bbb$
G$$ 29
Ab$$ 33
A$$ 38
Fx"""
G#$$$
Gx"""
Ebb""
Fb""
Ionian on 51, 29, 7, 38, 16
D""" 6
E""" 15
Lydian on 20, 51, 29, 7, 38, 16
Cx""""
Dx""""
Gx""""
(incomplete Lower
(incomplete Lower
(incomplete Lower
modes on 6)
modes on 15)
Mixolydian on 29, 7, 38, 16
(3 extensions required for
complete Upper modes)
45
Bbb""
Cbb!!!
Cb""" 51
B"" 51
Bb"" 46 !!!!!!!!!!!!!!!!!!!!!!!!!!!
Cb"" 50
A#"
B" 50
Cbb!!
Bb" 45 !!!!!!!!!!!!!!!!!!!!!!!!!!!
Cb" 49
A#
B 49
Cbb!
Bb 44 !!!!!!!!!!!!!!!!!!!!!!!!!!!
Cb 48
A#$
B$ 48
Cbb
Bb$ 43 !!!!!!!!!!!!!!!!!!!!!!!!!!!
Cb$ 47
A#$$
B$$ 47
Cbb"
Bb$$ 42 !!!!!!!!!!!!!!!!!!!!!!!!!!!
A#$$$
Cb$$ 46
B$$$ 46
Dbb!!!
C"" 2
B#!
#
!!
Dbb!!
#
!
C" 1
#
!!!!!!!!!!!!!!!!!
#
!
o
Dbb!
#
o
!
C0
#
!!!!!!!!!!!!!!!!! o
B#"
#
o
"
Dbb
#
"
o
C$ 52
#
"
Dbb"
#
""
C$$ 51
B#"""
modes on 37)
Dbb""
Gbb""
Abb""
Dbb""
First Extension
!
C""" 50
F""" 19
G""" 28
C""" 50
(optional)
!
B#""""
E#""""
Fx""""
B#""""
Fbb""
Bbbb""
Cbb""
!
Eb""" 10
Ab""" 32
Bb""" 41
(optional)
!
D#""""
G#""""
A#""""
Third Extension
!
(optional)
!
Db""" 1
C#"""" 1
Gb""" 23
F#"""" 23
Polychromatic Notation and Extended Tonality © Copyright Juhan Puhm 2016
Cb""" 45
B"""" 45
#
!!!!!!!!!!!!!!!!!
B#""
!
!
!
B#
A""" 37
Second Extension
!!
"
""
Complete Upper
and Lower modes
on all keys
*
Compendium Musica
Modes for 53Et
Complete Locrian to Lydian modes from Ebb to A#
Lower Modes
Upper Modes
(retrograde inversion around C)
! "
Gb# - 27
C Locrian (Db!+)
Eb - 13
Db! - 5
Bb - 44
Db# - 5
C Phrygian (Ab!+)
Bb - 44
Ab# - 36
C Aeolian (Eb!+)
Eb! - 14
C-0
F - 22
C Dorian 2 (Bb!+)
C-0
Eb# - 14
Bb! - 45
D-9
Bb - 44
F - 22
C-0
C Dorian 3
*
46
C# - 1
D$ - 8
Bb - 44
C Dorian 1 (Bb+)
D-9
A$ - 39
(not part of the above system of modes)
Polychromatic Notation and Extended Tonality © Copyright Juhan Puhm 2016
G - 31
D-9
B$ - 48
C-0
F - 22
A$ - 39
Bb - 44
G$ - 30
A - 40
F#$ - 26
E$ - 17
D$ - 8
Eb - 13
C$ - 52
Bb# - 45
G - 31
A$ - 39
G$ - 30
D-9
B$ - 48
C-0
F - 22
C Mixolydian (F+)
A - 40
G - 31
E$ - 17
D$ - 8
F# - 23
Eb# - 14
(self-symmetrical around C)
C Ionian (C+)
D-9
F# - 23
G - 31
*
(hybrid construct)
Bb# - 45
Bb# - 45
C-0
A$ - 39
G - 31
G - 31
C-0
C Lydian (G+)
C-0
Eb# - 14
Ab! - 36
F - 22
Eb# - 14
Ab# - 36
F - 22
G - 31
E$ - 17
C-0
F - 22
D$ - 8
A$ - 39
Compendium Musica
Polychromatic Layout for 53Et
Generalised Isomorphic Keyboard
Db!!!! 8
C#!!! 8
8
30
(optional)
52
Fb!!!!
E!!! 21
First Extension
21
43
12
34
3
(optional)
25
Necessary notes for
47
Fbb!!!!
complete Upper modes
16
Eb!!! 16
Bbbb!!!!
on top row keys
38
D#!!
Ab!!! 38
Enharmonic dual keys
29
Second Extension
(optional)
Ebb!!!!
Dbb!!!!
A!!! 43
Dx!!
D!!! 12
Abb!!!!
Cx!!
G!!! 34
Gx!!
Dbb!!!!
Gbb!!!!
B#!!
F!!! 25
Cbb!!!!
E#!!
Bb!!! 47
Db!!! 7
C#!! 7
C!!! 3
Fx!!
G#!!
Gb!!! 29
F#!! 29
Cb!!! 51
51
Fb!!!
20
E!! 20
Bbb!!!
note duplication
42
Dx!
A!! 42
Ebb!!!
11
D!! 11
Abb!!!
Cx!
G!! 33
B!!
Dbb!!!
2
C!! 2
Gbb!!!
24
B#!
F!! 24
Cbb!!!
E#!
Bb!! 46
Dbb!!!
C!! 2
Fx!
46
Fbb!!!
15
Eb!! 15
Bbbb!!!
37
D#!
Ab!! 37
Db!! 6
C#! 6
6
G#!
Gb!! 28
F#! 28
Cb!! 50
Fb!!
19
E! 19
Bbb!!
Dx
A! 41
Ebb!!
10
D! 10
Abb!!
Cx
G! 32
B!
Dbb!!
1
C! 1
Gbb!!
23
B#
F! 23
Cbb!!
E#
Bb! 45
Dbb!!
C! 1
Fx
45
Fbb!!
14
Eb! 14
Bbbb!!
36
D#
Ab! 36
Db! 5
C# 5
5
G#
Gb! 27
F# 27
Cb! 49
Fb!
18
E 18
Bbb!
Dx"
A 40
Ebb!
9
D9
Abb!
Cx"
G 31
B
Dbb!
0
C0
Gbb!
22
B#"
F 22
Cbb!
E#"
Bb 44
Dbb!
C0
Fx"
44
Fbb!
13
Eb 13
Bbbb!
35
D#"
Ab 35
4
Db 4
C#" 4
G#"
Gb 26
Polychromatic Notation and Extended Tonality © Copyright Juhan Puhm 2016
49
Gx"
31
26
B#
A#
49
40
50
Gx
32
27
B#!
A#!
50
41
51
Gx!
33
28
B#!!
A#!!
Start of 53Et
53 ET
Cb!!!! 52
B!!! 52
Bbb!!!!
C!!! 3
7
47
Gb!!!! 30
F#!!! 30
Third Extension
A#"
B#"
Compendium Musica
F#! 26
Cb 48
48
Fb
17
E! 17
Bbb
Dx!!
A! 39
39
Ebb
8
30
Dbb
52
C! 52
21
B#!!
D! 8
Abb
Cx!!
G! 30
Dbb
Fx!!
C! 52
Gbb
F! 21
Cbb
E#!!
Bb! 43
Fbb
12
Eb! 12
Bbbb
34
D#!!
Ab! 34
Db! 3
C#!! 3
3
G#!!
Gb! 25
F#!! 25
Cb! 47
Fb!
16
E!! 16
Bbb!
Dx!!!
A!! 38
Ebb!
7
29
Dbb!
51
C!! 51
20
B#!!!
Abb!
Cx!!!
G!! 29
Dbb!
Fx!!!
C!! 51
Gbb!
Fbb!
Gx!!!
F!! 20
Cbb!
E#!!!
Bb!! 42
Start of 53Et
11
Eb!! 11
Bbbb!
note duplication
33
D#!!!
Ab!! 33
Enharmonic dual keys
24
Db!! 2
C#!!! 2
G#!!!
Gb!! 24
F#!!! 24
Cb!! 46
46
Fb!!
15
E!!! 15
Bbb!!
complete Lower modes
Dx!!!!
A!!! 37
First Extension
37
6
28
50
C!!! 50
(optional)
19
B#!!!!
Ebb!!
Dbb!!
Abb!!
Cx!!!!
G!!! 28
Dbb!!
Fx!!!!
C!!! 50
Gbb!!
Gx!!!!
F!!! 19
Cbb!!
E#!!!!
Bb!!! 41
Fbb!!
Second Extension
10
Eb!!! 10
Bbbb!!
(optional)
32
D#!!!!
Ab!!! 32
Third Extension
23
(optional)
45
Db!!! 1
C#!!!! 1
B!!! 46
D!!! 6
41
1
B#!!!
A#!!!
Necessary notes for
on bottom row keys
B!! 47
D!! 7
42
2
B#!!
A#!!
47
38
B#!!!!
A#!!!!
G#!!!!
Gb!!! 23
F#!!!! 23
(foldout)
48
Gx!!
43
25
B! 48
Polychromatic Notation and Extended Tonality © Copyright Juhan Puhm 2016
Cb!!! 45
B!!!! 45
Compendium Musica
Isomorphic Keyboard Patterns for 53Et
C Lydian (G+)
A - 40
C Ionian (C+)
D-9
D-9
G - 31
C-0
G - 31
C-0
C-0
C-0
F - 22
F#! - 26
B! - 48
B! - 48
E! - 17
E! - 17
A! - 39
!
!
A! - 39
(retrograde inversion around C)
(retrograde inversion around C)
!
!
C Locrian (Db!+)
D! - 8
C Phrygian (Ab!+)
Eb" - 14
Bb" - 45
Ab" - 36
Eb" - 14
Db! - 5
Ab! - 36
Gb" - 27
Db" - 5
C-0
C-0
G - 31
C-0
F - 22
C-0
Bb - 44
F - 22
Eb - 13
Bb - 44
C Mixolydian (F+)
G - 31
C Dorian 1 (Bb+)
C-0
C-0
C-0
C-0
F - 22
F - 22
Bb - 44
Bb - 44
Eb - 13
E! - 17
A! - 39
A! - 39
D! - 8
D! - 8
G! - 30
!
!
G! - 30
(retrograde inversion around C)
(retrograde inversion around C)
!
!
C Aeolian (Eb!+)
C Dorian 2 (Bb!+)
F" - 23
C! - 52
C! - 52
C" - 1
C" - 1
Bb" - 45
F" - 23
Eb! - 14
Bb! - 45
Ab" - 36
Eb" - 14
D-9
A - 40
G - 31
C-0
D-9
C-0
49
G - 31
C-0
F - 22
Polychromatic Notation and Extended Tonality © Copyright Juhan Puhm 2016
C-0
Compendium Musica
*
C Dorian 3
Bb! - 45
Eb! - 14
(hybrid construct)
(self-symmetrical around C)
*
(not part of the above
D-9
G - 31
system of isomorphic patterns)
C-0
C-0
F - 22
Bb - 44
A" - 39
D" - 8
(foldout)
50
Polychromatic Notation and Extended Tonality © Copyright Juhan Puhm 2016
Compendium Musica
Mode Roots for Minimum Complete 53Et System
(53 notes + (2 x 8) duplicate notes = 69 Notes)
Complete Mode Roots
Complete Locrian to Lydian modes from Ebb! to A#"
Syntonic Comma range = !!!!, !!!, !!, !, o, ", "", """, """"
Cycle of Fifths Keys by
( 31n mod 53 )
Fifths
51
Lower Modes only
Upper
Lower
Complete Upper and Lower Modes
24
Dx#
F#!!!
Gb!!
46
Gx#
B!!!
Cb!!
15
Cx#
Dx!!!!
E!!!
(incomplete
37
Fx#
Gx!!!!
A!!!
Lower
Bbb!!
6
Bx
Cx!!!!
D!!!
modes)
Ebb!!
28
Ex
Fx!!!!
G!!!
Abb!!
Abb!!
Bx#
D#!!!
Eb!!
33
Ex#
G#!!!
Ab!!
2
Ax#
C#!!!
Db!!
C""
Fb!!
Fbb!
Cx"
Bbbb!
Fx"
E#"
(incomplete
Lower
modes)
Ax
B#!!!!
C!!!
Dbb!!
Dbb!!
19
Dx
E#!!!!
F!!!
Gbb!!
Gbb!!
Dx
41
Gx
A#!!!!
Bb!!!
Cbb!!
Cbb!!
Gx
10
Cx
D#!!!!
Eb!!!
Fbb!!
Fbb!!
Cx
32
Fx
G#!!!!
Ab!!!
Bbbb!!
Bbbb!!
1
B#
C#!!!!
Db!!!
C"
Fx
B#
23
E#
F#!!!!
Gb!!!
E#
45
A#
B!!!!
Cb!!!
14
D#
E!!!!
Fb!!!
36
G#
A!!!!
Bbb!!!
5
C#
D!!!!
Ebb!!!
27
F#
49
B
18
A#
D#
G#
C#
F#
B
E
A
D
G
C
F
Bb
Eb
Ab
Db
Gb
Cb
Fb
Bbb
Ebb
G!!!!
Abb!!!
C!!!!
Dbb!!!
E
F!!!!
Gbb!!!
Dx!
40
A
Bb!!!!
Cbb!!!
Gx!
9
D
Eb!!!!
Fbb!!!
Cx!
31
G
Ab!!!!
Bbbb!!!
Fx!
0
C
22
C!!!!
C
Db!!!!
B#!
F
Gb!!!!
E#!
44
Bb
Cb!!!!
13
Eb
Fb!!!!
A#!
D#!
G#!
C#!
F#!
B!
E!
A!
D!
G!
C!
F!
Bb!
Eb!
Ab!
Db!
Gb!
Cb!
Fb!
Bbb!
Ebb!
35
Ab
4
Db
26
Gb
Abb!!!!
48
Cb
Dbb!!!!
17
Fb
Gbb!!!!
Dx!!
Dx!!
39
Bbb
Cbb!!!!
Gx!!
Gx!!
8
Ebb
Fbb!!!!
Cx!!
Cx!!
Abb
52
Dbb
21
43
Bbb!!!!
C""""
Ebb!!!!
Fx!!
Fx!!
B#!!
B#!!
Gbb
E#!!
E#!!
Cbb
A#!!
D#!!
G#!!
C#!!
F#!!
B!!
E!!
A!!
D!!
G!!
C!!
F!!
A#!!
D#!!
G#!!
C#!!
F#!!
B!!
E!!
A!!
D!!
G!!
C!!
F!!
Bb!!
12
Fbb
34
Bbbb
3
Ebbb
Bbbb!!!!
C!
C"""
25
Abbb
47
Dbbb
16
Gbbb
Dx!!!
38
Cbbb
Gx!!!
7
Fbbb
Cx!!!
29
Bbbbb
Fx!!!
C!!
(incomplete
51
Ebbbb
B#!!!
Upper
20
Abbbb
E#!!!
modes)
42
Dbbbb
A#!!!
Bb!!
D""
G""
C""
F""
Bb""
Eb""
Ab""
Db""
Gb""
Cb""
Fb""
Bbb""
Ebb""
B#"
50
C!!!
Lower
Upper Modes only
Modes
Eb!!
Ab!!
Db!!
Gb!!
Cb!!
Fb!!
Bbb!!
Ebb!!
11
30
Upper
Modes
A#"
D#"
G#"
C#"
F#"
B"
E"
A"
D"
G"
C"
F"
Bb"
Eb"
Ab"
Db"
Gb"
Cb"
Fb"
Bbb"
Ebb"
Ebb"""
D""
G""
C""
F""
Bb""
Eb""
Ab""
Db""
Gb""
Cb""
Fb""
Bbb""
Ebb""
(incomplete
Abb"""
Lower
Dbb"""
modes)
Gbb"""
Cbb"""
Fbb"""
Bbbb"""
Abb""
Abb""
Dbb""
Dbb""
Gbb""
Gbb""
Dx""""
Cbb""
Cbb""
Gx""""
Fbb""
Fbb""
Cx""""
Bbbb""
Bbbb""
Fx""""
B#""""
E#""""
A#""""
D#""""
G#""""
C#""""
Abb"
F#""""
Dbb"
B""""
Gbb"
Dx"""
Cbb"
Gx"""
A""""
Fbb"
Cx"""
D""""
Fx"""
G""""
B#"""
C""""
Bbbb"
E""""
E#"""
F""""
A#"""
Bb""""
D#"""
Eb""""
G#"""
Ab""""
C#"""
Db""""
Abb
F#"""
Gb""""
Dbb
B"""
Cb""""
Gbb
Dx""
Dx""
E"""
Fb""""
Cbb
Gx""
Gx""
A"""
Bbb""""
Fbb
Cx""
Cx""
D"""
Ebb""""
Bbbb
Fx""
Fx""
G"""
Abb""""
B#""
B#""
C"""
Dbb""""
E#""
E#""
F"""
Gbb""""
A#""
D#""
G#""
C#""
F#""
B""
E""
A""
A#""
(incomplete
Bb"""
Cbb""""
D#""
Upper
Eb"""
Fbb""""
G#""
modes)
Ab"""
Bbbb""""
(incomplete
Upper
modes)
Abb!
Dbb!
Gbb!
Dx"
Cbb!
Gx"
Polychromatic Notation and Extended Tonality © Copyright Juhan Puhm 2016
C#""
Db"""
F#""
Gb"""
B""
Cb"""
E""
Fb"""
A""
Bbb"""
Compendium Musica
Modes for 53Et
Complete Modes on
Fbb!
!
o
"
Fbb!
Eb!!
Eb!!
Eb!!
Ab!!
Ab!!
Ab!!
Ab!!
Db!!
Db!!
Db!!
Db!!
Gb!!
Bbbb!
Gb!!
Gb!!
Gb!!
Cb!!
Cb!!
Cb!!
Cb!!
Fb!!
Fb!!
Fb!!
Fb!!
Bbb!!
Bbb!!
Bbb!!
Bbb!!
Ebb!!
Ebb!!
Ebb!!
Abb!!
Abb!!
Abb!!
Dbb!!
Dbb!!
Ebb!!
Complete Locrian to Lydian modes from Ebb! to A#"
roots
Cb""
Ebb""
Ebb""
Abb""
Abb""
Abb""
Dbb""
Dbb""
Dbb""
Dbb""
Gbb""
Gbb""
Gbb""
Gbb""
Cbb""
Cbb""
Cbb""
Fbb""
Fbb""
Ab"
Eb""
Ab""
Gb""
Gb""
Gb""
Cb""
Fb""
A"
Cb"
Cb"
Cb"
Fb"
Fb"
Fb"
Bbb"
Bbb"
Bbb"
Bbb"
Bbb"
Bbb"
Bbb"
Ebb"
Ebb"
Ebb"
Ebb"
Ebb"
Ebb"
Ebb"
Abb"
Abb"
Abb"
Abb"
Abb"
Abb"
Dbb"
Dbb"
Dbb"
Dbb"
Dbb"
D"
D"
D"
D"
D"
G"
G"
G"
G"
G"
C"
C"
C"
C"
C"
F"
Gbb"
Gbb"
Gbb"
Gbb"
Cbb"
Cbb"
Cbb"
Fbb"
Fbb"
A
C
Eb
D
D
G
G
C
C
C
C
C
C
F
Eb
Eb
Eb
Eb
Ab
Ab
Ab
Db
Db
Gb
Gb
Gb
Gb
Gb
Gb
Gb
Cb
Cb
Cb
Cb
Cb
Cb
Fb
Fb
Fb
Fb
Bbb
Bbb
Bbb
Ebb
Ebb
Ebb
Ebb
Ebb
Ebb
Ebb
Abb
Abb
Abb
Abb
Abb
Abb
Dbb
Dbb
Dbb
Dbb
Dbb
Gbb
Gbb
Gbb
Gbb
Cbb
Cbb
Cbb
Fbb
Fbb
A!
C!
Db!
Db!
Db!
Gb!
Gb!
Cb!
Fb!
Fb!
Fb!
Fb!
Fb!
Fb!
Fb!
Bbb!
Bbb!
Bbb!
Bbb!
Bbb!
Bbb!
Bbb!
Ebb!
Ebb!
Ebb!
Ebb!
Ebb!
Ebb!
Abb!
Gbb!
Gbb!
Gbb!
Gbb!
Cbb!
Cbb!
Cbb!
F!
Db!
Gb!
Abb!
D!
C!
Db!
Cb!
Dbb!
E!
A!
C!
Db!
Cb!
Abb!
E!
A!
Eb!
Gb!
Dbb!
E!
A!
F!
Cb!
Abb!
E!
A!
Bb!
Gb!
Dbb!
E!
A!
F!
Cb!
Abb!
E!
A!
Bb!
Gb!
Dbb!
E!
F!
Cb!
Abb!
B!
Bb!
Gb!
Dbb!
F#!
B!
F!
Cb!
E!!
C!!
Bb!!
F#!
F#!
C!
Db!
A#!!
G!!
G!!
G!!
C!!
C!!
F!!
F!!
F!!
F!!
Bb!!
Bb!!
A""
Polychromatic Notation and Extended Tonality © Copyright Juhan Puhm 2016
E#!!
E#!!
E#!!
A#!!
A#!!
A#!!
D#!!
C#!!
F#!!
A!!
C!!
E#!!
C#!!
E!!
G!!
B#!!
C#!!
E!!
A!!
B#!!
C#!!
E!!
D!!
B#!!
D#!!
B!!
A!!
Fx!!
B#!!
G#!!
F#!!
D!!
Cx!!
Fx!!
D#!!
B!!
A!!
Cx!!
Fx!!
G#!!
F#!!
Gx!!
Cx!!
D#!!
B!!
Gx!!
Fx!!
G#!!
F#!!
D!!
Dx!!
G#!!
B!!
D!!
Bb!!
G#!
B!
C!
Ab!
D#!
F#!
C!
Eb!
D#!
B!
C!
Eb!
D#!
C#!
G!
Ab!
D#!
G#!
D!
Ab!
D#!
G#!
D!
Eb!
D#!
C#!
G!
Ab!
A#!
D#!
C#!
D!
Eb!
E#!
A#!
G#!
G!
Ab!
E#!
A#!
C#!
D!
Eb!
E#!
A#!
G#!
G!
Ab!
E#!
F#!
Bb!
B#!
A#!
B!
F!
B#!
E#!
F#!
Bb!
B#!
A#!
B!
F!
B#!
E#!
F#!
Bb!
Fx!
B#!
A#!
B!
Bb!
Fx!
C#!
D!
Cx!
Cx!
Fx!
G#!
G!
Gx!
Cx!
C#!
D!
Gx!
Fx!
G#!
G!
Eb!
Dx!
C#!
G!
Ab!
Bbbb
D
D
Ab
Fb
A
G
Db
Bbb
B
D
Ab
Fb
F#
G
Db
Bbb
C#
F#
D
Ab
Fb
C#
F#
G
Db
Bbb
C#
F#
D
Ab
C#
C#
A
Eb
D#
F#
A
Eb
A#
C#
A
F
E#
A#
F#
A
Bb
E#
A#
C#
A
F
E#
A#
G#
E
Bb
E#
A#
D#
B
F
E#
A#
D#
B
Bb
E#
G#
E
F
B#
D#
E
Bb
B#
G#
B
F
B#
D#
E
Bb
B#
G#
B
F
Fx
B#
D#
E
Bb
Fx
G#
B
Cx
Cx
Fx
D#
E
Gx
Cx
G#
B
Gx
Fx
G#
E
Bb
Dx
F#
G
Db
Bbb
A#
G
Db
Bbbb"
D"
D"
G"
Db"
Cb"
F#"
E"
Eb"
Fb"
C#"
A"
Ab"
Cb"
G#"
C#"
E"
Eb"
Fb"
G#"
C#"
E"
Eb"
Cb"
G#"
C#"
A"
Ab"
Gb"
G#"
C#"
A"
Ab"
Db"
G#"
C#"
E"
Eb"
Db"
G#"
G#"
A"
Ab"
Gb"
D#"
E"
Eb"
Db"
D#"
A"
Ab"
Gb"
D#"
E"
Eb"
Db"
D#"
A"
Ab"
Gb"
D#"
E"
Eb"
Db"
D#"
B"
F"
Gb"
A#"
D#"
F#"
Bb"
Db"
E#"
A#"
B"
F"
Gb"
E#"
A#"
F#"
Bb"
B#"
E#"
B"
Bb"
B#"
A#"
F#"
F"
B#"
E#"
B"
Bb"
B#"
A#"
F#"
F"
Fx"
B#"
C#"
G"
Cx"
E#"
B"
Bb"
Fx"
A#"
F#"
F"
Cx"
Fx"
E#"
B"
Bb"
Cx"
Fx"
A#"
F#"
Bb"
Gb"
D""
B"
C"
F"
Fb"
52
Eb""
Cb""
Cb"
Ebb!
Eb""
Cb""
Fb"
Cb
Eb""
C"
Bbbb""
F""
F""
Gb""
Ebb""
Ebb""
C""
Bb""
Db""
Fb""
C""
F""
Ab""
Bbb""
C""
Bb""
Ab""
Fb""
G""
F""
Db""
Bbb""
D""
G""
Bb""
Ab""
Fb""
D""
G""
Bb""
Db""
Bbb""
Abb""
C""
Db""
Bbb""
Gbb!!
D""
G""
Eb!!
Fx""
E#""
E#""
E#""
A#""
A#""
A#""
D#""
D#""
D#""
D#""
G#""
G#""
G#""
G#""
C#""
C#""
C#""
F#""
F#""
F#""
F#""
B""
B""
B""
E""
E""
E""
E""
A""
A""
A""
B#""
A#""
C#""
B""
B#""
Dx"
Gx"
Gx"
Compendium Musica
Modes for 53Et
Complete modes on C
Complete Modes on C
Lower Modes
C Locrian (Db!+)
F!
Upper Modes
C Phrygian (Ab!+)
C Aeolian (Eb!+)
C Dorian 2 (Bb!+)
F! - 23
C! - 1
F! - 23
C! - 1
F! - 23
Bb! - 45
C!
F!
Bb! - 45
Bb! - 45
Bb! - 45
Eb! - 14
Eb! - 14
Eb! - 14
Eb! - 14
Ab! - 36
Ab! - 36
Ab! - 36
Db! Db!
Db! - 5
Db! - 5
Gb!
Gb! - 27
Bb!
Bb! Bb!
Eb! Eb!
Eb!
Eb!
Ab!
Ab! Ab!
A
D
G
G
D
G
G
D
G
C
C
C
C
C
C
F
F
F
F
F
F
Bb
Bb
Bb
Bb
Eb
C
Eb
A"
D"
G"
C"
E"
A"
D"
G"
C Lydian (G+)
All Modes on C
Eb! - 14
Db! - 5
G - 31
D-9
G - 31
C-0
C-0
C-0
F - 22
F - 22
F - 22
Bb - 44
Bb - 44
A - 40
D-9
G - 31
C-0
G - 31
D-9
G - 31
C-0
C-0
C-0
F - 22
F - 22
F - 22
Bb - 44
Bb - 44
A - 40
D-9
G - 31
C-0
A - 40
D-9
G - 31
C-0
F - 22
Bb - 44
Eb - 13
F#"
B"
E"
A"
Eb - 13
Gb! - 27
C" - 52
Db! - 5
Bb - 44
G" - 30
Keys: Db!+ / Ab!+ / Eb!+ / Bb!+
Ab! - 36
D" - 8
Eb! - 14
C-0
F - 22
A" - 39
Bb+ / F+ / C+ / G+
Polychromatic Notation and Extended Tonality © Copyright Juhan Puhm 2016
Bb! - 45
G - 31
E" - 17
F#" - 26
B" - 48
E" - 17
A" - 39
B" - 48
E" - 17
A" - 39
D" - 8
E" - 17
A" - 39
D" - 8
G" - 30
A" - 39
D" - 8
G" - 30
C" - 52
Eb - 13
53
C Ionian (C+)
Ab! - 36
Eb - 13
B"
E"
A"
D"
C Mixolydian (F+)
Gb! - 27
A
D
G
C Dorian 1 (Bb+)
F! - 23
D-9
B" - 48
C! - 1
A - 40
F#" - 26
F#" - 26
B" - 48
E" - 17
A" - 39
D" - 8
G" - 30
C" - 52
Compendium Musica
Modes for 53Et
Renaissance Polyphony
C Major plus Music Ficta
F Major plus Music Ficta
F! - 23
Bb! - 45
Bb! - 45
Eb! - 14
A - 40
D-9
D-9
G - 31
G - 31
C-0
C-0
F - 22
Bb - 44
F - 22
Bb - 44
Eb - 13
F#" - 26
B" - 48
E" - 17
A" - 39
D" - 8
B" - 48
E" - 17
A" - 39
D" - 8
G" - 30
G#"" - 34
C#"" - 3
F#"" - 25
C#"" - 3
F#"" - 25
B"" - 47
Bb! - 45
Bb - 44
C-0
F - 22
D" - 8
A" - 39
F#"" - 25
G - 31
E" - 17
C#"" - 3
F! - 23
D-9
B" - 48
Eb! - 14
A - 40
Eb - 13
F#" - 26
F - 22
Bb - 44
G" - 30
G#"" - 34
D" - 8
B"" - 47
C-0
A" - 39
F#"" - 25
Combined
Eb! - 14
Eb - 13
G" - 30
D" - 8
B"" - 47
54
F - 22
Bb - 44
C-0
A" - 39
F#"" - 25
Bb! - 45
G - 31
E" - 17
C#"" - 3
F! - 23
D-9
B" - 48
G#"" - 34
Polychromatic Notation and Extended Tonality © Copyright Juhan Puhm 2016
A - 40
F#" - 26
E" - 17
C#"" - 3
Bb! - 45
G - 31
D-9
B" - 48
Compendium Musica
5 Limit Just Intonation and 53 Tone Equal Temperament
(all 5 Limit ratios calculated by Perfect Fifths and Syntonic Commas)
Syntonic Comma = (81/80) = 21.51 cents
Holdrian Comma = 2^(1/53) = 22.64 cents
!,",!!,"",!!!,""" = Syntonic Comma sharp, flat
5 Limit Ratios
53ET
Ratio
Cents
+/- from 12ET
2^(53/53)
2^(52/53)
2^(51/53)
2^(50/53)
2^(49/53)
2^(48/53)
2^(47/53)
2^(46/53)
2^(45/53)
2^(44/53)
2^(43/53)
2^(42/53)
2^(41/53)
2^(40/53)
2^(39/53)
2^(38/53)
2^(37/53)
2^(36/53)
2^(35/53)
2^(34/53)
2^(33/53)
2^(32/53)
2^(31/53)
2^(30/53)
2^(29/53)
2^(28/53)
2^(27/53)
2^(26/53)
2^(25/53)
2^(24/53)
2^(23/53)
2^(22/53)
2^(21/53)
2^(20/53)
2^(19/53)
2^(18/53)
2^(17/53)
2^(16/53)
2^(15/53)
2^(14/53)
2^(13/53)
2^(12/53)
2^(11/53)
2^(10/53)
2^(9/53)
2^(8/53)
2^(7/53)
2^(6/53)
2^(5/53)
2^(4/53)
2^(3/53)
2^(2/53)
2^(1/53)
2^(0/53)
2
1.974014
1.948365
1.923050
1.898064
1.873402
1.849061
1.825036
1.801323
1.777918
1.754817
1.732017
1.709512
1.687301
1.665377
1.643739
1.622382
1.601302
1.580496
1.559960
1.539692
1.519686
1.499941
1.480452
1.461216
1.442231
1.423492
1.404996
1.386741
1.368723
1.350939
1.333386
1.316061
1.298961
1.282084
1.265426
1.248984
1.232756
1.216738
1.200929
1.185325
1.169924
1.154723
1.139720
1.124911
1.110295
1.095869
1.081630
1.067577
1.053705
1.040015
1.026502
1.013164
1
1200
1177.36
1154.72
1132.08
1109.43
1086.79
1064.15
1041.51
1018.87
996.23
973.58
950.94
928.30
905.66
883.02
860.38
837.74
815.09
792.45
769.81
747.17
724.53
701.89
679.25
656.60
633.96
611.32
588.68
566.04
543.40
520.75
498.11
475.47
452.83
430.19
407.55
384.91
362.26
339.62
316.98
294.34
271.70
249.06
226.42
203.77
181.13
158.49
135.85
113.21
90.57
67.92
45.28
22.64
0
0
-22.64
-45.28
32.08
9.43
-13.21
-35.85
41.51
18.87
-3.77
-26.42
-49.06
28.30
5.66
-16.98
-39.62
37.74
15.09
-7.55
-30.19
47.17
24.53
1.89
-20.75
-43.40
33.96
11.32
-11.32
-33.96
43.40
20.75
-1.89
-24.53
-47.17
30.19
7.55
-15.09
-37.74
39.62
16.98
-5.66
-28.30
49.06
26.42
3.77
-18.87
-41.51
35.85
13.21
-9.43
-32.08
45.28
22.64
0
55
Dbb!
*Dbb
Dbb"
65536/32805
1048576/531441
83886080/43046721
1198.05
1176.54
1155.03
53Et +/-
53Et +/-
from Just
from Just
1.95
0.82
-0.32
* (2^8) / (3/2)^12 deviates from 53Et by only 0.82 cents!
!Utonality
-----(25/24)
Cb!!!
243/125
1150.83
3.88
Cb""
Cb!
Cb
Cb"
Cb""
48/25
256/135
4096/2187
327680/177147
26214400/14348907
1129.33
1107.82
1086.31
1064.81
1043.30
2.75
1.61
0.48
-0.66
-1.79
Bbb!!!
216/125
946.92
4.02
Bbb""
Bbb!
Bbb
Bbb"
Bbb""
128/75
2048/1215
32768/19683
2621440/1594323
209715200/129140163
925.42
903.91
882.40
860.90
839.39
2.88
1.75
0.61
-0.52
-1.66
!
-----!
!
!
----------!
-----!
743.01
4.16
Abb!!
Abb!
Abb
Abb"
1024/675
16384/10935
262144/177147
20971520/14348907
721.51
700.00
678.49
656.99
3.02
1.89
0.75
-0.38
!
!
------
!
------ -------------------------------------------------------------------------------------------------------------- -----!
Gb!!!
729/500
652.79
3.81
Gb""
Gb"
Gb
Gb"
Gb""
36/25
64/45
1024/729
81920/59049
6553600/4782969
631.28
609.78
588.27
566.76
545.26
2.68
1.54
0.41
-0.73
-1.86
Fb!!!
162/125
448.88
3.95
Fb""
Fb!
Fb
Fb"
Fb""
32/25
512/405
8192/6561
655360/531441
52428800/43046721
427.37
405.87
384.36
362.85
341.35
2.82
1.68
0.55
-0.59
-1.72
Ebb"""
144/125
244.97
4.09
Ebb""
Ebb!
Ebb
Ebb"
Ebb""
256/225
4096/3645
65536/59049
5242880/4782969
419430400/387420489
223.46
201.96
180.45
158.94
137.44
2.95
1.82
0.68
-0.45
-1.59
4.22
Dbb!!
Dbb!
2048/2025
32768/32805
19.55
-1.95
3.09
1.95
C"""
1024000/531441
1135.48
-3.41
!
!
------
Bb!!!
59049/32000
1060.61
3.54
(25/24)
Bb!!
Bb"
Bb
Bb"
Bb""
729/400
9/5
16/9
1280/729
102400/59049
1039.10
1017.60
996.09
974.58
953.08
2.41
1.27
0.14
-1.00
-2.13
!
----------!
(16/15)
(16/15)
!
!
-----Ab!!!
6561/4000
856.70
3.68
(25/24)
Ab!!
Ab"
Ab
Ab"
Ab""
81/50
8/5
128/81
10240/6561
819200/531441
835.19
813.69
792.18
770.67
749.17
2.54
1.41
0.27
-0.86
-2.00
!
-----!
(16/15)
!
!
-----(25/24)
!
-----648/625
!
------
F!!!
177147/128000
562.56
3.47
!
!
------
F!!
F!
F
F"
F""
2187/1600
27/20
4/3
320/243
25600/19683
541.06
519.55
498.04
476.54
455.03
2.34
1.20
0.07
-1.07
-2.20
(25/24)
F"""
2048000/1594323
433.53
-3.34
!
!
------
!
-----!
(16/15)
!
!
-----!
----------!
!
!
------
!
-----!
(16/15)
Eb!!!
19683/16000
358.65
3.61
(25/24)
Eb!!
Eb"
Eb
Eb"
Eb""
243/200
6/5
32/27
2560/2187
204800/177147
337.15
315.64
294.13
272.63
251.12
2.48
1.34
0.20
-0.93
-2.07
!
-----!
Db!!!
2187/2000
154.74
3.75
(25/24)
Db!!
Db"
Db
Db"
Db""
27/25
16/15
256/243
20480/19683
1638400/1594323
133.24
111.73
90.22
68.72
47.21
2.61
1.48
0.34
-0.79
-1.93
!
-----!
(16/15)
!
------ -------------------------------------------------------------------------------------------------------------- ------
Polychromatic Notation and Extended Tonality © Copyright Juhan Puhm 2016
-----!
(16/15)
(25/24)
41.06
0
-1.14
-2.27
!
!
------ ------------------------------------------------------------------------------------------------------------------!
(25/24)
128/125
1200
1178.49
1156.99
(16/15)
(25/24)
Dbb"""
2/1
160/81
12800/6561
(16/15)
(25/24)
192/125
C
C"
C""
(16/15)
(25/24)
Abb"""
53Et +/from Just
(16/15)
!
!
-----------
(16/15)
!
!
------
Compendium Musica
Otonality!
Avg.->
1.8723 (75 values)
53Et +/-
53Et +/-
53Et +/-
from Just
from Just
from Just
53ET
Ratio
Cents
+/- from 12ET
2^(53/53)
2^(52/53)
2^(51/53)
2^(50/53)
2^(49/53)
2^(48/53)
2^(47/53)
2^(46/53)
2^(45/53)
2^(44/53)
2^(43/53)
2^(42/53)
2^(41/53)
2^(40/53)
2^(39/53)
2^(38/53)
2^(37/53)
2^(36/53)
2^(35/53)
2^(34/53)
2^(33/53)
2^(32/53)
2^(31/53)
2^(30/53)
2^(29/53)
2^(28/53)
2^(27/53)
2^(26/53)
2^(25/53)
2^(24/53)
2^(23/53)
2^(22/53)
2^(21/53)
2^(20/53)
2^(19/53)
2^(18/53)
2^(17/53)
2^(16/53)
2^(15/53)
2^(14/53)
2^(13/53)
2^(12/53)
2^(11/53)
2^(10/53)
2^(9/53)
2^(8/53)
2^(7/53)
2^(6/53)
2^(5/53)
2^(4/53)
2^(3/53)
2^(2/53)
2^(1/53)
2^(0/53)
2
1.974014
1.948365
1.923050
1.898064
1.873402
1.849061
1.825036
1.801323
1.777918
1.754817
1.732017
1.709512
1.687301
1.665377
1.643739
1.622382
1.601302
1.580496
1.559960
1.539692
1.519686
1.499941
1.480452
1.461216
1.442231
1.423492
1.404996
1.386741
1.368723
1.350939
1.333386
1.316061
1.298961
1.282084
1.265426
1.248984
1.232756
1.216738
1.200929
1.185325
1.169924
1.154723
1.139720
1.124911
1.110295
1.095869
1.081630
1.067577
1.053705
1.040015
1.026502
1.013164
1
1200
1177.36
1154.72
1132.08
1109.43
1086.79
1064.15
1041.51
1018.87
996.23
973.58
950.94
928.30
905.66
883.02
860.38
837.74
815.09
792.45
769.81
747.17
724.53
701.89
679.25
656.60
633.96
611.32
588.68
566.04
543.40
520.75
498.11
475.47
452.83
430.19
407.55
384.91
362.26
339.62
316.98
294.34
271.70
249.06
226.42
203.77
181.13
158.49
135.85
113.21
90.57
67.92
45.28
22.64
0
0
-22.64
-45.28
32.08
9.43
-13.21
-35.85
41.51
18.87
-3.77
-26.42
-49.06
28.30
5.66
-16.98
-39.62
37.74
15.09
-7.55
-30.19
47.17
24.53
1.89
-20.75
-43.40
33.96
11.32
-11.32
-33.96
43.40
20.75
-1.89
-24.53
-47.17
30.19
7.55
-15.09
-37.74
39.62
16.98
-5.66
-28.30
49.06
26.42
3.77
-18.87
-41.51
35.85
13.21
-9.43
-32.08
45.28
22.64
0
-----"
B""
B"
B
B"
B!!
1594323/819200
19683/10240
243/128
15/8
50/27
1152.79
1131.28
1109.78
1088.27
1066.76
1.93
0.79
-0.34
-1.48
-2.61
"
----------"
B!!!
4000/2187
1045.26
-3.75
(16/15)
A""
A"
A
A"
A!!
177147/102400
2187/1280
27/16
5/3
400/243
948.88
927.37
905.87
884.36
862.85
2.07
0.93
-0.20
-1.34
-2.48
"
-----"
A!!!
32000/19683
841.35
-3.61
(16/15)
(16/15)
G"""
1594323/1024000
766.47
3.34
"
"
------
G""
G"
G
G!
G!!
19683/12800
243/160
3/2
40/27
3200/2187
744.97
723.46
701.96
680.45
658.94
2.20
1.07
-0.07
-1.20
-2.34
(16/15)
"
"
-----(25/24)
"
"
-----(25/24)
(25/24)
"
-----648/625
"
-----(25/24)
256000/177147
637.44
"
"
-----(25/24)
"
-----"
A#""
A#"
A#
A#!
A#""
387420489/209715200
4782969/2621440
59049/32768
3645/2048
225/128
1062.56
1041.06
1019.55
998.04
976.54
1.59
0.45
-0.68
-1.82
-2.95
A#"""
125/72
955.03
-4.09
"
"
------
G#""
G#"
G#
G#!
G#""
43046721/26214400
531441/327680
6561/4096
405/256
25/16
858.65
837.15
815.64
794.13
772.63
1.72
0.59
-0.55
-1.68
-2.82
(25/24)
G#!!!
125/81
751.12
-3.95
F#""
F#"
F#
F#"
F#""
4782969/3276800
59049/40960
729/512
45/32
25/18
654.74
633.24
611.73
590.22
568.72
1.86
0.73
-0.41
-1.54
-2.68
F#!!!
1000/729
547.21
-3.81
(16/15)
"
"
-----------
531441/409600
6561/5120
81/64
5/4
100/81
450.83
429.33
407.82
386.31
364.81
2.00
0.86
-0.27
-1.41
-2.54
E!!!
8000/6561
343.30
-3.68
"
-----"
(16/15)
"
"
------ --------------------------------------------------------------------------------------------------------------------- ------
B#"""
125/64
1158.94
-4.22
E#"
E#
E#!
E#!!
14348907/10485760
177147/131072
10935/8192
675/512
543.01
521.51
500.00
478.49
0.38
-0.75
-1.89
-3.02
E#"""
125/96
456.99
-4.16
(25/24)
"
-----"
"
"
------
D#""
D#"
D#
D#!
D#""
129140163/104857600
1594323/1310720
19683/16384
1215/1024
75/64
360.61
339.10
317.60
296.09
274.58
1.66
0.52
-0.61
-1.75
-2.88
D#!!!
125/108
253.08
-4.02
14348907/13107200
177147/163840
2187/2048
135/128
25/24
156.70
135.19
113.69
92.18
70.67
1.79
0.66
-0.48
-1.61
-2.75
250/243
49.17
-3.88
(16/15)
59049/51200
729/640
9/8
10/9
800/729
246.92
225.42
203.91
182.40
160.90
2.13
1.00
-0.14
-1.27
-2.41
(25/24)
"
-----"
D!!!
64000/59049
139.39
-3.54
(16/15)
(16/15)
C"""
531441/512000
64.52
3.41
"
"
------
C#""
C#"
C#
C#!
C#""
"
"
------
C""
C"
C
6561/6400
81/80
1/1
43.01
21.51
0
2.27
1.14
0
(25/24)
C#!!!
56
-1.95
-3.09
"
-----"
D""
D"
D
D"
D!!
(25/24)
1201.95
1180.45
(25/24)
(16/15)
(16/15)
"
"
-----------
"
-----"
-3.47
E""
E"
E
E"
E!!
32805/16384
2025/1024
(25/24)
"
"
------------------------------------------------------------------------------------------------------------------- -----"
G!!!
"
-----"
(16/15)
------ --------------------------------------------------------------------------------------------------------------------- ------
B#!
B#!!
"
----------"
"
------
* (3/2)^12 / (2^7) deviates from 53Et by only -0.82 cents!
B#"
*B#
B#!
43046721/41943040
531441/524288
32805/32768
Polychromatic Notation and Extended Tonality © Copyright Juhan Puhm 2016
44.97
23.46
1.95
0.32
-0.82
-1.95
Compendium Musica
5 Limit Just Intonation and 53 Tone Equal Temperament
(all 5 Limit ratios calculated by Perfect Fifths and Syntonic Commas)
Syntonic Comma = (81/80) = 21.51 cents
Holdrian Comma = 2^(1/53) = 22.64 cents
!,",!!,"",!!!,""" = Syntonic Comma sharp, flat
5 Limit Ratios
!(3/2)^x
53ET
Ratio
Cents
+/- from 12ET
2^n
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
-1
-2
-3
-4
-5
-6
-7
-8
-9
-10
-11
-12
-13
-14
-15
-16
-17
-18
-19
-20
-21
-22
-23
-24
-25
-26
2^(11/53)
2^(33/53)
2^(2/53)
2^(24/53)
2^(46/53)
2^(15/53)
2^(37/53)
2^(6/53)
2^(28/53)
2^(50/53)
2^(19/53)
2^(41/53)
2^(10/53)
2^(32/53)
2^(1/53)
2^(23/53)
2^(45/53)
2^(14/53)
2^(36/53)
2^(5/53)
2^(27/53)
2^(49/53)
2^(18/53)
2^(40/53)
2^(9/53)
2^(31/53)
2^(0/53)
2^(22/53)
2^(44/53)
2^(13/53)
2^(35/53)
2^(4/53)
2^(26/53)
2^(48/53)
2^(17/53)
2^(39/53)
2^(8/53)
2^(30/53)
2^(52/53)
2^(21/53)
2^(43/53)
2^(12/53)
2^(34/53)
2^(3/53)
2^(25/53)
2^(47/53)
2^(16/53)
2^(38/53)
2^(7/53)
2^(29/53)
2^(51/53)
2^(20/53)
2^(42/53)
1.154723
1.539692
1.026502
1.368723
1.825036
1.216738
1.622382
1.081630
1.442231
1.923050
1.282084
1.709512
1.139720
1.519686
1.013164
1.350939
1.801323
1.200929
1.601302
1.067577
1.423492
1.898064
1.265426
1.687301
1.124911
1.499941
1
1.333386
1.777918
1.185325
1.580496
1.053705
1.404996
1.873402
1.248984
1.665377
1.110295
1.480452
1.974014
1.316061
1.754817
1.169924
1.559960
1.040015
1.386741
1.849061
1.232756
1.643739
1.095869
1.461216
1.948365
1.298961
1.732017
249.06
747.17
45.28
543.40
1041.51
339.62
837.74
135.85
633.96
1132.08
430.19
928.30
226.42
724.53
22.64
520.75
1018.87
316.98
815.09
113.21
611.32
1109.43
407.55
905.66
203.77
701.89
0
498.11
996.23
294.34
792.45
90.57
588.68
1086.79
384.91
883.02
181.13
679.25
1177.36
475.47
973.58
271.70
769.81
67.92
566.04
1064.15
362.26
860.38
158.49
656.60
1154.72
452.83
950.94
49.06
47.17
45.28
43.40
41.51
39.62
37.74
35.85
33.96
32.08
30.19
28.30
26.42
24.53
22.64
20.75
18.87
16.98
15.09
13.21
11.32
9.43
7.55
5.66
3.77
1.89
0
-1.89
-3.77
-5.66
-7.55
-9.43
-11.32
-13.21
-15.09
-16.98
-18.87
-20.75
-22.64
-24.53
-26.42
-28.30
-30.19
-32.08
-33.96
-35.85
-37.74
-39.62
-41.51
-43.40
-45.28
-47.17
-49.06
57
D#"""
53Et +/-
53Et +/-
53Et +/-
from Just
from Just
from Just
125/108
253.08
-4.02
G#"""
125/81
751.12
-3.95
C#"""
250/243
49.17
-3.88
F#"""
1000/729
547.21
-3.81
B"""
4000/2187
1045.26
-3.75
E"""
8000/6561
343.30
-3.68
A"""
32000/19683
841.35
-3.61
D"""
64000/59049
139.39
-3.54
G"""
256000/177147
637.44
-3.47
C"""
1024000/531441
1135.48
-3.41
F"""
2048000/1594323
433.53
-3.34
B#"""
125/64
1158.94
-4.22
E#"""
125/96
456.99
-4.16
A#"""
125/72
955.03
-4.09
Eb""
Ab""
Db""
Gb""
Cb""
Fb""
Bbb""
Ebb""
B#""
E#""
A#""
D#""
G#""
C#""
F#""
B""
E""
A""
D""
G""
C""
F""
Bb""
204800/177147
819200/531441
1638400/1594323
6553600/4782969
26214400/14348907
52428800/43046721
209715200/129140163
419430400/387420489
2025/1024
675/512
225/128
75/64
25/16
25/24
25/18
50/27
100/81
400/243
800/729
3200/2187
12800/6561
25600/19683
102400/59049
Polychromatic Notation and Extended Tonality © Copyright Juhan Puhm 2016
251.12
749.17
47.21
545.26
1043.30
341.35
839.39
137.44
1180.45
478.49
976.54
274.58
772.63
70.67
568.72
1066.76
364.81
862.85
160.90
658.94
1156.99
455.03
953.08
-2.07
-2.00
-1.93
-1.86
-1.79
-1.72
-1.66
-1.59
-3.09
-3.02
-2.95
-2.88
-2.82
-2.75
-2.68
-2.61
-2.54
-2.48
-2.41
-2.34
-2.27
-2.20
-2.13
B#"
E#"
A#"
D#"
G#"
C#"
F#"
B"
E"
A"
D"
G"
C"
F"
Bb"
Eb"
Ab"
Db"
Gb"
Cb"
Fb"
Bbb"
Ebb"
Abb"
Dbb"
32805/32768
10935/8192
3645/2048
1215/1024
405/256
135/128
45/32
15/8
5/4
5/3
10/9
40/27
160/81
320/243
1280/729
2560/2187
10240/6561
20480/19683
81920/59049
327680/177147
655360/531441
2621440/1594323
5242880/4782969
20971520/14348907
83886080/43046721
1.95
500.00
998.04
296.09
794.13
92.18
590.22
1088.27
386.31
884.36
182.40
680.45
1178.49
476.54
974.58
272.63
770.67
68.72
566.76
1064.81
362.85
860.90
158.94
656.99
1155.03
-1.95
-1.89
-1.82
-1.75
-1.68
-1.61
-1.54
-1.48
-1.41
-1.34
-1.27
-1.20
-1.14
-1.07
-1.00
-0.93
-0.86
-0.79
-0.73
-0.66
-0.59
-0.52
-0.45
-0.38
-0.32
Compendium Musica
Otonality!
53Et +/-
53Et +/-
53Et +/-
53Et +/-
from Just
from Just
from Just
from Just
53ET
4.09
2^(11/53)
2^(33/53)
2^(2/53)
2^(24/53)
2^(46/53)
2^(15/53)
2^(37/53)
2^(6/53)
2^(28/53)
2^(50/53)
2^(19/53)
2^(41/53)
2^(10/53)
2^(32/53)
2^(1/53)
2^(23/53)
2^(45/53)
2^(14/53)
2^(36/53)
2^(5/53)
2^(27/53)
2^(49/53)
2^(18/53)
2^(40/53)
2^(9/53)
2^(31/53)
2^(0/53)
2^(22/53)
2^(44/53)
2^(13/53)
2^(35/53)
2^(4/53)
2^(26/53)
2^(48/53)
2^(17/53)
2^(39/53)
2^(8/53)
2^(30/53)
2^(52/53)
2^(21/53)
2^(43/53)
2^(12/53)
2^(34/53)
2^(3/53)
2^(25/53)
2^(47/53)
2^(16/53)
2^(38/53)
2^(7/53)
2^(29/53)
2^(51/53)
2^(20/53)
2^(42/53)
"Utonality
B#
E#
A#
D#
G#
C#
F#
B
E
A
D
G
C
F
Bb
Eb
Ab
Db
Gb
Cb
Fb
Bbb
Ebb
Abb
Dbb
531441/524288
177147/131072
59049/32768
19683/16384
6561/4096
2187/2048
729/512
243/128
81/64
27/16
9/8
3/2
1/1
4/3
16/9
32/27
128/81
256/243
1024/729
4096/2187
8192/6561
32768/19683
65536/59049
262144/177147
1048576/531441
23.46
521.51
1019.55
317.60
815.64
113.69
611.73
1109.78
407.82
905.87
203.91
701.96
0
498.04
996.09
294.13
792.18
90.22
588.27
1086.31
384.36
882.40
180.45
678.49
1176.54
-0.82
-0.75
-0.68
-0.61
-0.55
-0.48
-0.41
-0.34
-0.27
-0.20
-0.14
-0.07
0
0.07
0.14
0.20
0.27
0.34
0.41
0.48
0.55
0.61
0.68
0.75
0.82
B#!
E#!
A#!
D#!
G#!
C#!
F#!
B!
E!
A!
D!
G!
C!
F!
Bb#
Eb#
Ab#
Db#
Gb#
Cb!
Fb!
Bbb!
Ebb!
Abb!
Dbb!
43046721/41943040
14348907/10485760
4782969/2621440
1594323/1310720
531441/327680
177147/163840
59049/40960
19683/10240
6561/5120
2187/1280
729/640
243/160
81/80
27/20
9/5
6/5
8/5
16/15
64/45
256/135
512/405
2048/1215
4096/3645
16384/10935
32768/32805
44.97
543.01
1041.06
339.10
837.15
135.19
633.24
1131.28
429.33
927.37
225.42
723.46
21.51
519.55
1017.60
315.64
813.69
111.73
609.78
1107.82
405.87
903.91
201.96
700.00
-1.95
0.32
0.38
0.45
0.52
0.59
0.66
0.73
0.79
0.86
0.93
1.00
1.07
1.14
1.20
1.27
1.34
1.41
1.48
1.54
1.61
1.68
1.75
1.82
1.89
1.95
D!!
G!!
C!!
F!!
Bb!!
Eb!!
Ab!!
Db!!
Gb##
Cb##
Fb##
Bbb##
Ebb##
Abb!!
Dbb!!
A#!!
D#!!
G#!!
C#!!
F#!!
B!!
E!!
A!!
58
59049/51200
19683/12800
6561/6400
2187/1600
729/400
243/200
81/50
27/25
36/25
48/25
32/25
128/75
256/225
1024/675
2048/2025
387420489/209715200
129140163/104857600
43046721/26214400
14348907/13107200
4782969/3276800
1594323/819200
531441/409600
177147/102400
Polychromatic Notation and Extended Tonality © Copyright Juhan Puhm 2016
246.92
744.97
43.01
541.06
1039.10
337.15
835.19
133.24
631.28
1129.33
427.37
925.42
223.46
721.51
19.55
1062.56
360.61
858.65
156.70
654.74
1152.79
450.83
948.88
2.13
2.20
2.27
2.34
2.41
2.48
2.54
2.61
2.68
2.75
2.82
2.88
2.95
3.02
3.09
1.59
1.66
1.72
1.79
1.86
1.93
2.00
2.07
Ebb###
144/125
244.97
Abb###
192/125
743.01
4.16
Dbb###
128/125
41.06
4.22
G!!!
1594323/1024000
766.47
3.34
C!!!
531441/512000
64.52
3.41
F!!!
177147/128000
562.56
3.47
Bb!!!
59049/32000
1060.61
3.54
Eb!!!
19683/16000
358.65
3.61
Ab!!!
6561/4000
856.70
3.68
Db!!!
2187/2000
154.74
3.75
Gb!!!
729/500
652.79
3.81
Cb!!!
243/125
1150.83
3.88
Fb!!!
162/125
448.88
3.95
Bbb!!!
216/125
946.92
4.02
Compendium Musica
7, 11 and 13 Limit Just Intonation and 53 Tone Equal Temperament
7 Limit Ratios
53ET
Ratio
Cents
+/- from 12ET
2^(53/53)
2^(52/53)
2^(51/53)
2^(50/53)
2^(49/53)
2^(48/53)
2^(47/53)
2^(46/53)
2^(45/53)
2^(44/53)
2^(43/53)
2^(42/53)
2^(41/53)
2^(40/53)
2^(39/53)
2^(38/53)
2^(37/53)
2^(36/53)
2^(35/53)
2^(34/53)
2^(33/53)
2^(32/53)
2^(31/53)
2^(30/53)
2^(29/53)
2^(28/53)
2^(27/53)
2^(26/53)
2^(25/53)
2^(24/53)
2^(23/53)
2^(22/53)
2^(21/53)
2^(20/53)
2^(19/53)
2^(18/53)
2^(17/53)
2^(16/53)
2^(15/53)
2^(14/53)
2^(13/53)
2^(12/53)
2^(11/53)
2^(10/53)
2^(9/53)
2^(8/53)
2^(7/53)
2^(6/53)
2^(5/53)
2^(4/53)
2^(3/53)
2^(2/53)
2^(1/53)
2^(0/53)
2
1.974014
1.948365
1.923050
1.898064
1.873402
1.849061
1.825036
1.801323
1.777918
1.754817
1.732017
1.709512
1.687301
1.665377
1.643739
1.622382
1.601302
1.580496
1.559960
1.539692
1.519686
1.499941
1.480452
1.461216
1.442231
1.423492
1.404996
1.386741
1.368723
1.350939
1.333386
1.316061
1.298961
1.282084
1.265426
1.248984
1.232756
1.216738
1.200929
1.185325
1.169924
1.154723
1.139720
1.124911
1.110295
1.095869
1.081630
1.067577
1.053705
1.040015
1.026502
1.013164
1
1200
1177.36
1154.72
1132.08
1109.43
1086.79
1064.15
1041.51
1018.87
996.23
973.58
950.94
928.30
905.66
883.02
860.38
837.74
815.09
792.45
769.81
747.17
724.53
701.89
679.25
656.60
633.96
611.32
588.68
566.04
543.40
520.75
498.11
475.47
452.83
430.19
407.55
384.91
362.26
339.62
316.98
294.34
271.70
249.06
226.42
203.77
181.13
158.49
135.85
113.21
90.57
67.92
45.28
22.64
0
0
-22.64
-45.28
32.08
9.43
-13.21
-35.85
41.51
18.87
-3.77
-26.42
-49.06
28.30
5.66
-16.98
-39.62
37.74
15.09
-7.55
-30.19
47.17
24.53
1.89
-20.75
-43.40
33.96
11.32
-11.32
-33.96
43.40
20.75
-1.89
-24.53
-47.17
30.19
7.55
-15.09
-37.74
39.62
16.98
-5.66
-28.30
49.06
26.42
3.77
-18.87
-41.51
35.85
13.21
-9.43
-32.08
45.28
22.64
0
59
C!!
C!
35/18
63/32
Cb
B
28/15
B"
40/21
27/14
Bb""
64/35
53Et +/-
Avg.->
53Et +/-
Avg.->
from Just
4.72
from Just
8.75
1172.74
1151.23
1137.04
1115.53
1080.56
4.62
3.49
-4.96
-6.10
6.24
1044.86
-3.35
968.83
947.32
933.13
4.76
3.62
-4.83
C!!
C!
125/63
96/49
Bb""
Bb
Bb!!
Bb!
140/81
7/4
A"
12/7
25/14
A
A!!
105/64
Ab!
Ab
14/9
G#
63/40
45/28
G"
G!!
G""
32/21
54/35
35/24
Gb
F#
7/5
10/7
F""
48/35
857.09
3.28
821.40
786.42
764.92
750.73
729.22
-6.30
6.03
4.90
-3.56
-4.69
653.18
3.42
617.49
582.51
-6.17
6.17
546.82
-3.42
Ab
42/25
100/63
G""
G!!
F!
35/27
21/16
Fb
E
56/45
E"
80/63
470.78
449.27
435.08
413.58
378.60
4.69
3.56
-4.90
-6.03
6.30
128/105
342.91
-3.28
81/70
266.87
252.68
231.17
4.83
-3.62
-4.76
9/7
Eb""
F!!
7/6
D"
D""
8/7
E
35/32
Db!
Db
28/27
C#
21/20
15/14
C"
C""
64/63
36/35
155.14
3.35
119.44
84.47
62.96
48.77
27.26
-6.24
6.10
4.96
-3.49
-4.62
Polychromatic Notation and Extended Tonality © Copyright Juhan Puhm 2016
D!!
63/50
25/21
D
D!!
F""
49/36
168/125
64/49
Eb
Eb!
49/32
G!
125/84
72/49
F"
F!!
49/27
28/25
54/49
C"
C""
49/48
126/125
1186.21
1164.30
-8.85
-9.59
1031.79
9.72
1003.80
-7.58
898.15
7.51
799.89
-7.44
737.65
9.52
688.16
666.26
-8.91
-9.65
533.74
511.84
9.65
8.91
462.35
-9.52
400.11
7.44
301.85
-7.51
196.20
7.58
168.21
-9.72
35.70
13.79
9.59
8.85
Compendium Musica
11 Limit Ratios
C!!
49/25
1165.02
Avg.->
53Et +/-
Avg.->
53Et +/-
Avg.->
from Just
10.82
from Just
8.07
from Just
3.45
-10.31
B
Bb""
A!!
90/49
1052.57
49/30
Ab""
849.38
848.66
80/49
-11.06
10.99
-10.93
Bb""
A!!
Ab
G!!
F""
E!!
49/40
Eb""
D!!
351.34
350.62
60/49
49/45
147.43
10.93
-10.99
11.06
60
50/49
34.98
21/11 1119.46
-10.03
11/6
-7.85
18/11
11/7
16/11
11/8
1049.36 20/11 1035.00
852.59
782.49
648.68
551.32
22/15
15/11
663.05
536.95
7.92
-7.92
417.51
-9.96
Eb""
11/9
347.41
-7.79
12/11
22/21
150.64
80.54
25/13
1132.10
-0.02
Cb!
24/13
1061.43
Bb!!
26/15
952.26
-1.32
A
22/13
910.79
-5.13
G#"
13/8
840.53
G""
20/13
745.79
1.38
F#"
13/9
636.62
-2.66
Gb!
18/13
563.38
2.66
F!!
13/10
454.21
-1.38
Fb!
16/13
359.47
Eb
13/11
289.21
5.13
D""
15/13
247.74
1.32
C#"
13/12
138.57
Db!
26/25
67.90
13/7
1071.70
2.72
-7.55
6.51
21/13
830.25
-2.79
7.48
9.96
14/11
D!!
B"
7.79
E
Db
C""
13 Limit Ratios
53Et +/-
11/10
165.00
7.85
-6.45
6.45
26/21
369.75
2.79
-7.48
-6.51
14/13
128.30
-2.72
10.03
10.31
Polychromatic Notation and Extended Tonality © Copyright Juhan Puhm 2016
0.02
7.55
53ET
2^(53/53)
2^(52/53)
2^(51/53)
2^(50/53)
2^(49/53)
2^(48/53)
2^(47/53)
2^(46/53)
2^(45/53)
2^(44/53)
2^(43/53)
2^(42/53)
2^(41/53)
2^(40/53)
2^(39/53)
2^(38/53)
2^(37/53)
2^(36/53)
2^(35/53)
2^(34/53)
2^(33/53)
2^(32/53)
2^(31/53)
2^(30/53)
2^(29/53)
2^(28/53)
2^(27/53)
2^(26/53)
2^(25/53)
2^(24/53)
2^(23/53)
2^(22/53)
2^(21/53)
2^(20/53)
2^(19/53)
2^(18/53)
2^(17/53)
2^(16/53)
2^(15/53)
2^(14/53)
2^(13/53)
2^(12/53)
2^(11/53)
2^(10/53)
2^(9/53)
2^(8/53)
2^(7/53)
2^(6/53)
2^(5/53)
2^(4/53)
2^(3/53)
2^(2/53)
2^(1/53)
2^(0/53)
Compendium Musica
7, 11 and 13 Limit Just Intonation and 53 Tone Equal Temperament
7 Limit Ratios
!(3/2)^x
2^n
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
-1
-2
-3
-4
-5
-6
-7
-8
-9
-10
-11
-12
-13
-14
-15
-16
-17
-18
-19
-20
-21
-22
-23
-24
-25
-26
61
53ET
Ratio
Cents
+/- from 12ET
2^(11/53)
2^(33/53)
2^(2/53)
2^(24/53)
2^(46/53)
2^(15/53)
2^(37/53)
2^(6/53)
2^(28/53)
2^(50/53)
2^(19/53)
2^(41/53)
2^(10/53)
2^(32/53)
2^(1/53)
2^(23/53)
2^(45/53)
2^(14/53)
2^(36/53)
2^(5/53)
2^(27/53)
2^(49/53)
2^(18/53)
2^(40/53)
2^(9/53)
2^(31/53)
2^(0/53)
2^(22/53)
2^(44/53)
2^(13/53)
2^(35/53)
2^(4/53)
2^(26/53)
2^(48/53)
2^(17/53)
2^(39/53)
2^(8/53)
2^(30/53)
2^(52/53)
2^(21/53)
2^(43/53)
2^(12/53)
2^(34/53)
2^(3/53)
2^(25/53)
2^(47/53)
2^(16/53)
2^(38/53)
2^(7/53)
2^(29/53)
2^(51/53)
2^(20/53)
2^(42/53)
1.154723
1.539692
1.026502
1.368723
1.825036
1.216738
1.622382
1.081630
1.442231
1.923050
1.282084
1.709512
1.139720
1.519686
1.013164
1.350939
1.801323
1.200929
1.601302
1.067577
1.423492
1.898064
1.265426
1.687301
1.124911
1.499941
1
1.333386
1.777918
1.185325
1.580496
1.053705
1.404996
1.873402
1.248984
1.665377
1.110295
1.480452
1.974014
1.316061
1.754817
1.169924
1.559960
1.040015
1.386741
1.849061
1.232756
1.643739
1.095869
1.461216
1.948365
1.298961
1.732017
249.06
747.17
45.28
543.40
1041.51
339.62
837.74
135.85
633.96
1132.08
430.19
928.30
226.42
724.53
22.64
520.75
1018.87
316.98
815.09
113.21
611.32
1109.43
407.55
905.66
203.77
701.89
0
498.11
996.23
294.34
792.45
90.57
588.68
1086.79
384.91
883.02
181.13
679.25
1177.36
475.47
973.58
271.70
769.81
67.92
566.04
1064.15
362.26
860.38
158.49
656.60
1154.72
452.83
950.94
49.06
47.17
45.28
43.40
41.51
39.62
37.74
35.85
33.96
32.08
30.19
28.30
26.42
24.53
22.64
20.75
18.87
16.98
15.09
13.21
11.32
9.43
7.55
5.66
3.77
1.89
0
-1.89
-3.77
-5.66
-7.55
-9.43
-11.32
-13.21
-15.09
-16.98
-18.87
-20.75
-22.64
-24.53
-26.42
-28.30
-30.19
-32.08
-33.96
-35.85
-37.74
-39.62
-41.51
-43.40
-45.28
-47.17
-49.06
D!!
G!!
C!!
F!!
Bb!!
Eb!!
B!
E!
A!
D!
G!
C!
81/70
54/35
36/35
48/35
64/35
128/105
27/14
9/7
12/7
8/7
32/21
64/63
53Et +/-
Avg.->
53Et +/-
Avg.->
from Just
4.72
from Just
8.75
252.68
750.73
48.77
546.82
1044.86
342.91
-3.62
-3.56
-3.49
-3.42
-3.35
-3.28
1137.04
435.08
933.13
231.17
729.22
27.26
-4.96
-4.90
-4.83
-4.76
-4.69
-4.62
G#
C#
F#
B
E
45/28
15/14
10/7
40/21
80/63
821.40
119.44
617.49
1115.53
413.58
-6.30
-6.24
-6.17
-6.10
-6.03
C
1/1
0.00
0.00
Ab
Db
Gb
Cb
Fb
63/40
21/20
7/5
28/15
56/45
C"
F"
Bb"
Eb"
Ab"
Db"
63/32
21/16
7/4
7/6
14/9
28/27
A""
D""
G""
C""
F""
Bb""
105/64
35/32
35/24
35/18
35/27
140/81
786.42
84.47
582.51
1080.56
378.60
6.03
6.10
6.17
6.24
6.30
1172.74
470.78
968.83
266.87
764.92
62.96
4.62
4.69
4.76
4.83
4.90
4.96
857.09
155.14
653.18
1151.23
449.27
947.32
3.28
3.35
3.42
3.49
3.56
3.62
Polychromatic Notation and Extended Tonality © Copyright Juhan Puhm 2016
G!!
C!!
F!!
Bb!!
C!
F!
E
A
D
63/50
42/25
28/25
Bb
Eb
Ab
25/14
25/21
100/63
G"
C"
49/32
49/48
49/36
49/27
126/125
168/125
125/84
125/63
D""
G""
C""
F""
54/49
72/49
96/49
64/49
737.65
35.70
533.74
1031.79
9.52
9.59
9.65
9.72
13.79
511.84
8.85
8.91
400.11
898.15
196.20
694.24
7.44
7.51
7.58
7.64
505.76
1003.80
301.85
799.89
-7.64
-7.58
-7.51
-7.44
688.16
1186.21
-8.91
-8.85
168.21
666.26
1164.30
462.35
-9.72
-9.65
-9.59
-9.52
Compendium Musica
11 Limit Ratios
C!!
50/49
Bb!!
Eb!!
Ab!!
90/49
60/49
80/49
13 Limit Ratios
53Et +/-
Avg.->
53Et +/-
Avg.->
53Et +/-
Avg.->
from Just
10.82
from Just
8.07
from Just
3.45
34.98
10.31
1052.57
350.62
848.66
-11.06
-10.99
-10.93
F!!
Bb!!
Eb!!
11/8
11/6
11/9
551.32
1049.36
347.41
15/11
20/11
536.95
1035.00
-7.92
-7.85
-7.79
Ab
Db
21/11
14/11
11/7
22/21
1119.46
417.51
C""
62
49/25
49/40
49/30
49/45
351.34
849.38
147.43
10.93
10.99
11.06
1165.02
-10.31
A""
D""
G""
18/11
12/11
16/11
782.49
80.54
852.59
150.64
648.68
247.74
745.79
1.32
1.38
G#!
C#!
F#!
13/8
13/12
13/9
840.53
138.57
636.62
1132.10
A
22/13
910.79
-5.13
Eb
13/11
289.21
5.13
Gb"
Cb"
Fb"
18/13
24/13
16/13
67.90
563.38
1061.43
359.47
0.02
2.66
2.72
2.79
F""
Bb""
13/10
26/15
454.21
952.26
25/13
21/13
14/13
830.25
128.30
-2.79
-2.72
-2.66
-0.02
7.48
7.55
-10.03
-9.96
9.96
10.03
Db"
E""
A""
D""
15/13
20/13
6.45
6.51
B!
B
E
D!!
G!!
11/10
22/15
165.00
663.05
7.79
7.85
7.92
26/25
13/7
26/21
1071.70
369.75
-6.51
-6.45
Polychromatic Notation and Extended Tonality © Copyright Juhan Puhm 2016
-1.38
-1.32
-7.55
-7.48
53ET
2^(11/53)
2^(33/53)
2^(2/53)
2^(24/53)
2^(46/53)
2^(15/53)
2^(37/53)
2^(6/53)
2^(28/53)
2^(50/53)
2^(19/53)
2^(41/53)
2^(10/53)
2^(32/53)
2^(1/53)
2^(23/53)
2^(45/53)
2^(14/53)
2^(36/53)
2^(5/53)
2^(27/53)
2^(49/53)
2^(18/53)
2^(40/53)
2^(9/53)
2^(31/53)
2^(0/53)
2^(22/53)
2^(44/53)
2^(13/53)
2^(35/53)
2^(4/53)
2^(26/53)
2^(48/53)
2^(17/53)
2^(39/53)
2^(8/53)
2^(30/53)
2^(52/53)
2^(21/53)
2^(43/53)
2^(12/53)
2^(34/53)
2^(3/53)
2^(25/53)
2^(47/53)
2^(16/53)
2^(38/53)
2^(7/53)
2^(29/53)
2^(51/53)
2^(20/53)
2^(42/53)
Compendium Musica
Just Third and Fifth Tuning
(all 5 Limit ratios calculated by Perfect Fifths and Syntonic Commas)
Deviations +/- from 12ET
C = 1/1
Syntonic Comma = (81/80) = 21.51 cents
!,",!!,"",!!!,""" = Syntonic Comma sharp, flat
Dbb! 65536/32805
-1.95
Bbb 32768/19683
-17.60
Fb 8192/6561
-33.24
Db" 20480/19683
-31.28
-46.92
F"" 25600/19683
Ebb!+
Dbb 1048576/531441
-23.46
Abb 262144/177147
-21.51
-37.15
Cb" 327680/177147
Ebb+
Fb" 655360/531441
Cb"+
-52.79
2048000/1594323
-35.19
Gb" 81920/59049
Fb+
Cb+
Db"+
Ab"" 819200/531441
-50.83
Eb""+
F"""
-19.55
Bbb+
Gb"+
Db"" 1638400/1594323
Ebb 65536/59049
Ab"+
Eb"" 204800/177147
-48.88
Bb""+
-66.47
C"""
1024000/531441
Bb"" 102400/59049
Eb"+
F""+
-64.52
G"""
256000/177147
C""+
-62.56
D"""
41.06
Abb!!! 192/125
Dbb!!! 128/125
64000/59049
Ebb!!!+
Abb!! 1024/675
21.51
Ebb!! 256/225
Bbb!!+
5.87
Cb! 256/135
Gb!+
23.46
7.82
Gb! 64/45
Db!+
Ab 128/81
-7.82
-23.46
C" 160/81
Eb 32/27
-5.87
-21.51
G" 40/27
-37.15
-52.79
C#""" 250/243
Fb!! 32/25
Cb!!+
9.78
11.73
-3.91
-19.55
D" 10/9
-35.19
-50.83
G#""" 125/81
B""+
Cb!! 48/25
Ab! 8/5
13.69
F 4/3
-1.96
C 1/1
-17.60
A" 5/3
-15.64
Bb!+
C+
G+
A"+
E"" 100/81
27.37
Eb!+
Bb 16/9
43.01
Bbb!!!+
Gb!!+
Db! 16/15
F+
D"+
A"" 400/243
25.42
Ab!+
Bb+
G"+
Bbb!! 128/75
Fb!!+
-60.61
E"+
B"+
B"" 50/27
-33.24
F#"" 25/18
-31.28
C#"" 25/24
-48.88
D#""" 125/108
-46.92
A#""" 125/72
-44.97
54.74
Ab!!!
F#""+
52.79
Db!!!
2187/2000
Ab!!!+
56.70
6561/4000
Eb!!!+
Bb!! 729/400
39.10
23.46
D! 729/640
F!! 2187/1600
41.06
25.42
A! 2187/1280
C!!+
A!+
9.78
-5.87
D#" 1215/1024
B#"" 2025/1024
-19.55
19683/16000
11.73
-3.91
A#" 3645/2048
F!!!+
43.01
27.37
E! 6561/5120
29.33
13.69
G# 6561/4096
E#" 10935/8192
-0.0013
G!!+
F# 729/512
58.65
C!! 6561/6400
G!! 19683/12800
D!!+
E!+
B 243/128
Eb!!!
Bb!!!+
C#
B!+
2187/2048
F#!+
C#+
63
Polychromatic Notation and Extended Tonality © Copyright Juhan Puhm 2016
-1.96
Compendium Musica
Dbb!! 2048/2025
19.55
Ebb!!+
Abb! 16384/10935
0.0013
Ebb! 4096/3645
Bbb!+
-15.64
Cb 4096/2187
-13.69
-29.33
-44.97
C"" 12800/6561
-43.01
G"" 3200/2187
44.97
E"""
8000/6561
Bbb!!! 216/125
Gb!! 36/25
15.64
0.00
G 3/2
31.28
F" 320/243
-41.06
D"" 800/729
C"+
-56.70
B"""
46.92
Fb!!! 162/125
-13.69
-29.33
Db!! 27/25
E#""" 125/96
1.96
D 9/8
F#"""
48.88
Cb!!! 243/125
-11.73
G#"" 25/16
-27.37
D#"" 75/64
-43.01
B#""" 125/64
-41.06
1000/729
33.24
50.83
Ab!! 81/50
35.19
Eb!! 243/200
37.15
C! 81/80
21.51
G! 243/160
5.87
E 81/64
7.82
Bb!!+
F! 27/20
19.55
3.91
A 27/16
Gb!!! 729/500
Db!!!+
G!+
F!!+
D!+
E+
B" 15/8
60.61
E""+
-54.74
Eb!!+
17.60
F#"+
-39.10
Gb!!!+
A+
E" 5/4
4000/2187
C!+
D+
Eb+
A""+
Bb! 9/5
F!+
44.97
-25.42
Ab!!+
Eb! 6/5
59049/32000
Bb" 1280/729
Cb!!!+
Db!!+
Bb!!!
-9.78
F"+
Fb!!!+
29.33
Db 256/243
-11.73
D""+
-58.65
Fb! 512/405
Ab+
-27.37
G""+
Ebb!!! 144/125
Gb 1024/729
Eb" 2560/2187
Bb"+
3.91
Cb!+
Db+
Ab" 10240/6561
32000/19683
Bbb! 2048/1215
Fb!+
Gb+
A"""
1.96
B+
F#+
F#" 45/32
-9.78
C#" 135/128
-7.82
G#" 405/256
-25.42
A#"" 225/128
-23.46
E#"" 675/512
-21.51
C#"+
F!!!
177147/128000
62.56
C!!!
531441/512000
64.52
G!!!
1594323/1024000
66.47
D!! 59049/51200
46.92
A!! 177147/102400
48.88
E!! 531441/409600
50.83
B!! 1594323/819200
B! 19683/10240
31.28
F#! 59049/40960
33.24
C#! 177147/163840
35.19
G#! 531441/327680
37.15
15.64
D# 19683/16384
17.60
A# 59049/32768
19.55
E# 177147/131072
21.51
B# 531441/524288
B#" 32805/32768
1.95
52.79
A!!+
64
Polychromatic Notation and Extended Tonality © Copyright Juhan Puhm 2016
23.46
Compendium Musica
Partch 43 Tone Scale
1/1 =G =392 * 2^n
A½!
12/11
150.64
1/11 Limit
C5/6!
14/11
417.51
B"#
9/7
435.08
A"#
8/7
231.17
1/7 Limit
C#1/6#
10/7
617.49
Utonality
# 81/80
1/5 Limit
G#
81/80
21.51
Ab#
16/15
111.73
(81/80)
1/3 Limit
3 Limit
G
1/1
0
A
9/8
203.91
Otonality
7 Limit
11 Limit
65
Ab1/6!
21/20
84.47
Eb#
8/5
813.69
(81/80)
(81/80)
Bb
32/27
294.13
C
4/3
498.04
F
16/9
996.09
B!
5/4
386.31
G½#
33/32
53.27
C"!
21/16
470.78
Bb½#
11/9
347.41
Db1/6!
7/5
582.51
C½#
11/8
551.32
G½!
64/33
1146.73
½!
/3!
5
/6!
approximately 1/2 semitone flat
"#
/6#
approximately 1/3 semitone sharp
2
F#1/6#
40/21
1115.53
(81/80)
Bb"!
7/6
266.87
Ab2/3#
11/10
165.00
E"#
12/7
933.13
C#
27/20
519.55
A!
10/9
182.40
! 81/80
D"#
32/21
729.22
F# 2/3!
20/11
1035.00
Bb#
6/5
315.64
(81/80)
5 Limit
E½!
18/11
852.59
D½!
16/11
648.68
1
F#
9/5
1017.60
D
3/2
701.96
E
27/16
905.87
(81/80)
(81/80)
D!
40/27
680.45
E!
5/3
884.36
approximately 5/6 semitone flat
approximately 1/6 semitone sharp
#
syntonic comma (81/80) sharp
!
syntonic comma (81/80) flat
G
2/1
1200
(81/80)
F#!
15/8
1088.27
G!
160/81
1178.49
/6!
"!
1
Eb"!
14/9
764.92
approximately 2/3 semitone flat
F"!
7/4
968.83
D5/6#
11/7
782.49
Polychromatic Notation and Extended Tonality © Copyright Juhan Puhm 2016
approximately 1/3 semitone flat
/6#
/3#
approximately 5/6 semitone sharp
½#
approximately 1/2 semitone sharp
5
2
F½#
11/6
1049.36
approximately 1/6 semitone flat
approximately 2/3 semitone sharp
Compendium Musica
Partch 43 Tone Scale
Harmonic Structure (from which can be derived scales, modes, chords and tonalities)
1/1 =G =392 * 2^n
-(also the Roots of 3 incomplete Tertiary Tonalities
Primary Ratios
Otonality!
(Bold = Roots)
Secondary
Ratios
D"# (O)
32/21
729.22
F#1/6#
40/21
1115.53
Partch Tuning Notes = 8/7, 11/8
C
D
4/3
3/2
498.04
701.96
(7/6)
(9/7)
(7/6)
B"#
A"# (O)
E"# (O)
8/7
12/7
(3/2)
9/7
231.17
933.13
435.08
1
(U) C# /6#
(7/6)
(9/7)
(11/9)
D5/6#
10/7
G
617.49
1/1
11/7
Otonality!
0
782.49
(11/9)
(9/7)
(7/6)
Bb½#
(U) F½#
11/9
(3/2)
11/6
347.41
1049.36
(11/6)
(11/9)
C
D
4/3
3/2
498.04
701.96
without the 5 Limit major third)
$Utonality
Ab2/3#
11/10
165.00
Secondary
Ratio
(U) C½#
11/8
551.32
(3/2)
(U) G½#
33/32
53.27
$Utonality
(Bold = Roots)
Primary Ratios
Secondary
Otonality!
Primary Ratios
Ab! (O)
16/15
111.73
Ratios
(Bold = Roots)
Bb (O)
32/27
294.13
G%
160/81
1178.49
F (O)
16/9
996.09
(U) D"
40/27
680.45
Eb! (O)
8/5
813.69
Bb! (O)
6/5
315.64
(U) C (O)
4/3
498.04
(U) A"
10/9
182.40
(U) G (O)
1/1
0
(U) E"
5/3
884.36
F! (O)
9/5
1017.60
(U) D (O)
3/2
701.96
(U) B"
5/4
386.31
C! (O)
27/20
519.55
(U) A
9/8
203.91
(U) F#"
15/8
1088.27
$Utonality
(Bold = Roots)
G½% (O)
64/33
1146.73
(3/2)
D½% (O)
16/11
648.68
F#2/3%
20/11
1035.00
Secondary
Ratio
Otonality!
Partch Tuning Notes = 7/4, 16/11
Primary Ratios
-The 8/7, 11/8, 7/4 and 16/11 intervals are prescribed by Partch to tune the 7 and 11 limit ratios. Without a doubt a difficult task to tune these intervals accurately by ear.
After which the rest of the 7 and 11 limit ratios present no problems. The 5 limit ratios present no tuning problems.
66
Polychromatic Notation and Extended Tonality © Copyright Juhan Puhm 2016
(Bold = Roots)
Ratios
-(also the Roots of 3 incomplete Tertiary Tonalities
Primary Ratios
C
D
4/3
3/2
498.04
701.96
(11/9)
(11/6)
E½%
A½% (O)
12/11
(3/2)
18/11
150.64
852.59
(7/6)
(9/7)
(11/9)
5
C /6%
G
$Utonality
Db1/6% (O)
14/11
1/1
417.51
0
7/5
(11/9)
(9/7)
(7/6)
582.51
Eb"%
(U) Bb"%
(U) F"%
14/9
(3/2)
7/6
7/4
764.92
266.87
968.83
(7/6)
(9/7)
(7/6)
C
D
4/3
3/2
498.04
701.96
(U) E
27/16
905.87
Secondary
Primary Ratios
Otonality!
G#
81/80
21.51
without the 5 Limit major third)
Ab1/6%
21/20
84.47
(U) C"%
21/16
470.78
Secondary
Ratios
$Utonality
(Bold = Roots)
Compendium Musica
Partch 43 Tone Scale
1/1 =G =392 * 2^n
0
Primary
Ratios
1 - 42
2 - 41
3 - 40
4 - 39
56/55
64/63
45/44
121/120
21.51
31.77
31.19
27.26
38.91
14.37
G
1/1
0
G"
81/80
21.51
G½"
33/32
53.27
Ab1/6!
21/20
84.47
Ab"
16/15
111.73
Secondary
Ratios
G!
160/81
1178.49
G½!
64/33
1146.73
F#1/6"
40/21
1115.53
F#!
15/8
1088.27
G
2/1
1200
False Ionian Scales
67
7 - 36
49/48
17.40
21.51
27.26
35.70
A½!
12/11
150.64
2
Ab /3"
11/10
165.00
A!
10/9
182.40
A
9/8
203.91
A#"
8/7
231.17
Bb#!
7/6
266.87
F½"
11/6
1049.36
F#2/3!
20/11
1035.00
F"
9/5
1017.60
F
16/9
996.09
F#!
7/4
968.83
E#"
12/7
933.13
1/1 =G =392 * 2^n
D#"
32/21
729.22
9/8
E#"
12/7
933.13
10/9
F#1/6"
40/21
1115.53
63/55
A½!
12/11
150.64
22/21
A#"
8/7
231.17
E½!
18/11
852.59
10/9
F#2/3!
20/11
1035.00
16/15
G½!
64/33
1146.73
9/8
A½!
12/11
150.64
121/
Bb½"
11/9
347.41
9/8
C½"
11/8
551.32
128/
D½!
16/11
648.68
9/8
E
27/16
905.87
10/9
F#!
15/8
1088.27
16/15
G
1/1
0
9/8
A
9/8
203.91
10/9
B!
5/4
386.31
8/7
C#1/6"
10/7
617.49
21/20
D
3/2
701.96
9/8
Bb"
6/5
315.64
10/9
C
4/3
498.04
21/20
Db1/6!
7/5
582.51
8/7
Eb"
8/5
813.69
10/9
F
16/9
996.09
9/8
G
1/1
0
16/15
Ab"
16/15
111.73
9/8
D
3/2
701.96
Ab"
16/15
111.73
108
10 - 33
64/63
16/15
D½!
16/11
648.68
9 - 34
81/80
C#1/6"
10/7
617.49
B#"
9/7
435.08
8 - 35
100/99
10/9
9/8
A#"
8/7
231.17
6 - 37
55/54
Secondary
Ratios
Primary
Ratios
5 - 38
81/80
Polychromatic Notation and Extended Tonality © Copyright Juhan Puhm 2016
121
Compendium Musica
11 - 32
12 - 31
13 - 30
14 - 29
15 - 28
64/63
81/80
55/54
45/44
56/55
27.26
21.51
31.77
38.91
31.19
Bb!
6/5
315.64
Bb½!
11/9
347.41
16 - 27
18 - 25
19 - 24
20 - 23
21 - 22
49/48
64/63
81/80
55/54
56/55
17.58
35.70
27.26
21.51
31.77
31.19
C /6"
14/11
417.51
5
B"
5/4
386.31
17 - 26
99/98
B#!
9/7
435.08
C½!
11/8
551.32
C
4/3
498.04
C#"
21/16
470.78
Bb
32/27
294.13
Db1/6"
7/5
582.51
C!
27/20
519.55
50/49
34.98
D#!
32/21
729.22
E
27/16
905.87
E"
5/3
884.36
E½"
18/11
852.59
False Ionian Scales
D#!
32/21
729.22
9/8
Bb
32/27
294.13
9/8
Db1/6"
7/5
582.51
C!
27/20
519.55
68
E#!
12/7
933.13
D5/6!
11/7
782.49
Eb!
8/5
813.69
D"
40/27
680.45
Eb#"
14/9
764.92
D
3/2
701.96
1/1 =G =392 * 2^n
10/9
C
4/3
498.04
10/9
10/9
Eb#"
14/9
764.92
10/9
D
3/2
701.96
F#1/6!
40/21
1115.53
1701/
1600
G!
81/80
21.51
640/
D5/6!
11/7
782.49
112/
567
A#!
8/7
231.17
49/44
C5/6"
14/11
417.51
55/49
C#1/6!
10/7
617.49
16/15
D#!
32/21
729.22
F
16/9
996.09
10/9
G"
160/81
1178.49
9/8
A"
10/9
182.40
16/15
Bb
32/27
294.13
D"
40/27
680.45
297/
9/8
F#"
7/4
968.83
15/14
F#"
15/8
1088.27
28/25
Ab1/6"
21/20
84.47
10/9
Bb#"
7/6
266.87
9/8
C#"
21/16
470.78
16/15
Db1/6"
7/5
582.51
9/8
E
27/16
905.87
16/15
F!
9/5
1017.60
9/8
G!
81/80
21.51
10/9
A
9/8
203.91
8/7
B#!
9/7
435.08
21/20
C!
27/20
519.55
280
99
Polychromatic Notation and Extended Tonality © Copyright Juhan Puhm 2016
D½"
16/11
648.68
C#1/6!
10/7
617.49
Compendium Musica
Partch 43 Tone Scale
1/1 =C
D½!
12/11
150.64
1/11 Limit
F5/6!
14/11
417.51
E"#
9/7
435.08
D"#
8/7
231.17
1/7 Limit
F#1/6#
10/7
617.49
Utonality
# 81/80
1/5 Limit
C#
81/80
21.51
Db#
16/15
111.73
(81/80)
1/3 Limit
3 Limit
C
1/1
0
D
9/8
203.91
Otonality
7 Limit
11 Limit
69
Db1/6!
21/20
84.47
Ab#
8/5
813.69
(81/80)
(81/80)
Eb
32/27
294.13
F
4/3
498.04
Bb
16/9
996.09
E!
5/4
386.31
C½#
33/32
53.27
F"!
21/16
470.78
Eb½#
11/9
347.41
Gb1/6!
7/5
582.51
F½#
11/8
551.32
C½!
64/33
1146.73
½!
/3!
5
/6!
approximately 1/2 semitone flat
"#
/6#
approximately 1/3 semitone sharp
2
B1/6#
40/21
1115.53
(81/80)
Eb"!
7/6
266.87
Db2/3#
11/10
165.00
A"#
12/7
933.13
F#
27/20
519.55
D!
10/9
182.40
! 81/80
G"#
32/21
729.22
B2/3!
20/11
1035.00
Eb#
6/5
315.64
(81/80)
5 Limit
A½!
18/11
852.59
G½!
16/11
648.68
1
Bb#
9/5
1017.60
G
3/2
701.96
A
27/16
905.87
(81/80)
(81/80)
G!
40/27
680.45
A!
5/3
884.36
approximately 5/6 semitone flat
approximately 1/6 semitone sharp
#
syntonic comma (81/80) sharp
!
syntonic comma (81/80) flat
C
2/1
1200
(81/80)
B!
15/8
1088.27
C!
160/81
1178.49
/6!
"!
1
Ab"!
14/9
764.92
approximately 2/3 semitone flat
Bb"!
7/4
968.83
G5/6#
11/7
782.49
Polychromatic Notation and Extended Tonality © Copyright Juhan Puhm 2016
Bb½#
11/6
1049.36
approximately 1/6 semitone flat
approximately 1/3 semitone flat
5
/6#
approximately 5/6 semitone sharp
2
/3#
approximately 2/3 semitone sharp
½#
approximately 1/2 semitone sharp
Compendium Musica
Partch 43 Tone Scale
Harmonic Structure (from which can be derived scales, modes, chords and tonalities)
1/1 =C
-(also the Roots of 3 incomplete Tertiary Tonalities
Primary Ratios
Otonality!
(Bold = Roots)
Secondary
Ratios
G"# (O)
32/21
729.22
B1/6#
40/21
1115.53
Partch Tuning Notes = 8/7, 11/8
F
G
4/3
3/2
498.04
701.96
(7/6)
(9/7)
(7/6)
E"#
D"# (O)
A"# (O)
8/7
12/7
(3/2)
9/7
231.17
933.13
435.08
1
(U) F# /6!
(7/6)
(9/7)
(11/9)
G5/6#
10/7
C
617.49
1/1
11/7
Otonality!
0
782.49
(11/9)
(9/7)
(7/6)
Eb½#
(U) Bb½#
11/9
(3/2)
11/6
347.41
1049.36
(11/6)
(11/9)
F
G
4/3
3/2
498.04
701.96
without the 5 Limit major third)
$Utonality
Db2/3#
11/10
165.00
Secondary
Ratio
(U) F½#
11/8
551.32
(3/2)
(U) C½#
33/32
53.27
$Utonality
(Bold = Roots)
Primary Ratios
Secondary
Otonality!
Primary Ratios
Db! (O)
16/15
111.73
Ratios
(Bold = Roots)
Eb (O)
32/27
294.13
C%
160/81
1178.49
Bb (O)
16/9
996.09
(U) G"
40/27
680.45
Ab! (O)
8/5
813.69
Eb! (O)
6/5
315.64
(U) F (O)
4/3
498.04
(U) D"
10/9
182.40
(U) C (O)
1/1
0
(U) A"
5/3
884.36
Bb! (O)
9/5
1017.60
(U) G (O)
3/2
701.96
(U) E"
5/4
386.31
F! (O)
27/20
519.55
(U) D
9/8
203.91
(U) B"
15/8
1088.27
$Utonality
(Bold = Roots)
C½% (O)
64/33
1146.73
(3/2)
G½% (O)
16/11
648.68
B2/3%
20/11
1035.00
Secondary
Ratio
Otonality!
Partch Tuning Notes = 7/4, 16/11
Primary Ratios
-The 8/7, 11/8, 7/4 and 16/11 intervals are prescribed by Partch to tune the 7 and 11 limit ratios. Without a doubt a difficult task to tune these intervals accurately by ear.
After which the rest of the 7 and 11 limit ratios present no problems. The 5 limit ratios present no tuning problems.
70
Polychromatic Notation and Extended Tonality © Copyright Juhan Puhm 2016
(Bold = Roots)
Ratios
-(also the Roots of 3 incomplete Tertiary Tonalities
Primary Ratios
F
G
4/3
3/2
498.04
701.96
(11/9)
(11/6)
A½%
D½% (O)
12/11
(3/2)
18/11
150.64
852.59
(7/6)
(9/7)
(11/9)
5
F /6%
C
$Utonality
Gb1/6% (O)
14/11
1/1
417.51
0
7/5
(11/9)
(9/7)
(7/6)
582.51
Ab"%
(U) Eb"%
(U) Bb"%
14/9
(3/2)
7/6
7/4
764.92
266.87
968.83
(7/6)
(9/7)
(7/6)
F
G
4/3
3/2
498.04
701.96
(U) A
27/16
905.87
Secondary
Primary Ratios
Otonality!
C#
81/80
21.51
without the 5 Limit major third)
Db1/6%
21/20
84.47
(U) F"%
21/16
470.78
Secondary
Ratios
$Utonality
(Bold = Roots)
Compendium Musica
Partch 43 Tone Scale
1/1 =C
0
Primary
Ratios
1 - 42
2 - 41
3 - 40
4 - 39
7 - 36
9 - 34
10 - 33
64/63
45/44
121/120
100/99
81/80
64/63
49/48
21.51
31.77
31.19
27.26
38.91
14.37
17.40
21.51
27.26
35.70
C
1/1
0
C½"
33/32
53.27
Db1/6!
21/20
84.47
Db"
16/15
111.73
Secondary
Ratios
C!
160/81
1178.49
C½!
64/33
1146.73
B1/6"
40/21
1115.53
B!
15/8
1088.27
C
2/1
1200
False Ionian Scales
D½!
12/11
150.64
Db2/3"
11/10
165.00
D!
10/9
182.40
D
9/8
203.91
D#"
8/7
231.17
Eb#!
7/6
266.87
Bb½"
11/6
1049.36
B2/3!
20/11
1035.00
Bb"
9/5
1017.60
Bb
16/9
996.09
Bb#!
7/4
968.83
A#"
12/7
933.13
1/1 =C
D#"
8/7
231.17
9/8
E#"
9/7
435.08
10/9
F#1/6"
10/7
617.49
16/15
G#"
32/21
729.22
9/8
A#"
12/7
933.13
10/9
B1/6"
40/21
1115.53
63/55
D½!
12/11
150.64
22/21
D#"
8/7
231.17
G½!
16/11
648.68
9/8
A½!
18/11
852.59
10/9
B2/3!
20/11
1035.00
16/15
C½!
64/33
1146.73
9/8
D½!
12/11
150.64
121/
Eb½"
11/9
347.41
9/8
F½"
11/8
551.32
128/
G½!
16/11
648.68
G
3/2
701.96
9/8
A
27/16
905.87
10/9
B!
15/8
1088.27
16/15
C
1/1
0
9/8
D
9/8
203.91
10/9
E!
5/4
386.31
8/7
F#1/6"
10/7
617.49
21/20
G
3/2
701.96
9/8
Eb"
6/5
315.64
10/9
F
4/3
498.04
21/20
Gb1/6!
7/5
582.51
8/7
Ab"
8/5
813.69
10/9
Bb
16/9
996.09
9/8
C
1/1
0
16/15
Db"
16/15
111.73
Db"
16/15
111.73
8 - 35
56/55
C"
81/80
21.51
71
6 - 37
55/54
Secondary
Ratios
Primary
Ratios
5 - 38
81/80
108
Polychromatic Notation and Extended Tonality © Copyright Juhan Puhm 2016
121
Compendium Musica
11 - 32
12 - 31
13 - 30
14 - 29
15 - 28
16 - 27
17 - 26
18 - 25
19 - 24
20 - 23
21 - 22
64/63
81/80
55/54
45/44
56/55
99/98
49/48
64/63
81/80
55/54
56/55
27.26
21.51
31.77
38.91
31.19
17.58
35.70
27.26
21.51
31.77
31.19
Eb!
6/5
315.64
Eb½!
11/9
347.41
F5/6"
14/11
417.51
E"
5/4
386.31
E#!
9/7
435.08
F½!
11/8
551.32
F
4/3
498.04
F#"
21/16
470.78
Eb
32/27
294.13
Gb1/6"
7/5
582.51
F!
27/20
519.55
50/49
34.98
G#!
32/21
729.22
A
27/16
905.87
A"
5/3
884.36
A½"
18/11
852.59
False Ionian Scales
G#!
32/21
729.22
9/8
Eb
32/27
294.13
9/8
Gb1/6"
7/5
582.51
F!
27/20
519.55
72
G5/6!
11/7
782.49
Ab!
8/5
813.69
G"
40/27
680.45
Ab#"
14/9
764.92
G
3/2
701.96
1/1 =C
B1/6!
40/21
1115.53
1701/
G"
40/27
680.45
297/
9/8
Bb#"
7/4
968.83
9/8
A
27/16
905.87
A#!
12/7
933.13
10/9
F
4/3
498.04
10/9
10/9
Ab#"
14/9
764.92
10/9
G
3/2
701.96
D#!
8/7
231.17
49/44
F5/6"
14/11
417.51
55/49
F#1/6!
10/7
617.49
16/15
G#!
32/21
729.22
Bb
16/9
996.09
10/9
C"
160/81
1178.49
9/8
D"
10/9
182.40
16/15
Eb
32/27
294.13
28/25
Db1/6"
21/20
84.47
10/9
Eb#"
7/6
266.87
9/8
F#"
21/16
470.78
16/15
Gb1/6"
7/5
582.51
9/8
C!
81/80
21.51
10/9
D
9/8
203.91
8/7
E#!
9/7
435.08
21/20
F!
27/20
519.55
C!
81/80
21.51
640/
G5/6!
11/7
782.49
112/
15/14
B"
15/8
1088.27
16/15
Bb!
9/5
1017.60
1600
280
567
99
Polychromatic Notation and Extended Tonality © Copyright Juhan Puhm 2016
G½"
16/11
648.68
F#1/6!
10/7
617.49