Towards an Expanded Definition of Consonance:
Tuneable Intervals on Horn, Tuba and Trombone
by Marc Sabat and Robin Hayward
PLAINSOUND MUSIC EDITION
for James Tenney
(1934-2006)
Towards an Expanded Definition of Consonance:
Tuneable Intervals on Horn, Tuba and Trombone
by Marc Sabat and Robin Hayward
July 2006, Berlin
ABSTRACT
Three classes of musical intervals may be distinguished: tuneable intervals which may be
tuned precisely by ear; intervals which cannot be directly tuned but may be reached through
a finite series of tuneable interval steps; and intervals which cannot be precisely tuned, even
if they can be closely approximated by a trained listener. The process of tuning precisely is
accomplished by listening to the periodicity of the composite sound and by paying attention
to combination tones and beats. Tuneable intervals are always expressible as frequency
ratios of natural numbers.
This distinction offers the possibility of new, phenomenologically based definitions: a
tuneable interval is called a consonance; an interval which cannot be precisely tuned is
called a dissonance. The boundary between these classes will vary according to individual
experience. Since some intervals are more readily tuned than others, relative consonance is
proportional to the degree of difficulty in precisely tuning an interval.
The valved brass instruments derive their pitches from the overtone series of different
lengths of tubing. By tuning these lengths in rational proportions to each other, the overtone
series become related by frequency ratios of natural numbers, implying the possibility of
producing tuneable intervals without having to alter the natural pitches. This also increases
the potential number of tuneable intervals between two such instruments. Various options for
horn, tuba and trombone are discussed in order to determine the most useful microtonal
gamut, including conventional valve configurations as well as possible modifications of the
instruments’ design.
The set of available tuneable intervals within this large range of pitches is most readily
determined by computational methods. The resulting tables offer an overview of new
instrumental possibilities and establish a framework for composition within a 23-limit subset
of harmonic space. The ensuing discussion includes some general aspects of modulation
and enharmonic near-equivalence in this space.
2
1. TUNEABLE SOUNDS
1.1 Definitions
When a sound exhibits a periodic or near-periodic pattern of vibrations, its spectral structure
resembles a harmonic series, consisting of integer multiples of a fundamental frequency. A
sound of this form may be called harmonic. Harmonic sounds which are perceived as having
a single predominant pitch-height (the fundamental frequency) will be referred to as pitches.
This paper describes research about the perceptibility of relationships between pitches.
Computational methods allow musically significant subsets to be determined and compared
with available microtonal pitch-sets on acoustic brass instruments.
The relationship between two pitches is called a musical interval. There is a perceptual
identity between intervals having the same frequency ratio because of the analogy between
their composite vibration patterns and their composite spectral structures. If a frequency ratio
is represented by a ratio of natural numbers, the composite sound is also harmonic and is
called a natural interval. The melodic distance between the pitches of an interval is
commonly measured in cents proportional to the logarithm of the frequency ratio.
1.2 Ordering intervals melodically and harmonically
These two numerical representations (as a ratio of frequencies and as a distance in cents)
suggest two ways in which natural intervals may be ordered. Let us assume two pitches P0
and P1, with frequencies F0 and F1. If P0 is allowed to remain fixed (for example as A4
440Hz) and P1 varies within the range of the piano, from A01 27.5 Hz up to C8 4186 Hz, then
the melodic distance
(1)
¢(P1) = (1200·log2(F1/F0))
varies between -4800¢ and +3900¢. As this value increases, the pitch rises. If it is negative,
P1 is below P0; if positive, P1 is above P0. Given any set of pitches {p1,p2,...,pn}, their melodic
distances to P0 allow us to deduce a rising order {P1,P2,...,Pn} with melodic distances to P0
{¢(P1),¢(P2),...,¢(Pn)} such that ¢(Pb) ≥ ¢(Pa)) if (b > a). Note also that since (by logarithmic
identities)
(2)
¢(Fb/Fa) = ¢((Fb/F0)(F0/Fa)) = ¢(Fb/F0) - ¢(Fa/F0)
the set of melodic distances also allows us to determine all intervals between pitches in the
set. In general, given two pitch sets {Pn} and {Qm} it is possible to form an array MD{PxQ}
such that the elements of P and Q are in rising order and MDi,j = ¢(FPi/FQj). This array
evaluates all possible intervals between the pitch sets as melodic distances.
A second possible ordering may be deduced from the representation by frequency ratios.
Just as cents enable us to represent the percept melodic distance between pitches, the
ratios will enable us to deduce harmonic distance, a mathematical measure of relative
consonance (more precisely defined below).
In general, frequency ratios made up of smaller numbers (in lowest terms), such as 1/1, 2/1,
1
The pitch notation used in this paper is based on MIDI note number 60 = middle C = C4.
3
3/1, 3/2, 4/3, represent more easily perceived intervals, and frequency ratios made up of
larger numbers (in lowest terms), such as 19/16 or 32/27, represent intervals which are more
difficult to perceive. One attempt to generalize this mathematically is Leonhard Euler’s
gradus suavitatis (consonance grade) function2 (1739). Given a ratio R in lowest terms,
expressed as a product of primes in the form:
(3)
R = p1k1·p2k2·...·pnkn
Euler’s function G(R) is defined as:
(4)
G(R) = 1 + ∑|ki|·(pi - 1)
Ratios which produce lower consonance grade values are usually easier to tune than ratios
with higher values.
1.3 Tuneable Intervals
Consider two pitches, melodically close to each other, sounding simultaneously. As they
approach unison, we perceive amplitude modulation or beating which gradually slows down.
At a certain point, this beating is replaced by a phenomenon of spectral fusion (perceptible
periodicity), which may be likened to a focussing of the sound of the interval 1/1. Any interval
that may be determined by ear in a similar manner is called a tuneable interval. Note that one
of the characteristics of tuneability is a region of tolerance in which the interval is perceived
as being nearly in tune. The existence of such regions, together with the fact that the
frequency range of human hearing is limited, implies that there can only be a finite number of
tuneable intervals.
Tuneability is not proposed as an absolute property. It varies depending on the register,
relative volume and timbre of the sounds, as well as on the experience of the listener.
Nevertheless, it suggests a precise perceptual definition of consonance: namely, a
consonant interval is one which may be precisely tuned by ear. Relative consonance may be
described as the degree of difficulty in achieving a precise intonation.
It should be noted that we distinguish tuneable intervals from intervals which may played by
memorizing their sound and approximating their size. There are many ratios which familiar
usage allows us to play with acceptable accuracy, but it is not possible to say from their
sound alone when they are exactly centered. Several examples which come to mind include
the semitone 16/15, the wholetone 9/8, and the minor seventh 16/9. Similarly, many
instrumentalists can learn to reproduce the pitches of Equal Temperament, for example
when playing with a piano. By contrast, the septimal wholetone 8/7 and the minor sevenths
7/4 and 9/5 are directly tuneable.
1.4 Prime limits and harmonic space
Let us consider the set of positive rational numbers in lowest terms. Each such number may
be considered to represent a unique natural interval and (as above) may be represented as a
product of the form
(3)
2
R = p1k1·p2k2·...·pnkn
Martin Vogel: “On the Relations of Tone” (Orpheus Verlag, Bonn, 1993), pp. 139–147
4
where the pi are unique and increasing prime numbers. Following the musical terminology
established by Harry Partch3, we will define pn as the limit of the interval. For example, the
interval 5/3 will be called a “5-limit interval”, all natural intervals generated by the primes 2, 3,
and 5 will be called the “5-limit set”, and so on. The set of all pn-limit intervals may be
modelled using James Tenney’s concept of harmonic space4 as an n-dimensional lattice with
the primes p1,p2,...,pn as axes. The interval R is located by the coordinates (k1,k2,...,kn). A
progression from one interval to another can be modeled as a movement in harmonic space.
Tenney proposes a refinement of Euler’s gradus suavitatis with his harmonic distance
function:
(5)
HD(R) = C · ∑|ki|log2(pi)
where C is a constant.
Since the number of possible tuneable intervals is finite, they must be contained within some
prime limit. The subset of harmonic space which may be traversed by tuneable interval steps
will be called the perceptible harmonic space. We will attempt to establish a provisional set of
tuneable intervals and examine some characteristics of the resulting space.
To determine the prime limit of tuneable intervals, it is necessary to examine all intervals of
the form p/a, where p is a prime and a is an integer less than p. If a tuneable interval exists
for each prime less than p, and if p is also found to be tuneable, all intervals within the limit p
may be reached by a finite number of tuneable interval steps.
As an example, for the prime 23 the tuneability of each interval 23/1, 23/2, up to 23/22 is
examined. Note that the interval 23/1 is itself between four and five octaves 5 in size (circa
5428.3¢). Because the phenomena which establish tuneability depend largely on the
interaction of spectral components, wider intervals are increasingly difficult to tune precisely.
It is easier to judge the tuneability of narrower intervals, such as 23/3 or 23/4. Earlier
research on string instruments 6 limited the intervals for the purpose of musical practicability
to a three-octave range. Nonetheless, a brief test of larger intervals indicates that ratios as
high as 23/1 could be tuneable and examination of these larger intervals merits further
study.7
1.5 Analysis of the 23-limit set of tuneable intervals within three octaves
Let us assume a given prime p is tuneable. In the set of intervals {p/a} mentioned above, as
the denominator rises the interval becomes narrower and the resulting periodic pattern
increases in complexity. It is musically relevant to consider the smallest tuneable interval
resulting from this series. Empiric testing indicates that for the seven primes
{2,3,5,7,11,13,23}, at least one tuneable interval is 1200¢ (2/1) or less. In the case of 17 and
19, the smallest tuneable interval is between 1200¢ and 1902¢ (3/1). For each prime p up to
23, we can also define a least generative tuneable interval. It is the smallest p-limit tuneable
interval with a power of 2 in the denominator: {2/1, 3/2, 5/4, 7/4, 11/8, 13/8, 17/4, 19/8, 23/8}.
3
Harry Partch: “Genesis of a Music” (Second Edition, Da Capo Press, New York, 1974), p. 75.
James Tenney: “John Cage and the Theory of Harmony” (1983), reprinted in “Soundings 13: The
music of James Tenney” (Soundings Press, Santa Fe, 1984), p. 69.
5
n
The musical term octave refers to the interval 2/1 or 1200¢, n octaves is 2 /1 or (1200n)¢.
6
Marc Sabat: “Analysis of Tuneable Intervals on Violin and Cello” (Plainsound Music Edition, Berlin,
2004)
7
It was determined that the intervals 9/1, 19/2, 10/1, 21/2, 11/1, 12/1, 13/1, 14/1 and 15/1 within the
fourth octave may be easily tuned on horn and tuba.
4
5
The existence of these intervals implies that we can determine a finite tuneable path
connecting any two points within a 23-limit harmonic space.
Having established that we are primarily interested in intervals composed of the first nine
prime numbers, the next step is to generate a set of possibly tuneable ratios and to subject
them all to empirical testing. Intervals produced by the integers {1,2,...,28} are considered.
(Although it is possible that a few tuneable intervals may exist with ratios involving higher
numbers, experience indicates that this is quite unlikely.) This set generates 784 ratios, 28 of
which of the form n/n. The remaining 756 ratios may be paired into two groups of 378,
depending on whether the interval is measured downwards or upwards, for example 2/1 and
1/2. By reducing the set to intervals measured upwards (numerator ≥ denominator) 406
possibilities remain. For each interval, its melodic distance from 1/1 (0¢) is calculated and the
set is sorted in rising order. Considering only ratios in lowest terms up to 8/1 (3600¢), there
are 213 possible tuneable intervals.
Each of these intervals was tested in several ways. Initially, the sounds were generated
using a sampled reed organ timbre on computer and the pitch of the upper note was detuned
in units of 0.1¢ to see if the exact interval could be determined by ear. A second test involved
listening for tuneable intervals as double-stops above the open A and D strings of a violin
(slowly rising glissando). Whenever an interval could be tuned by eliminating beating or by
periodicity recognition, its cents value would be measured using an electronic tuner and
compared with the rising table. A similar test was performed above the open D string of a
cello.
The resulting tuneable intervals were tested once again several months later, at which point
92 intervals were identified (including the unison 1/1). The 183 intervals above and below A
were notated and their tuneability was further verified between two string instruments (violin
and contrabass for the lower intervals and violin and violin for the upper intervals).
In the table below, the results of these tests are tabulated. The relative difficulty of tuning the
sounds is shown in four degrees – easy (3), somewhat more difficult (2), quite difficult (1), not
possible (0). Euler’s consonance grade values and Tenney’s harmonic distance values are
included for comparison with the results of this empiric evaluation.
6
Table 1
Limit
Ratio
1
1/1
Inversion
Cents
0
Tuneable Degree
x
Euler
Tenney
3
1
0
7
28/27
27/28
62.960904
0
15
9.56224242
13
27/26
26/27
65.337341
0
20
9.45532722
13
26/25
25/26
67.900234
0
22
9.34429591
5
25/24
24/25
70.672427
0
14
9.22881869
23
24/23
23/24
73.680654
0
28
9.10852446
23
23/22
22/23
76.956405
0
34
8.98299357
11
22/21
21/22
80.537035
0
20
8.85174904
7
21/20
20/21
84.467193
0
15
8.71424552
19
20/19
19/20
88.800698
0
25
8.56985561
19
19/18
18/19
93.603014
0
24
8.41785251
17
18/17
17/18
98.954592
0
22
8.25738784
17
17/16
16/17
104.95541
0
21
8.08746284
5
16/15
15/16
111.73129
0
11
7.9068906
7
15/14
14/15
119.44281
0
14
7.71424552
13
14/13
13/14
128.29824
0
20
7.50779464
5
27/25
25/27
133.23757
0
15
9.39874369
13
13/12
12/13
138.57266
0
17
7.28540222
23
25/23
23/25
144.35308
0
31
9.16741815
11
12/11
11/12
150.63706
0
15
7.04439412
23
23/21
21/23
157.49344
0
31
8.91587938
11
11/10
10/11
165.00423
0
16
6.78135971
19
21/19
19/21
173.26789
0
27
8.64024494
5
10/9
9/10
182.40371
0
10
6.4918531
19
19/17
17/19
192.55761
0
35
8.33539035
7
28/25
25/28
196.19848
0
17
9.45121111
3
9/8
8/9
203.91
0
8
6.169925
23
26/23
23/26
212.25331
0
36
9.22400167
17
17/15
15/17
216.68669
0
23
7.99435344
11
25/22
22/25
221.30949
0
20
9.10328781
7
8/7
7/8
231.17409
2
10
5.80735492
23
23/20
20/23
241.96063
x
0
29
8.84549005
13
15/13
13/15
247.74105
0
19
7.60733031
19
22/19
19/22
253.80493
7
7/6
6/7
266.87091
23
27/23
23/27
17
20/17
13
13/11
19
7
0
30
8.70735913
3
10
5.39231742
277.59066
0
29
9.27844946
17/20
281.3583
0
23
8.40939094
11/13
289.20972
0
23
7.15987134
19/16
16/19
297.51302
0
23
8.24792751
25/21
21/25
301.84652
0
17
9.03617361
5
6/5
5/6
315.64129
3
8
4.9068906
23
23/19
19/23
330.76133
0
41
8.77148947
17
17/14
14/17
336.1295
0
24
7.89481776
23
28/23
23/28
340.55156
0
31
9.33091688
x
x
11
11/9
9/11
347.40794
1
15
6.62935662
11
27/22
22/27
354.54706
x
0
18
9.21431912
13
16/13
13/16
359.47234
0
17
7.70043972
17
21/17
17/21
365.8255
0
25
8.47978026
13
26/21
21/26
369.74675
0
22
9.09275714
7
5
5/4
4/5
386.31371
3
7
4.32192809
19
24/19
19/24
404.44198
x
0
24
8.83289001
19
19/15
15/19
409.2443
0
25
8.15481811
11
14/11
11/14
417.50796
0
18
7.26678654
23
23/18
18/23
424.36435
0
28
8.69348696
3
11
5.97727992
0
28
8.54689446
7
9/7
7/9
435.0841
17
22/17
17/22
446.36253
x
13
13/10
10/13
454.21395
2
18
7.02236781
17
17/13
13/17
464.42775
0
29
7.78790256
7
21/16
16/21
470.78091
0
13
8.39231742
19
25/19
19/25
475.11441
3
4/3
3/4
498.045
5
27/20
20/27
23
23/17
19
19/14
11
19
x
0
27
8.8917837
3
7
3.5849625
519.55129
0
13
9.0768156
17/23
523.31894
0
39
8.6110248
14/19
528.68711
0
26
8.05528244
15/11
11/15
536.95077
0
17
7.36632221
26/19
19/26
543.01465
0
32
8.94836723
11
11/8
8/11
551.31794
1
14
6.45943162
13
18/13
13/18
563.38234
0
18
7.87036472
5
25/18
18/25
568.71743
0
14
8.81378119
7
7/5
5/7
582.51219
3
11
5.12928302
17
24/17
17/24
596.99959
0
22
8.67242534
17
17/12
12/17
603.00041
0
21
7.67242534
19
27/19
19/27
608.35199
7
10/7
7/10
617.48781
23
23/16
16/23
628.27435
x
x
x
x
0
25
9.00281502
2
12
6.12928302
0
27
8.52356196
13
13/9
9/13
636.61766
x
1
17
6.87036472
11
16/11
11/16
648.68206
x
1
15
7.45943162
19
19/13
13/19
656.98535
0
31
7.94836723
11
22/15
15/22
663.04923
0
18
8.36632221
17
25/17
17/25
667.67202
0
25
8.73131903
19
28/19
19/28
671.31289
3
3/2
2/3
701.955
17
26/17
17/26
23
23/15
13
20/13
17
17/11
0
27
9.05528244
3
4
2.5849625
735.57225
0
30
8.78790256
15/23
740.00563
0
29
8.43045255
13/20
745.78605
0
19
8.02236781
11/17
753.63747
0
27
7.54689446
1
12
6.97727992
0
13
8.64385619
7
14/9
9/14
764.9159
5
25/16
16/25
772.62743
x
x
11
11/7
7/11
782.49204
2
17
6.26678654
19
19/12
12/19
795.55802
x
0
23
7.83289001
17
27/17
17/27
800.90959
0
23
8.84235034
3
8
5.32192809
0
21
8.09275714
2
16
6.70043972
5
8/5
5/8
813.68629
13
21/13
13/21
830.25325
x
13
13/8
8/13
840.52766
11
18/11
11/18
852.59206
0
16
7.62935662
23
23/14
14/23
859.44844
0
30
8.33091688
17
28/17
17/28
863.8705
5
5/3
3/5
884.35871
3
27/16
16/27
13
22/13
13/22
x
0
25
8.89481776
3
7
3.9068906
905.865
0
11
8.7548875
910.79028
0
24
8.15987134
x
8
17
17/10
10/17
918.6417
7
12/7
7/12
933.12909
0
22
7.40939094
2
11
6.39231742
19
19/11
11/19
13
26/15
15/26
946.19507
0
29
7.70735913
952.25895
0
20
7
7/4
4/7
968.82591
8.60733031
3
9
4.80735492
23
23/13
13/23
3
16/9
9/16
987.74669
0
35
8.22400167
996.09
0
9
7.169925
7
25/14
14/25
1003.8015
5
9/5
5/9
1017.5963
11
20/11
11/20
1034.9958
11
11/6
6/11
1049.3629
13
24/13
13/24
1061.4273
13
13/7
7/13
1071.7018
7
28/15
15/28
1080.5572
5
15/8
8/15
1088.2687
x
x
0
16
8.45121111
3
9
5.4918531
0
17
7.78135971
2
14
6.04439412
0
18
8.28540222
x
2
19
6.50779464
0
15
8.71424552
x
1
10
6.9068906
x
x
17
17/9
9/17
1101.0454
0
21
7.25738784
19
19/10
10/19
1111.1993
0
24
7.56985561
11
21/11
11/21
1119.463
0
19
7.85174904
23
23/12
12/23
1126.3193
1
27
8.10852446
13
25/13
13/25
1132.0998
0
21
8.34429591
7
27/14
14/27
1137.0391
0
14
8.56224242
x
2
2/1
1/2
1200
3
2
1
13
27/13
13/27
1265.3373
x
0
19
8.45532722
5
25/12
12/25
1270.6724
0
13
8.22881869
23
23/11
11/23
1276.9564
0
33
7.98299357
7
21/10
10/21
1284.4672
0
14
7.71424552
19
19/9
9/19
1293.603
0
23
7.41785251
17
17/8
8/17
1304.9554
0
20
7.08746284
7
15/7
7/15
1319.4428
0
13
6.71424552
13
28/13
13/28
1328.2982
13
13/6
6/13
1338.5727
11
24/11
11/24
1350.6371
11
11/5
5/11
1365.0042
5
20/9
9/20
1382.4037
0
21
8.50779464
x
2
16
6.28540222
0
16
8.04439412
x
2
15
5.78135971
0
11
7.4918531
3
9/4
4/9
1403.91
3
7
5.169925
11
25/11
11/25
1421.3095
x
0
19
8.10328781
7
16/7
7/16
1431.1741
0
11
6.80735492
23
23/10
10/23
1441.9606
7
7/3
3/7
1466.8709
0
28
7.84549005
x
3
9
4.39231742
13
26/11
11/26
1489.2097
0
24
8.15987134
19
19/8
8/19
1497.513
x
1
22
7.24792751
5
12/5
5/12
1515.6413
x
3
9
5.9068906
17
17/7
7/17
1536.1295
x
1
23
6.89481776
11
22/9
9/22
1547.4079
0
16
7.62935662
11
27/11
11/27
1554.5471
0
17
8.21431912
3
6
3.32192809
0
19
8.26678654
5
5/2
2/5
1586.3137
11
28/11
11/28
1617.508
x
23
23/9
9/23
1624.3643
0
27
7.69348696
7
18/7
7/18
1635.0841
x
2
12
6.97727992
13
13/5
5/13
1654.2139
x
2
17
6.02236781
7
21/8
8/21
1670.7809
0
12
7.39231742
9
3
8/3
3/8
1698.045
5
27/10
10/27
1719.5513
19
19/7
7/19
1728.6871
11
11/4
4/11
1751.3179
5
25/9
9/25
1768.7174
x
x
3
6
4.5849625
0
12
8.0768156
0
25
7.05528244
3
13
5.45943162
0
13
7.81378119
7
14/5
5/14
1782.5122
x
3
12
6.12928302
17
17/6
6/17
1803.0004
x
2
20
6.67242534
7
20/7
7/20
1817.4878
x
2
13
7.12928302
23
23/8
8/23
1828.2743
x
2
26
7.52356196
13
26/9
9/26
1836.6177
0
18
7.87036472
3
3/1
1/3
1901.955
x
3
3
1.5849625
7
28/9
9/28
1964.9159
x
1
13
7.97727992
5
25/8
8/25
1972.6274
x
1
12
7.64385619
11
22/7
7/22
1982.492
x
1
18
7.26678654
19
19/6
6/19
1995.558
x
1
22
6.83289001
5
16/5
5/16
2013.6863
x
3
9
6.32192809
13
13/4
4/13
2040.5277
x
3
15
5.70043972
23
23/7
7/23
2059.4484
0
29
7.33091688
5
10/3
3/10
2084.3587
x
3
9
5.4918531
3
27/8
8/27
2105.865
x
1
10
7.7548875
17
17/5
5/17
2118.6417
x
2
21
6.40939094
7
24/7
7/24
2133.1291
x
1
12
7.39231742
7
7/2
2/7
2168.8259
x
3
8
3.80735492
7.45121111
7
25/7
7/25
2203.8015
0
15
5
18/5
5/18
2217.5963
x
3
10
6.4918531
11
11/3
3/11
2249.3629
x
3
13
5.04439412
7.50779464
13
26/7
7/26
2271.7018
0
20
5
15/4
4/15
2288.2687
x
3
9
5.9068906
19
19/5
5/19
2311.1993
x
2
23
6.56985561
23
23/6
6/23
2326.3193
x
1
26
7.10852446
7
27/7
7/27
2337.0391
x
1
13
7.56224242
2
4/1
1/4
2400
x
3
5
2
5
25/6
6/25
2470.6724
x
1
12
7.22881869
7
21/5
5/21
2484.4672
x
1
13
6.71424552
17
17/4
4/17
2504.9554
x
2
19
6.08746284
13
13/3
3/13
2538.5727
x
3
15
5.28540222
11
22/5
5/22
2565.0042
x
1
16
6.78135971
3
9/2
2/9
2603.91
x
3
6
4.169925
23
23/5
5/23
2641.9606
x
2
27
6.84549005
7
14/3
3/14
2666.8709
x
3
10
5.39231742
19
19/4
4/19
2697.513
x
2
21
6.24792751
5
24/5
5/24
2715.6413
x
2
10
6.9068906
5
5/1
1/5
2786.3137
x
3
5
2.32192809
13
26/5
5/26
2854.2139
x
1
18
7.02236781
7
21/4
4/21
2870.7809
x
2
11
6.39231742
3
16/3
3/16
2898.045
x
3
7
5.5849625
5
27/5
5/27
2919.5513
0
11
7.0768156
11
11/2
2/11
2951.3179
x
3
12
4.45943162
7
28/5
5/28
2982.5122
x
2
13
7.12928302
17
17/3
3/17
3003.0004
x
3
19
5.67242534
23
23/4
4/23
3028.2743
x
2
25
6.52356196
10
3
6/1
1/6
3101.955
x
3
4
2.5849625
5
25/4
4/25
3172.6274
x
2
11
6.64385619
19
19/3
3/19
3195.558
x
3
21
5.83289001
13
13/2
2/13
3240.5277
x
3
14
4.70043972
5
20/3
3/20
3284.3587
x
3
9
5.9068906
3
27/4
4/27
3305.865
x
1
9
6.7548875
7
7/1
1/7
3368.8259
x
3
7
2.80735492
11
22/3
3/22
3449.3629
x
3
14
6.04439412
5
15/2
2/15
3488.2687
x
3
8
4.9068906
23
23/3
3/23
3526.3193
x
3
25
6.10852446
2
8/1
1/8
3600
x
3
4
3
The set of tuneable intervals above A4, written in musical staff notation, follows as Table 2.
The accidentals, which provide exact graphical representations of frequency ratios, are from
the Extended Helmholtz-Ellis JI Pitch Notation8. A legend of the accidentals is included as
Table 3. The three degrees of tuneability are notated by using large open notes (3), small
open notes (2) and small black notes (1) respectively.
These intervals represent a set of microtonal interval-classes which are congruent to the
ability to perceive relationships between harmonic sounds and serve as a basis for further
analysis. Note that although the subset above 1/1 and the subset below 1/1 mirror each other
exactly, different classes of intervals are present in each octave (for example, 13/3, 3/13,
13/6 and 6/13 are included, but 13/12 and 12/13 are not). This is a departure from the
classical convention of allowing pitch-class set representations in which elements may be
freely transposed between octaves.
Tables 2 and 3 (overleaf)
8
Marc Sabat and Wolfgang von Schweinitz: “The Extended Helmholtz-Ellis JI Pitch Notation”
(Plainsound Music Edition, Berlin, 2004)
INTERVALS TUNEABLE BY EAR (up to 3 Octaves wide)
11
below the violin A-string
tested and notated by Marc Sabat
with assistance from Wolfgang von Schweinitz (cello); Beltane Ruiz (bass); Anaïs Chen (violin)
1/8

 j


3/23

1/6

n
-2.0

\v
+2.5
±e
-28.3

j


<e
-37.0

-18.6

-28.3


+13.7

o
;<u
-36.1


f

u

n
-3.0

5<



5
9/14
>v


n
-65.0

m


9m
7/9

<
-35.1
+13.7

<f
9
+63.4

o

5v
-47.4
9/11
28/9

>v
7/18

<
-35.1

9
1/2
j
3/5
o



8/13
9
+59.5

>u

u
-15.6
5/6

5
-51.3
6/7

>v
+33.1

8/11
5/7
+17.5
-17.5
4/5

+35.1
+61.4
+15.6

7/24
6/13
7/10
9/13
11/16
4
<
g
-33.1
-17.6
+51.3


>
+31.2


7/12
4/7
5/9

2/3
-3.9


6/25
+29.3
-33.1
9m
5u

-15.6
5/13
+45.8

<
5/11

n
-2.0
10/13
+45.8
4/9

g
3/8

u
25/8
+27.4
n
-51.3

>

+2.0

2/7
+31.2
5<

6/11
>u
+15.5
g
5/24
5/21
22/7
-82.5
+27.4
o
-17.6


+13.7
5/18
4/11
5



>v
+33.1
;u
4/25
1/5

m
3/7
\v
9<e
+35.1

>u
7/13
7/11
\v
\v
4/17
-5.0

14/5
+17.5

+4.4
9m
19/6
+4.4
8/19
+2.5
-15.6
-49.4

+45.8
3/11
5
-13.7

5/12
-82.5

u

5/26

9
3/19
9
3/13
+61.4

+59.5
>
16/5
-49.4
3/4
+2.0
f
6/17
;u
+28.3


+29.2
4/15
+11.7

-26.3
-13.7



2/13
4/21
n
+2.0
-65.0

7/20
u

5u
9
8/15
+11.7
8/5


3/16
13/4
<f
12/23
±e
-11.2
+59.5
7/17
2/5

Æó
o
2/11
5/19


+15.6
5/22
-3.9

-17.5
-5.9
-51.3
n
3/20
n
2/9
10/3
o

5
6/23
+15.6
±e
-2.0

-42.0
8/23
1/3

>u
±m
-26.3
-5.9
 n
+17.5
4/27
>
5/23
±e


+31.2
5/28

27/8
n
;t
-3.0
1/7
-49.4
7/27
17/5

5
3/17
;u
>v
+33.1
3/22

3/14
1/4

f
4/23
4/19


+11.7
±e
-26.3
2/15

<
-31.2
7/8
j
1/1


INTERVALS TUNEABLE BY EAR (up to 3 Octaves wide)
12
above the violin A-string
notated using the Extended Helmholtz-Ellis JI Pitch Notation
microtonal accidentals designed by Marc Sabat and Wolfgang von Schweinitz, 2004
1/1
 j
8/7



4

<f


-59.5
u


18/7

>v
0

-35.1

-27.4
t
24/7
>v
 t

-31.2
+65.0
4f
o
n

11/5
o
8/3
 -27.4 t

25/4

25/8

<
7/2


+17.6
22/7

<
/

+13.7
11/3
-11.7

u


:f
12/5
o
u

17/6

0v
-59.5
+11.2

15/4

-15.6

19/5
/f
n
+3.9

u


0o
-13.7
-45.8
19/3

0v
-59.5

n
-29.2

-15.6

u


4
+51.3
-2.0
20/3
13/2
11/2
16/3
<
v
+5.9
17/7

∞v

∞v
10/3
v
+5.9

+26.3
9/2
22/5
4f
21/4
∞v
+26.3

+28.3
+42.0
23/6
u

23/5
∞o

27/4

<
-31.2
7/1

<f
-17.5
28/5

4
+49.4
+3.0
:f
-11.7


23/12
5/2



:g


-33.1
-2.5
∞ò
+26.3

23/3



/

4/1
19/4
<
23/4
17/5
j
14/3
∞v

3/1
+18.6
27/7
+28.3
15/2
u
n
+2.0

>v
17/3

22/3

23/8
27/8
+37.0
-61.4
26/5

15/8
:>o
+36.1
20/7
>u
+17.5
13/4
16/5

o
-11.7

+15.6
8/5
+13.7
-13.7
+3.0

+65.0
0
:f
14/5
o
13/3
17/4
+5.0
19/8

4>
13/7

-2.0
-2.5
<f

4
+49.4

0>v

n
11/7
+82.5
-35.1

4/3
-45.8
14/9
11/6
4
+49.4
7/3
-17.5

-4.4
18/5
11/4
19/6
/

o
u
-4.4

<

4

4>
+82.5
5/1
/
9/5


+51.3
-15.5

n
0o
-28.3
9/4
n
+3.9
-2.0
21/5
24/5
+15.6

13/5
<f
-29.3

-33.1
-45.8
25/6



0o
28/9
<
+33.1


<
-61.4
+35.1


>v
+17.6

>v
3/2

13/10
+35.1
+2.0
-51.3
7/4

-13.7
5
-63.4
9/7
-31.2
13/6

u
16/11
0
-15.6
2/1
j

+33.1
4


12/7
5/3
+47.4
13/9
>u
5/4


10/7
+17.5
-17.5
11/9
o
-33.1
7/5
13/8
0v

>
11/8
6/5
+15.6
<
+31.2
+51.3
7/6
6/1

n
+2.0
8/1
j
Berlin, Oct. 2004 - April 2005
13
The Extended Helmholtz-Ellis JI Pitch Notation
microtonal accidentals designed by Marc Sabat and Wolfgang von Schweinitz, 2004
3-LIMIT (PYTHAGOREAN) INTERVALS
d
k
r
y
FUNCTION OF THE ACCIDENTALS
Ä
notate 35 pitches from the series of untempered perfect fifths
(3/2) ≈ ± 702.0 cents;
perfect fifth (3/2); perfect fourth (4/3); major wholetone (9/8)
5-LIMIT (PTOLEMAIC) INTERVALS
c
j
q
x
U
e
l
s
w
Å
notate an alteration by one syntonic comma (81/80) ≈ ± 21.5 cents;
major third (5/4); minor third (6/5); major sixth (5/3); minor sixth (8/5);
minor wholetone (10/9)
b
i
p
w
T
f
m
t
x
X
notate an alteration by two syntonic commas
(81/80)·(81/80) ≈ ± 43.0 cents;
augmented fifth (25/16); diminished fourth (32/25)
a
h
o
v
S
g
n
u
|
É
notate an alteration by three syntonic commas
(81/80)·(81/80)·(81/80) ≈ ± 64.5 cents;
minor diesis (128/125)
7-LIMIT (SEPTIMAL) INTERVALS
<
• or <<
>
¶
or >>
notate an alteration by one septimal comma (64/63) ≈ ± 27.3 cents;
natural seventh (7/4); septimal wholetone (8/7);
septimal diminished fifth (7/5); septimal tritone (10/7);
septimal minor third (7/6); septimal quartertone (36/35)
notate an alteration by two septimal commas
(64/63)·(64/63) ≈ ± 54.5 cents;
septimal sixthtone (49/48)
11-LIMIT (UNDECIMAL) INTERVALS
C
B
notate an alteration by one undecimal quartertone
(33/32) ≈ ± 53.3 cents;
undecimal augmented fourth (11/8); undecimal diminished fifth (16/11)
9
notate an alteration by one tridecimal thirdtone (27/26) ≈ ± 65.3 cents;
tridecimal neutral sixth (13/8); tridecimal neutral third (16/13)
13-LIMIT (TRIDECIMAL) INTERVALS
0
PRIMES IN THE HARMONIC SERIES OCTAVE 16 - 32 (5-limit signs are given here relative to “A”)
notate an alteration of the 5-limit accidental by one 17-limit schisma
(16/17)·(16/15) = (256/255) ≈ ± 6.8 cents;
:l
;x
Galileo’s “equal-tempered” semitone (18/17);
17-limit diminshed seventh chord 10:12:14:17
/r
\r
notate an alteration by one 19-limit schisma
(19/16)·(27/32) = (513/512) ≈ ± 3.4 cents;
19-limit minor third (19/16); 19-limit minor triad 16:19:24
^y
±k
notate an alteration by one 23-limit comma
(23/16)·(8/9)·(8/9)·(8/9) ≈ ± 16.5 cents;
raised leading tone (23/12)
14
\s
]q
notate an alteration of the 5-limit accidental by one 29-limit comma
(29/16)·(5/9) = (145/144) ≈ ± 12.0 cents
¥B
µC
notate an alteration of the 11-limit accidental by one 31-limit schisma
(32/31)·(32/33) = (1024/1023) ≈ ± 1.7 cents
PRIMES IN THE HARMONIC SERIES OCTAVE 32 - 64 (5-limit signs are given here relative to “A”)
∂C
∑B
notate an alteration of the 11-limit accidental by one 37-limit schisma
(36/37)·(33/32) = (297/296) ≈ ± 5.8 cents
{w}
{j}
notate an alteration of the 5-limit accidental by one 41-limit schisma
(32/41)·(81/64)·(81/80) = (6561/6560) ≈ ± 0.3 cents
∏
π
notate an alteration by one 43-limit comma
(43/32)·(3/4) = (129/128) ≈ ± 13.5 cents
∫y or Ωy
ªk or æk
notate an alteration of the 7-limit accidental by one 47-limit schisma
(32/47)·(48/49)·(3/2) = (2304/2303) ≈ ± 0.8 cents
¡x
¬l
notate an alteration of the 5-limit accidental by one 53-limit comma
(32/53)·(5/3) = (160/159) ≈ ± 10.9 cents
√
ƒ
notate an alteration of the 13-limit accidental by one 59-limit schisma
(32/59)·(24/13) = (768/767) ≈ ± 2.3 cents
≈x
∆l
notate an alteration of the 7-limit accidental by one 61-limit schisma
(61/32)·(21/40) = (1281/1280) ≈ ± 1.4 cents
IRRATIONAL AND TEMPERED INTERVALS
A
a
j
z
Z
notate the respective Equal Tempered Semitone;
may be combined with a cents indication to notate any pitch
NOTE ABOUT CENTS INDICATIONS
optional cents indications may be placed above or below the respective accidentals and are always
understood in reference to Equal Tempered semitones, as implied by the Pythagorean accidentals
15
1.6 Analytic applications of the tuneable intervals
Before moving to a discussion about the realization of tuneable intervals on acoustic brass
instruments, we conclude this section by mentioning several properties of the intervals which
merit further investigation.
It was noted above that given any two pitches within 23-limit harmonic space, it is possible to
determine a finite tuneable path between them using least generative tuneable intervals of
the form (p/2n). This is easily demonstrated as follows. The two pitches are products of
primes up to 23. Thus, the interval between them is also a product of primes up to 23. Each
such prime may be written as a product of a generative tuneable interval and 2n/1. These
products may be combined into a new representation of the interval, in which factors of the
form 2n/1 are reduced to (2/1)m. This result is the desired tuneable path.
Since we are moving by discrete steps in harmonic space, there must also be a shortest
tuneable path. For example, all directly tuneable intervals have a shortest tuneable path of 1.
The interval 9/8 has a shortest tuneable path of 2 since (9/8) = (9/4)·(1/2). The set of 23-limit
intervals with a shortest tuneable path ≥ 2 may be defined as the set of intervals indirectly
tuneable by construction. Evaluating the shortest tuneable path and grouping intervals based
on this value indicates one way of developing the concept of harmonic distance.
It would be conceivable to use this concept to develop a generalized model of tonality and
harmonic regions in the sense Schoenberg proposes in his “Structural Functions of
Harmony”9. Given an arbitrary 23-limit pitch P0, consider the set of pitches within tuneable
path 1 and call it the tuneable harmonic region {THR(P0)}. The set of all chords generated by
these pitches may be parsed into classes of similar structures (inverted, transposed) built on
different degrees. The process of modulation might be well defined as a movement to a new
tuneable harmonic region.
Another significant application of tuneable intervals is the principle of tuneable melodic steps.
A melodic step is defined as being tuneable if it is the difference between two tuneable
intervals. In musical terms, if a pitch is sustained in one voice while a second voice moves
from one tuneable interval to another, both pitches may be precisely tuned by definition and
thus the melodic step between them may be accurately tuned. By evaluating the set of all
tuneable melodic steps, it has been determined that the largest tuneable melodic step is 8/1
(3600¢) and the smallest is 144/143 (12.1¢). Each such step may have several realizations,
which represent different musical possibilities. For example, the melodic interval 9/8 can be
tuned in 19 distinct ways as the difference between two tuneable intervals.
Finally, these intervals can serve as a basis for analyzing more complex tuneable structures
consisting of more than two pitches. In the case of three voices, a triad (P1:P2:P3) always has
the three component intervals P1:P2, P2:P3 and P1:P3. It is possible to define a set of triads in
which each component interval is tuneable and to examine the resulting pairwise consonant
chords.
9
Arnold Schoenberg: “Structural Functions of Harmony” (Second edition, revised, 1954, reprinted by
Faber and Faber, London, 1983), pp. 19–20
16
1.7 Array representation of tuneable intervals (overtones and undertones)
For the purposes of the following discussion, it is useful to examine an array representation
of the tuneable intervals (the element (i,j) is simply i/j in lowest terms):
Table 4
3
4
5
6
7
8
1 1/1 1/2
1
1/3
1/4
1/5
1/6
1/7
1/8
2 2/1 1/1
2/3
1/2
2/5
1/3
2/7
3 3/1 3/2
1/1
3/4
3/5
1/2
4 4/1 2/1
4/3
1/1
4/5
2/3
5 5/1 5/2
5/3
5/4
1/1
6 6/1 3/1
2/1
3/2
6/5
7 7/1 7/2
7/3
7/4
8 8/1 4/1
8/3
9
3/1
10
11
12
13
14
15
16
17
18
2
9
10
11
12
13
1/4
2/9
1/5
2/11
1/6
2/13
3/7
3/8
1/3
3/10
3/11
1/4
3/13 3/14 1/5
4/7
1/2
4/9
2/5
4/11
1/3
4/13
5/6
5/7
5/8
5/9
1/2
5/11
5/12
1/1
6/7
3/4
2/3
3/5
6/11
1/2
6/13
3/7
7/5
7/6
1/1
7/8
7/9
7/10
7/11
7/12
7/13
1/2
2/1
8/5
4/3
8/7
1/1
4/5
8/11
2/3
8/13
4/7 8/15
9/4
9/5
3/2
9/7
9/11
3/4
9/13 9/14 3/5
5/1 10/3 5/2
2/1
5/3 10/7 5/4
9/2
6/1
4/1
3/1 12/5 2/1 12/7 3/2
4/3
7/1 14/3 7/2 14/5 7/3
15/2 5/1 15/4 3/1
2/1
5/2
7/4 14/9
15/8 5/3
8/1 16/3 4/1 16/5 8/3
2/1
1/7 2/15
2/7 4/15
5/13 5/14 1/3
2/5
10/13 5/7
2/3
1/1
6/7
4/5
1/1
6/5
6/1
3/1
20
21
22
23
24
25
3/16 3/17 1/6 3/19 3/20
1/4
1/7 3/22 3/23
7/6
3/2
5/4
16/11
27
28
1/5
4/21 2/11 4/23
5/16 5/17 5/18 5/19
1/4
5/21 5/22 5/23 5/24 1/5
3/8
6/17 1/3 6/19 3/10
2/7 3/11 6/23
7/17 7/18
1/3 7/22
1/2
7/20
4/9 8/19
2/5
1/2
5/8
1/2
2/3
3/5
5/26
5/28
1/4 6/25 3/13
2/9 3/14
7/24
4/11 8/23
3/7
5/9
1/6 4/25 2/13 4/27 1/7
7/27 1/4
1/3 8/25 4/13 8/27 2/7
3/8
5/11
1/3 9/28
5/12 2/5
5/13
5/14
1/2
3/4
4/7 6/11 12/23 1/2
6/13
4/9
3/7
1/2
1/1
7/8
1/1
4/3
26
1/8
4/17 2/9 4/19
1/1
9/5
3/2
2/1
5/3
8/7
7/9
7/10
5/6
3/4
1/1
2/3 7/11
7/12
5/7
4/5
5/8
7/13
3/5
1/2
5/9
8/11
2/3
8/13
6/7 9/11
3/4
9/13
4/7
9/7
6/5
1/1
22/3 11/2 22/5 11/3 22/7 11/4
23/3 23/4 23/5 23/6
11/5
23/8
6/1 24/5 4/1 24/7 3/1
2/3 9/14
1/1
7/3
23
2/1
10/7 4/3
7/4
3/2
11/6
11/7
5/4
7/5
1/1
7/6
11/8
5/6
1/1
11/9
8/3
12/5
25/4 5/1 25/6
25/8
5/2
26
13/2 26/5 13/3
13/4
13/5
27
27/4
9/2 27/7 27/8 3/1
7/2 28/9 14/5
12/7 8/5
3/2
4/3
5/3
2/1
13/7
9/4
7/3
5/7
7/9
3/4
1/1
2/1
13/6
4/5 10/13
7/8
1/1
23/12
25
7/1 28/5 14/3 4/1
19
1/8
19/8
22
8/1
2/1
4/1 10/3 20/7 5/2
7/1 21/4 21/5 7/2
18
1/1
9/2 18/5 3/1 18/7 9/4
20/3 5/1
17
11/16
7/5
8/5
16
17/3 17/4 17/5 17/6 17/7
19/3 19/4 19/5 19/6
28
15
5/6
13/2 13/3 13/4 13/5 13/6 13/7 13/8 13/9 13/10
20
24
1/1
11/2 11/3 11/4 11/5 11/6 11/7 11/8 11/9
19
21
1/1
14
8/7
1/1
5/4
13/8
13/9
7/4
14/9
9/5
2/1
6/5
6/7
1/1
13/10
3/2
1/1
9/7
7/5
4/3
1/1
7/6
1/1
The ratios in this representation may be regarded as a set of pitches in relation to 1/1. Note
that each column is a set of overtones (a harmonic series) and each row is a set of
undertones (a subharmonic series). The interlocking structure of overtones and undertones
is fundamental to understanding how acoustic instruments generate subsets of the total
harmonic space and how these subsets may be traversed by perceptible harmonic intervals.
String instruments, for example, produce overtones (harmonics) on each string at nodal
points dividing the string length into integer parts. A string vibrating at frequency F has
partials with frequencies {F,2F,3F,...}. The nth partial (with frequency nF) divides a string of
unit length into n parts, generating (n - 1) possible10 nodal points with associated string
lengths {1/n,2/n,...,(n-1)/n}. Because string length is inversely proportional to frequency 11, the
possible nodal points for the nth partial have frequencies {nF,(n/2)F,...,(n/(n-1))F}. This is a
10
11
th
The string length ratio (k/n) must be in lowest terms to produce the n partial at that node.
Here it is assumed that all other parameters (tension, thickness, etc.) remain constant.
17
set of undertones (subharmonics) of the nth partial. Thus, the array of pitches consisting of
possible nodal points for each partial is a set of interlocking overtones and undertones
generated by the string fundamental frequency F. (Consider the ratios below the diagonal in
the array above.)
18
2. MICROTONALITY ON VALVED BRASS INSTRUMENTS
2.1 The combined range of the horn, trombone and tuba
The combined range of the horn, trombone and tuba is 5 octaves, descending from F5 to F0
(704 Hz and 22 Hz respectively assuming A4 = 440 Hz). As the very lowest pitches of the
tuba lie at the border between pitch and noise, and since the timbre of the very highest horn
and trombone notes becomes increasingly thin, the practical range in terms of tuneability
extends over four-and-a-half octaves from D5 down to A0.
2.2 The purpose of the valve
Before the introduction of the valve in the early 19th century, all of the pitches available on
natural horns were derived from overtones of a single length of tubing. Various lengths of
tubing known as crooks were used to transpose the instrument into different keys, and each
length also gave a distinctive colour to the sound. The natural pitches of the overtone series
could be altered by hand-stopping, a technique which inevitably had a strong influence on
the timbre of each note. Valves were invented in order to make natural horns fully chromatic
and enable a homogenous sound throughout all pitches.
A valved horn may also be regarded as a collection of natural horns, each with its own
overtone row. By tuning the valve-slides in whole number proportions to one another, these
overtone rows become harmonically related and a network of natural intervals emerges.
Pitches differing by very small melodic steps in this microtonal gamut may be played on
widely varying lengths of tubing, creating alternations of tone-colour reminiscent of natural
horn technique.
2.3 Utonal valve tuning
If the second valve is tuned to add 1/15 of the open horn length, the resulting interval when
the valve is depressed is the semitone 15:16. Tuning the first valve to 2/15 of the open horn
length results in the interval 15:17. Combining these valves creates the minor third 15:18. If
the third valve is tuned to add 4/15 of the open horn length the resulting interval is 15:19.
Combining the second and third valves produces 15:20; combining the first and third valves
produces 15:21, and a combination of all three valves (1+2+3) results in 15:22.
Valves
2
1
1+2
3
2+3
1+3
1+2+3
Proportion added
1/15
2/15
3/15
4/15
5/15 = 1/3
6/15 = 2/5
7/15
Ratio to open horn
15:16
15:17
15:18 = 5:6
15:19
15:20 = 3:4
15:21 = 5:7
15:22
The valve combinations thus form an undertone row descending from 15 through 22; in the
just intonation terminology proposed by Harry Partch it is an utonal tuning of the valves.
19
In the 1970s the German music theorist Martin Vogel proposed such a tuning method for
valved brass instruments 12. In his system the second valve may be tuned to 1/14, 1/15, 1/16,
or 1/18 of the overall tube length. The instruments are equipped with a trigger system
enabling the valve-slides to be simultaneously adjusted during performance whilst
maintaining their proportion to one another – (1/n:2/n:4/n) where n = 14,15,16 or 18. Note
that the set of all valve-combinations tuned in this ‘binary’ fashion is {1/n,2/n,...7/n}. Adjusting
between the four different tuning systems makes extremely virtuosic demands on the player.
The slide action is mechanically intricate, and there is no possibility of easily changing the
instrument back to a conventional valve tuning.
2.4 Conventional tuning of the valves
The first three valves of a brass instrument lower the pitch by a tone, semitone and a minor
third. Combinations of the three valves create a descending chromatic scale over a
diminished 5th: for example, F – E – Eb – D – Db – C – B.
Let H be the length of the open horn, and V1 and V2 be the lengths of tubing added by two of
the valves. Then, since H < (H + V2),
1/H > 1/(H + V2)
V1/H > V1/(H + V2)
1 + (V1/H) > 1 + (V1/(H + V2))
(H/H) + (V1/H) > (H + V2)/(H + V2) + (V1/(H + V2))
and thus (H + V1)/H > (H + V2 + V1)/(H + V2).
Therefore, each valve lowers the pitch by a larger interval when used on its own than when
used in combination with another valve. In the descending scale above, the semitone
between F and E is larger than that from C to B, though both steps are brought about by
depressing the second valve.
Each of the three valves may be tuned individually so as to lower the pitch by a tempered
tone, semitone, and minor third, but their combinations will not coincide with tempered
tuning. Also, since brass instruments are played by sounding the overtone series, many of
the available pitches will deviate from Equal Temperament even when the fundamental is
tuned to a tempered pitch.
2.5 Rational approximation of the tempered tuning
The closest approximation to Equal Temperament on the horn is attained by the following
valve tuning. Note that the four traditionally employed combinations (1, 2, 1+2, 2+3) produce
tempered fundamentals within a tolerance of 5 cents.
Valves
1
2
1+2
3
2+3
1+3
1+2+3
12
Proportion added
1/8 = 10/80
1/16 = 5/80
3/16 = 15/80
1/5 = 16/80
21/80
13/40 = 26/80
31/80
Ratio to open horn
8:9
16:17
16:19
5:6
101:80
53:40
111:80
Lowers by (cents)
203.9¢
105¢
297.5¢
315.6¢
403.5¢
487.2¢
567.0¢
Martin Vogel: “On the Relations of Tone” (Orpheus Verlag, Bonn, 1993), pp. 373–385
20
The tuba may be tuned similarly. Fundamentals near tempered pitches are indicated in
boldface.
Table 5
Valves
Proportion added
Ratio to open horn
Lowers by (cents)
2
1/16 = 15/240
17/16
105
6
1/12 = 20/240
13/12
138.6
1
1/8 = 30/240
9/8
203.9
2+6
7/48 = 35/240
55/48
235.7
5
1/6 = 40/240
7/6
266.9
1+2
3/16 = 45/240
19/16
297.5
3
1/5 = 48/240
6/5
315.6
1+6
5/24 = 50/240
29/24
327.6
2+5
11/48 = 55/240
59/48
357.2
5+6
1/4 = 60/240
5/4
386.3
2+3
21/80 = 63/240
101/80
403.5
1+2+6
13/48 = 65/240
61/48
414.9
3+6
17/60 = 68/240
77/60
431.9
1+5
7/24 = 70/240
31/24
443.1
2+5+6
5/16 = 75/240
21/16
470.8
1+3
13/40 = 78/240
53/40
487.2
4
1/3 = 80/240
4/3
498
2+3+6
83/240 = 83/240
323/240
514.2
1+2+5
17/48 = 85/240
65/48
524.9
3+5
11/30 = 88/240
41/30
540.8
1+5+6
3/8 = 90/240
11/8
551.3
1+2+3
31/80 = 93/240
111/80
567
2+4
19/48 = 95/240
67/48
577.4
1+3+6
49/120 = 98/240
169/120
592.8
4+6
5/12 = 100/240
17/12
603
2+3+5
103/240 = 103/240
343/240
618.2
1+2+5+6
7/16 = 105/240
23/16
628.3
3+5+6
9/20 = 108/240
29/20
643.3
1+4
11/24 = 110/240
35/24
653.2
1+2+3+6
113/240 = 113/240
353/240
668
2+4+6
23/48 = 115/240
71/48
677.7
1+3+5
59/120 = 118/240
179/120
692.3
4+5
1/2 = 120/240
3/2
702
2+3+5+6
41/80 = 123/240
121/80
716.3
1+2+4
25/48 = 125/240
73/48
725.8
3+4
8/15 = 128/240
23/15
740
1+4+6
13/24 = 130/240
37/24
749.4
1+2+3+5
133/240 = 133/240
373/240
763.4
2+4+5
9/16 = 135/240
25/16
772.6
1+3+5+6
23/40 = 138/240
63/40
786.4
4+5+6
7/12 = 140/240
19/12
795.6
2+3+4
143/240 = 143/240
383/240
809.2
1+2+4+6
29/48 = 145/240
77/48
818.2
3+4+6
37/60 = 148/240
97/60
831.6
1+4+5
5/8 = 150/240
13/8
840.5
21
1+2+3+5+6
51/80 = 153/240
131/80
853.8
2+4+5+6
31/48 = 155/240
79/48
862.6
1+3+4
79/120 = 158/240
199/120
875.7
2+3+4+6
163/240 = 163/240
403/240
897.3
1+2+4+5
11/16 = 165/240
27/16
905.9
3+4+5
7/10 = 168/240
17/10
918.6
1+4+5+6
17/24 = 170/240
41/24
927.1
1+2+3+4
173/240 = 173/240
413/240
939.7
1+3+4+6
89/120 = 178/240
209/120
960.6
2+3+4+5
61/80 = 183/240
141/80
981.1
1+2+4+5+6
37/48 = 185/240
85/48
989.3
3+4+5+6
47/60 = 188/240
107/60
1001.5
1+2+3+4+6
193/240 = 193/240
433/240
1021.6
1+3+4+5
33/40 = 198/240
73/40
1041.5
2+3+4+5+6
203/240 = 203/240
443/240
1061.1
1+2+3+4+5
71/80 = 213/240
151/80
1099.8
1+3+4+5+6
109/120 = 218/240
229/120
1118.8
1+2+3+4+5+6 233/240 = 233/240
473/240
1174.6
The complete microtonal pitch set available in this tuning consists of the harmonic series
over these fundamentals. In the following example, commonly played tuba fingerings for the
chromatic scale are compared with their theoretical intonation (see note below about the
notation of fingerings). Alternate fingerings for the four notes which deviate from tempered
tuning are marked below the standard fingerings. Since the common denominator used is
240, this tuning is referred to as the 240-utonal tuba tuning.
Table 6
240-Utonal Tuba: Chromatic Scale With Conventional Fingerings
Tempered F


0
0


kk
kk
k
k
+2

-2 hk
hk
h
h
hh
kh
h
h
0
+4
+4



hk
hk
k
h
kk
hh
h
h
+4



 





-12 kh
kh
k
k
-18 kh
kk
k
k
-19 kk
kh
k
k
-14 kk
kk
k
k
+4
-2 kh
kk
k
k
-3 kk
kh
k
k
-2 kk
kh
h
k
kh
kh
k
k
kk
kk
k
k
 
-1 kk
hk
k
h
kk
-1 hk
k
h

+4
kk
kk
k
h
kk
kk
k
h

kk
-2 kh
h
k

kh
kh
k
k
kk
-3 hk
k
h
-2 kh
kk
k
k

+2


-2 hk
kk
k
h



+2
-3 kk
kh
k
k


kk
kk
k
k
-4 kk
kh
h
k


kk
kk
k
k
-4 kk
kh
h
k
+2
kh
kh
k
k
kk
-4 kh
h
k
kk
kk
k
h
+2





kh
kh
k
k
-4 kh
kk
k
k
-5 kk
kh
k
k
+2

+2
kh
kh
k
k


kh
-4 kk
k
k


-4 kh
kk
k
k
-5 kk
kh
k
k
kk
-5 kh
k
k

0

0
kk
kk
k
k
kk
kk
k
k
22
The notation of fingerings is adapted from woodwind notation. For the tuba: the column of
four circles indicates valves 1 – 4, operated by the right hand, and the column of two circles
valves 5 and 6, operated by the left hand. For the horn: the column of three circles indicates
valves 1 – 3, and the single circle the thumb valve which switches between the Bb and F
horn. All horn valves are operated by the right hand. Note that these fingerings are written for
tubas on which the fifth and sixth valves are a large tone and semitone respectively and for
horns on which Bb-side is played with the thumb valve depressed. For instruments with
these valves reversed, fingerings must be adapted accordingly.
2.6 Utonal horn tunings
Tuning the valve slides according to an undertone row incorporates the valve combinations
and overtone rows as integral parts of an expanded microtonal tuning system. Of the four
semitone tunings mentioned earlier (14:15, 15:16, 16:17, and 18:19), only 15:16 and 16:17
may be arrived at by adjusting the valve-slides of a standard horn, without making physical
alterations to the instrument itself.
The two resulting sets of undertones are notated below with valve fingerings using the
Extended Helmholtz-Ellis JI Pitch Notation.
Table 7
Double Horn 15 and 16: Fundamental Pitches
 f15

+12
Bb Horn
(15)
hk
k
k
 15
 o
+14
F Horn
(15)
kk
k
k
 f16

+12
Bb Horn
(16)
hk
k
k
 16
 o
+14
F Horn
(16)
kk
k
k
0

16
n
hk
h
k
17

¨u
18

n
hh
-5 k
k
hh
-4 h
k
17
18
+2

n
kk
h
k
¨

17
hk
h
k
+9
¨
19

Æv
hk
k
h
+4
16
+7
+2

¨u
kh
-3 k
k
+8

18
f
hh
k
k
+10
17

kk
h
k

n
kh
-2 h
k
+14
19

Æo
hh
h
k
+16
18

f
kh
k
k

Æo
kh
h
k

hk
h
h
+16

Æv
kk
k
h
+25
20
o
19
20

g
hk
k
h
+27
19
+14
+29
21
§

®
hh
k
h
hh
-51 h
h
21
22
+31
20

o
kk
h
h
+41

22
21

§o
hk
h
h
§


®
kh
k
h
kh
-49 h
h
22
23

®o
-40 hh
k
h

±f
-17 hh
h
h
+43
20

g
kk
k
h
21

§o
kk
h
h
22

®o
-38 kh
k
h
23

±f
-15 kh
h
h
23
In our work to date as composers, we have prioritized the 1/15 over the 1/16 tuning because
it offers more musically useful pitches when combining brass instruments with strings (note
that the Bb and F raised by a comma have D and A as fifth partials, matching open strings of
a violin, viola, cello or bass). Rather than using a trigger system for retuning the slides while
playing, we apply this single fixed utonal tuning to a modern double horn in Bb and F.
However, we do incorporate Vogel’s proposal to make custom longer slides for the third
valve (4/15 instead of 3/15 of the open horn lengths), eliminating the synonym 1+2 = 3.
The 1/15 tuning is called the 45-utonal horn tuning, because all resulting undertones on both
sides of the instrument can be represented as part of a single series:
Bb-horn
45:48:51:54:57:60:63:66
F-horn
60:64:68:72:76:80:84:88
The following table shows the available pitches on a double horn tuned in this manner,
organized as sixteen harmonic series over the sixteen undertone fundamentals produced by
the set of all valve combinations.
Table 8 (overleaf)
24
Double Horn in Bb and F / Bb-Horn : Tuning in Just Intonation
valves 1, 2, 3* (custom-built) are tuned to rational proportions (2/15, 1/15, 4/15) of the respective open horn’s length producing, in various combinations,
two Utonal Series of fundamental pitches with wavelengths in the proportions 15 : 16 : 17 : 18 : 19 : 20 : 21 : 22
with the two horns tuned a just perfect fourth (3:4) apart
* Alternately, the normal 3rd valve may be used, tuned to the proportion 3/15, so that the valve combination 1+2 and 3 are synonymous.
In this case, the undertone tuning “22” will not be available, and the fingerings indicated for undertones 19 - 22 apply instead to 18 - 21
!
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notated AT SOUNDING PITCH
using the Extended Helmholtz-Ellis JI Pitch Notation
accidentals designed by Marc Sabat and Wolfgang von Schweinitz, 2004
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Double Horn in Bb and F / F-Horn : Tuning in Just Intonation
valves 1, 2, 3* (custom-built) are tuned to rational proportions (2/15, 1/15, 4/15) of the respective open horn’s length producing, in various combinations,
two Utonal Series of fundamental pitches with wavelengths in the proportions 15 : 16 : 17 : 18 : 19 : 20 : 21 : 22
with the two horns tuned a just perfect fourth (3:4) apart
notated AT SOUNDING PITCH
using the Extended Helmholtz-Ellis JI Pitch Notation
accidentals designed by Marc Sabat and Wolfgang von Schweinitz, 2004
* Alternately, the normal 3rd valve may be used, tuned to the proportion 3/15, so that the valve combination 1+2 and 3 are synonymous.
In this case, the undertone tuning “22” will not be available, and the fingerings indicated for undertones 19 - 22 apply instead to 18 - 21
15
16
17
18
19
20
21
22
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25
The preceding table indicates pitches based on a theoretical model. To evaluate the
accuracy of this model, sensors were constructed for each valve and the horn and tuba were
connected to a computer. A MAX/MSP program generated exact overtone rows over each
theoretical fundamental using sinewaves. The instrumental sounds were ring-modulated with
these sinewaves to test the accuracy of the valve combinations. It was possible to determine
valve-slide positions which allowed the desired pitches to be produced within a very
acceptable tolerance. As expected, naturally occurring inharmonicities demand that the
players sometimes make fine corrections with their lips. However, once measured, the valveslide positions can be set before playing and the results remain consistent once the
instruments have been warmed up.
2.7 Possible extensions of the horn tuning
In order to further extend the pitch-gamut of the horn, players may employ half-valve and
hand-stopping techniques. To gain even more useful pitches it is necessary to make
alterations to the physical structure of the horn.
Many horns come equipped with a stopping valve, designed to lower the pitch by a semitone.
By altering this to produce the quartertone ratio 30:31, the set of available fundamentals
becomes 30:31:32:33:34:35:36:37:38:39:40:41:42:43:44:45. Notably, this includes the 13limit undertone 39, and would link the horn to more of the higher-prime tuba fundamentals
discussed below.
2.8 Utonal tuba tunings
In contrast to the horn, tuba designs are anything but standardized. Bass tubas may be
pitched in F or Eb, contrabass tubas in C or Bb. Instruments may have between 3 and 6
valves, and a variety of compensating systems are available, designed to adjust the valve
combinations closer to tempered tuning. For the purposes of this study the 6-valved F tuba
has been chosen, as it is the most closely related to the horn and offers more microtonal
possibilities than tubas with less valves.
The length of the F tuba is identical to that of the F horn, and the first three valves produce
the same intervals. The fourth valve lowers the fundamental by a perfect fourth to C. The fifth
and sixth valves are intended as tone and semitone valves in proportion to the length of the
tuba with the fourth valve depressed.
This 6-valve system may also be tuned utonally. Following the example of the horn, the first
three valves may be tuned to produce 15:16, 15:17, and 15:18, bringing about an undertone
series from 15 to 21. The fourth valve may be regarded as transposing the F tuba into a C
tuba, in effect creating a double tuba in F and C. The fifth and sixth valves act as the first and
second valves of this C tuba, and may be tuned to produce the intervals 15:16 and 15:17.
26
Valves
0 = F tuba
1
2
1+2 [= 3]
Proportion added
none
2/15 = 6/45
1/15 = 3/45
3/15 = 9/45
Ratio to open horn
15:15
15:17
15:16
15:18
4 = C tuba
5
6
5+6 [= 2+3]
1/3 = 15/45
(2/15)(4/3) = 8/45
(1/15)(4/3) = 4/45
(3/15)(4/3) = 12/45
(3:4)(15:15)
(3:4)(15:17)
(3:4)(15:16)
(3:4)(15:18)
Since 2+3 is synonymous with 5+6, this combination acts a “third valve” for the C tuba.
Combinations of 2+3 with the fifth and sixth valves thus extend its undertone row down to 21.
Table 9
Double Tuba in F and C: Fundamental Pitches
 15
 o
+14
+2

n
kk
kk
k
k
 15
 o
+16
+4
16
kk
kh
k
k

17
¨u
kh
-3 kk
k
k
+4
kk
kk
k
h

18
n
-2 kh kk
kh kk
k h
k k
0
16

n
kk
hk
k
h
17

¨u
hk
-1 kk
k
h
+16
19

Æv
kk
kh
h
k
+6
18

n
hk kk
hk kh
k h
h h
+31
20

o
kh
kk
h
k
+18
19

Æv
kk
hh
h
h
21

§
+33
20

o
hk
kh
h
h
kh
kh
h
k
21

§v
hk
hh
h
h
Calculating all 64 possible valve combinations results in the following table of pitches, with 40
distinct fundamentals descending one octave from F1 to F0, tuned as undertones between
45 and 90, called the 45-utonal tuba tuning.
Table 10 (overleaf)
27
6-Valve F Tuba Tuning in Just Intonation
valves 1, 2, 3, 4, 5, 6 tuned to rational proportions (6/45, 3/45, 9/45, 15/45, 8/45, 4/45) of the open tuba’s length producing, in various combinations,
a Utonal Series of fundamental pitches with wavelengths in the proportions 45 : 48 : 49 : 51 : 52 : 53 : 54 : 55 : 56 : 57 : 58 : 59 : 60 :
61 : 62 : 63 : 64 : 65 : 66 : 67 : 68 : 69 : 70 : 71 : 72 : 73 : 74 : 75 : 76 : 77 : 78 : 79 : 80 : 81 : 82 : 83 : 84 : 86 : 87 : 90
45
48
49
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
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30
This table effectively multiplies the 15-utonal tuning by 3, dividing each semitone step into
three unequal parts. An analogous procedure might be considered with higher multiples of
15:16. If the second valve is considered to be producing the undertone relationship 60:64,
the sixth valve may be tuned to 60:65 and the fifth valve 60:70. The resulting undertone row
would now descend from 60 to 119, dividing the semitones into four unequal parts (60-utonal
tuba tuning). For example, 16:17 may be divided into three parts 48:49:50:51 and into four
parts 64:65:66:67:68. Because no valve adds a unit proportion (1/45 or 1/60 of the length),
both tunings have gaps in their undertone rows. In the 45-utonal table the missing
undertones are 46, 47, 50, 85, 88, 89. The 60-utonal table is missing 61, 62, 63, 66, 67, 71,
108, 112, 113, 116, 117, 118.
Complete undertone rows may be generated by valves in the proportions 1:2:4:8:16:32. This
‘binary’ tuning could be realised by shortening the sixth and fifth valves to 1/60 and 2/60 of
the open tube, leaving the second and first valves tuned to 4/60 and 8/60, and extending the
third and fourth valves to 16/60 and 32/60:
Valve
1
2
3
4
5
6
Proportion added
2/15 = 8/60
1/15 = 4/60
4/15 = 16/60
8/15 = 32/60
1/30 = 2/60
1/60
Ratio to open horn
60:68
60:64
60:76
60:92
60:62
60:61
This would complete the 60-utonal tuning, giving 64 unique undertone fundamentals
descending from 60 through 123. The addition of a trigger attached to the main slide could
further extend the binary principle to include 1/120 and 1/240 of the open tube. This utopian
microtonal tuba would include as subsets all tunings discussed above (240, 45, 60).
2.9 Comparison of the horn, tuba and trombone
In the case of the double horn, two independent instruments are combined in one: a Bb-horn
with three valves and an F-horn with three valves. A thumb trigger valve redirects the airflow
from one side of the instrument to the other. There are therefore six separate valve slides,
operated by three valves. In this way, any valve combination on the F-side of the horn is
always tuned a perfect fourth below the same valve combination on the Bb-side.
The tenorbass trombone consists of two instruments as well, the tenor trombone in Bb and
the bass trombone in F (tuned to the same fundamental pitches as the two sides of the open
horn). A slide may be extended continuously from 0 cm to around 60 cm, adding between 0
and 120 cm to the open tube. This allows the Bb-side to descend in pitch by approximately
600¢. Switching between the two sides is accomplished similarly to the horn, with a thumb
trigger valve. In this case, however, the valve adds a fixed length of tubing, rather than
redirecting the airflow to an independent system. Thus, the interval between the two sides
decreases as the slide position increases.
In the case of the tuba, a single length of tubing tuned to F (the same fundamental as the
horn and trombone F-sides) is extended with combinations of six valves. The first three are
identical to the horn, and the fourth valve is tuned to lower the pitch by a perfect fourth (1/3 of
the open tube length, producing a 3:4 ratio). The fourth valve on the tuba, when used to
alternate between otherwise identical valve combinations, functions identically to the
trombone trigger valve, lowering by decreasing amounts as the length of tubing increases.
31
2.10 Slide division of the tenorbass trombone
Traditionally, trombone players learn seven positions on the Bb-side of their instrument,
corresponding to the tempered semitones descending from Bb1 down to E1. On the F-side,
there are six positions, ranging from a somewhat low F1 descending down to C1. The
chromatic fundamental B0 is not present. The trigger valve-slide is tuned to an interval
between a fourth and tritone, allowing the slide13 to reach down to C1 at full extension. The
trigger valve-slide has a great degree of flexibility: in the first position it may be tuned as
narrow as 1/3 of the Bb side, producing a 3:4 relationship between the two sides (498¢). It
may be pulled out to 3/8 of the Bb side, producing an 8:11 relationship (551.3¢), and in some
cases to as much as 2/5 of the Bb side, producing a 5:7 relationship (582.5¢).
Each of these rational possibilities suggests an undertone division of the slide into
equidistant steps. Note that a slide extension of n centimeters extends the overall tube length
by 2n. Thus, an extension of 48 cm lowers the Bb comma up on the Bb side by a perfect
fourth to F comma up (from 288 cm to 384 cm). If the distance between Bb and F is divided
into n integer parts, a descending utonal series {3n,3n+1,..,4n} is obtained. In the case of a
trigger valve-slide tuned to 1/3 of the Bb side, the corresponding points on the F-side will
then be tuned to {4n,4n+1,...,5n} in the same utonal series. Depending on the choice of n,
the series on both sides may be extended as far as the slide allows. Since its maximum
extension is 60 cm, and the step size is (48/n), the maximum number of steps is the greatest
integer less than or equal to (5n/4).
To maximize congruence with the available fundamentals of the horn and tuba, a division of
the 48 cm length into 45 parts (steps of 1,1 cm) is used. This produces steps ranging from
135:136 (12.8¢) at the top of the Bb-side to 235:236 (7.4¢) at the bottom of the F-side.
Roughly described, it divides each conventional position into nine to ten parts. The following
horn and tuba fundamentals may be played by the trombone at unison or one octave higher:
Bb-Horn:
(unison) 45, 48, 51, 54, 57, 60, 63, 66
F-Horn:
(unison) 60, 64, 68, 72, 76
(80, 84 and 88 are below the range of the trombone’s F-side)
Tuba:
(unison) 45, 48, 49, 51, 52, 53, 54, 55, 56, 57, 58, 59
(60 through 67 are below the range of the trombone’s F-side)
(octave higher) 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 86, 87, 90
Of course, it is clear that trombonists can technically reach any fundamental within the
available range (C through Bb). Nevertheless, equal division of the slide gives a conceptual
framework for combining the trombone with the utonal tunings of the horn and tuba. The
following table indicates the first twelve steps of the trombone 45-division. Pitches with boxed
text above indicate corresponding horn and tuba undertones. Bracketed pitches are
enharmonic simplifications (within a 5-cent tolerance range) of frequency ratios including
primes above 13.
Table 11 (overleaf)
13
In this section, there are two uses of the word “slide”. The trigger valve-slide is set to a fixed interval
and is used to alternate between the Bb and F sides of the tenorbass trombone. The slide refers to the
continuously moveable tubing which allows for different fundamentals to be played on each side of the
instrument.
32
Undertone Slide Divisions of the Tenorbass Trombone
Trigger Valve tuned to a perfect Fourth 3:4
Bb-side descends 135 -> 191 / F-side descends 180 -> 236
H 45
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Marc Sabat, August 2006
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33
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34
2.11 Two musical examples
Wonderful Scatter (2005) is a composition by Marc Sabat for tuba in the 45-utonal tuning,
computer and loudspeaker (see score example 1 below). Sensors are installed under each of
the six valves, continually measuring their positions. Movements produced by the tubist’s
fingers (and thus information about the valve-slide lengths being used at any given time) are
transmitted in real-time across an ethernet cable to the computer. The sensor mechanism
was developed in Berlin by Sukandar Kartadinata.
The tubist plays all available pitches in the first four octaves of the instrument’s range,
organized in an ascending microtonal gamut. The lengths of notes are inversely proportional
to the respective tube lengths (a tube length of 45 is played for one pulse at 45 bpm and a
tube length of 90 is played for one pulse at 90 bpm). Notes are separated in time by
durations proportional to the melodic distance between them (measured in cents) so that
each semitone contains one minute of non-playing pause. After the second, third and fourth
octaves, the player repeats the lowest octave (pedal tones).
A MAX/MSP patch analyzes the valve combinations. Corresponding to the 40 possible tubelengths there are 40 loops, and each loop is allocated a duration proportional to its respective
fundamental wavelength. Using information from the sensors, each new note is routed to its
corresponding loop. Over the course of the performance, the computer acts as a memorysieve, sorting the pitches played into 40 different overtone-series loops, which are activated
one-by-one as the tubist changes fingerings. Depending on the lengths of the loops and of
the notes played, the resulting textures range from a spectral chord to a chance-determined
overtone melody.
A subwoofer and corresponding loudspeaker amplify the output of the computer program,
which eliminates higher frequencies (above 352 Hz), producing a soft, round sound
resembling sinewaves. As the music progresses, each new note gradually merges with
harmonically related tones and the movement from tone to tone is transformed from a
relationship based on melodic distance to various possible relationships of harmonic
distance. The title is a literal English translation taken from the Latin Requiem Mass text
Tuba mirum, spargens sonum – War-trumpet wonderful, scatter sound.
35
Score Example 1
Wonderful Scatter
for Robin Hayward
Marc Sabat
The first 2 pages are to be repeated after each new octave (indicated by signs in the score). Time between notes is measured in quarter notes at 100 bpm, following the rule 100 cents = 1 minute.
Black noteheads are to be recorded as quarter-notes at the indicated tempo (exact bpm may be somewhat exaggerated to delineate the tempo relationships — sempre poco rubato).
White noteheads indicate a connected (quasi legato) sequence of quarter-notes, one for each of the notated synonymous valve-combinations.
!
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q = 100
q = 80
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q = 100
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q = 79
q = 100
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q = 75
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q = 62
28 q
#!
#!
"!
"#
q = 81
1/81
n
1/62
q = 100
1/57
\v
#
41 q
21 q
22 q
!!
!#
"#
"!
25 q
!!
!!
"#
"#
"
!!
#!
"!
"!
1/73
q = 100
1/60
o
#
#!
#!
q = 100 "!
"!
1/63
+15.6
9
q = 100
1/77
!!
##
"#
"! q = 100
"
5
26 q
##
!!
"#
"!
q = 100
27 q
q = 100
1/66
1/64
+3.9
q = 78
#!
!#
"#
"! q = 66
q = 100
q = 100
"
1/70
1/67
26 q
"
-7.6
#!
!!
"!
"!
1/74
#!
##
"!
"!
#$
20 q
1/78
+61.4
q = 74
"
!!
!#
"! q = 82
"!
-25.2
22 q
#!
!#
q = 100 "!
"!
1/75
24 q
!#
!#
"#
"!
q = 86
1/86
{m}
21 q
!!
##
"!
"!
1/71
##
#!
"!
"!
"
+53.9
1/79
-60.6
##
!!
"! q = 100
"!
q = 72
!!
!#
"!
"#
"
q = 100
1/82
{¥9}u
21 q
{¥0}$
22 q
!#
#!
"!
"!
q = 100
1/83
!#
!!
"!
"!
1/76
+6.4
q = 83
1/84
1/80
! 1/72
!
j#
!
!
"
o
>v
!
!
!
!
q = 87
1/87
+33.1
!
!
!
!
q = 100
!!
!!
"!
"!
!
!
!
!
q = 90
1/90
36 q
!
(on the final repeat, after 112 beats release all fingers to produce the fingering "45" without playing a note; slowly fade out loop)
36
$
$ (when repeating the pedal tones, play until the cut sign and then return to the respective higher octave)
!
q = 90
q = 45
q = 100
q = 87
2/90 1/45
+13.7
#
o
!
!"
59 q
##
##
"#
"#
!!
!!
"!
"!
-27.6
q = 84
q = 100
q = 80
"
!
q = 76
"
{¥0}$
22 q
!#
#!
"!
"!
-60.6
q = 75
2/76
\v
+6.4
! 2/72
j#
!!
!#
"!
"#
"
23 q
+29.3
q = 100
!#
!#
"#
"!
#
.>
24 q
##
#!
"!
"!
+24.2
#
%5
23 q
#!
##
"!
"!
!#
!!
"#
"!
q = 100
#
{5<}
#!
!!
"!
"#
#
n
!#
#!
"#
"!
q = 70
>o
+48.8
q = 100
q = 100
+31.2
##
!#
"#
"!
q = 60
#
o
#!
##
"!
"#
q = 90
q = 57
!#
!!
"#
"#
##
##
"#
"!
\v
#
##
#!
"!
"#
q = 100
"
o
22 q
!
+8.4
!#
#!
"!
"!
q = 76
q = 100
"
q = 72
q = 100
2/48 3/72
#
##
#!
"#
"#
!!
!#
"!
"#
##
#!
"!
"!
!#
!#
"#
"!
24 q
q = 56
#
q = 66
#
!!
!!
"#
"#
#!
#!
"#
"!
q = 100
q = 79
"
{¥0}$
11 q-58.7
q = 75
26 q
#!
##
"#
"!
+77.1
!#
!#
"!
"#
#
q = 71
#
##
!#
"!
"!
q = 100
q = 68
2/68
;u
25 q
!# ##
!! ##
"! "!
"# "!
q = 65
"
#!
!#
"!
"!
#!
##
"!
"!
!#
!!
"#
"!
+55.8
q = 100
"
9f
#
µ4
&o
28 q
q = 83
"
q = 100
q = 100
25 q
q = 55
5o
10 q
%5
q = 52
q = 100
"
!!
##
"#
"!
12 q
"
#!
!#
"#
"!
#
#!
!!
"#
"#
q = 100
q = 100
q = 100
#
#!
#!
"!
"!
+66.3
!!
!#
"#
"!
q = 49
"
##
!#
"#
"#
#$
20 q
-5.7
"
{m}
"
10 q
#
q = 77
5>v
-14.3
q = 100
q = 73
#
!!
#!
"#
"!
q = 100
3/73
12 q
#
{¥m}
#!
!!
"#
"!
-21.9
q = 69
25 q
3/69
±n
-24.4
#
#!
#!
"#
"!
##
!#
"!
"!
24 q
q = 100
!#
!!
"!
"#
##
##
"!
"!
q = 81
3/81
-2.0 #!
#!
"#
"#
q = 100
3/77
22 q
q = 54
2/54
n
21 q
#!
!!
"!
"!
-23.2
q = 100
!!
#!
"!
"!
q = 100
q = 100
2/49
¶$
q = 86
3/86
!!
!#
"!
"!
q = 82
29 q
3/82
11 q
q = 78
##
!#
"!
"#
q = 100
3/70
>o
3/87
2/52 3/78
##
!!
"#
"#
q = 74
"
-33.7 #!
!#
"#
"#
9
22 q
q = 70
q = 87
2/55
+63.4
-45.5
q = 58
2/58
"
##
!!
"!
"#
-13.0
!#
##
"!
"#
3/83
!!
##
"!
"!
+50.7
!#
##
"#
"!
27 q
q = 61
!"
{¥9}u
23 q
#
!!
##
"!
"#
-1.0
2/61
3/74
q = 100
!#
#!
"#
"!
24 q
#!
!!
"#
"!
q = 100
-25.7
q = 100
3/71
!!
#!
"!
"#
23 q
!#
##
"!
"!
!#
#!
"!
"#
30 q
3/75
!!
!!
"!
"#
#
!!
#!
"#
"!
2/65
q = 100
21 q
!#
!!
"!
"!
3/79
!#
##
"#
"#
.>
q = 100
q = 53 q = 100
g
+26.2
q = 84
"
!!
##
"#
"#
2/56 3/84
2/53
¬'
23 q
##
!!
"!
"!
n
+30.4
+31.3
3/76
\v
q = 48
11 q
#
±e
!!
#!
"#
"#
ƒv
!#
#!
"#
"#
q = 80 q = 100
#
q = 69
-26.3
+58.9
-55.3
>v
3/80
q = 100
{¥m}
q = 100
##
#!
"#
"!
q = 59
22 q
!#
!#
"!
"!
2/69
28 q
#!
#!
"!
"#
+35.1
+19.6
q = 73
2/59
31 q
!#
!#
"#
"#
5>v
-16.2
q = 62
q = 100
q = 100
q = 100
2/62
29 q
!!
!!
"!
"!
q = 77
22 q
!!
!#
"#
"!
-23.9
q = 100
3/90
2/57
!
+2.0
#
>
!!
!#
"#
"#
2/60
+4.4
-49.4
q = 63
#
n
-3.9 !!
!!
"#
"!
2/73
2/63
+15.6
#
#!
#!
"!
"!
25 q
5
26 q
##
!!
"#
"!
q = 100
2/77
24 q
"
#!
!#
"#
"!
q = 81
21 q
2/66
27 q
#!
!#
"!
"#
q = 100
2/70
2/64
+3.9
9
q = 100
"
!!
##
"#
"!
-47.4
2/67
!
!
q = 74
25 q
!!
#!
"!
"#
q = 67
q = 100
q = 64
q = 78
+61.4
#!
!#
"!
"!
41 q
2/81
"
2/74
!!
!!
"!
"#
q = 71
-75.4
!
g
-7.6
#!
!!
"!
"!
-25.2
22 q
2/71
26 q
!
q = 100
"
"
!!
#!
"!
"!
q = 100
2/78
2/75
##
!!
"!
"!
q = 72
q = 100
q = 100
#$
20 q
{m}
21 q
!!
##
"!
"!
2/79
o
q = 100
"
+53.9
q = 79
2/80
+17.6
q = 86
2/86
2/82
{¥9}u
21 q
!#
!!
"!
"!
q = 100
q = 100
"
!!
!#
"!
"! q = 82
2/83
>v
+33.1
!
q = 83
2/84
!
q = 100
2/87
25 q
q = 51
##
##
"!
"#
!!
!!
"#
"!
!#
!#
"!
"!
q = 100
2/51
!#
##
"!
"!
11 q
"
;u
-3.0
#!
##
"#
"#
11 q
37
Score Example 2
Monophones
for double horn and 6-valved F tuba*
q = 40
 

Horn**
  

 
+14
o
+14
o
p


    o
 

kk
k
k
+16
o
Tuba
 
p
9
 


o 

kh
k
h

 
 

 


o 

o


o



 o 
 
 
 
hh
k
k



    


 o 


+16
kk
kk
k
h
hk
hk
k
h
kk
kk
k
k


   n 
+14


   ¨u
+2
kk
k
k

-5

   n 
 


kk
kk
k
k
+16
o
kk
kh
k
k
o

-3
kk
h
k

  


Æv
n
  


+4



kh
kk
h
k

+2

  Æv 

+6
kk
hh
h
h
  
  
+4

n
kk
hk
k
h
+14
hk
h
h

hh
k
h

   o  



+16
o
kk
kk
k
h




  o


kh
kh
k
k
    n      Æv        o 


    



hk
h
k


kh
k
k
-2
kk
kh
h
k
29



+14
+14
o
n

-3
kk
h
h
+2
kh
k
h

 o   o
  

+14
hh
k
h

  


+16
+14


  
 
kh
h
k
     ¨u 
 

 
o 

o 
kh
k
h
kk
kk
k
k
+16

+16
kh
k
h
kk
kk
k
k
20
-2
¨u


+4
kh
kk
k
k
+16
+14
o


   Æv 


   n
kh
kk
h
k
+16
+14

  
kk
k
k
+16
kh
kk
h
k
Robin Hayward
+16
kh
kk
h
k
o
hh
k
h

   
+16
o
+14
+14
kk
k
k

kh
kk
h
k
* microtonal accidentals from The Extended Helmholtz-Ellis JI Pitch Notation, designed by Marc Sabat and Wolfgang von Schweinitz
** notated in F, sounding a 5th lower than written in both bass and treble clefs


38
 o        
 n        
 Æv   
  
  
  
   n 
 
37
+12
+2

  
 
kk
h
k
+4
Æv
+2
-4
hk
k
k

 §v 
+33

kk
kh
h
k
Æv 
  
+6

hk
hh
h
h
 o   o  o   
   
+14
+14
hh
k
h
+14
hh
k
h
hh
k
h
o

hk
hk
k
h
54
    o  o 

 

+18
+18
kh
k
h
-3
+4
kk
hk
k
h
60
 
 
+4
 o
 

+18
kh
kk
h
k
kk
hk
k
h
£f
-16
 
  
£f 
  
 
 
hh
k
h
-17
kk
kk
k
k
¨£u
-34
kh
kk
h
k

+16
£f
-17


kk
kk
k
k
+16

hh
k
h
  
   
o


  
hh
k
h
kk
h
k
+16
o
+16
n
 


 

+2
   
 

£n
-29
 

   
kk
h
k
£n

-31
hk
hk
k
h

+4
 
  



   
   
hk
h
h



kk
kh
k
k


+4
kk
k
k
Æv

hk
h
k


   
    
kk
kk
k
h
kh
kk
k
k
  o
 
o


+4
+16

hk
kh
h
h
kk
kk
k
h

kh
k
k
    o

 
67
hh
k
k
kk
hk
k
h
  ¨u
-1
kh
kh
h
k
+16
-1
hk
kk
k
h
o
  

 ¨u         o        
o
 
  


kh
k
h

kk
h
h
+18
kh
k
h
+4
 §n
 


+6
kh
h
k
+31
kk
hk
k
h

     n  n  n 
¨u
   Æv 
   

kk
k
k
+4
hk
hk
k
h


hk
kk
k
h
o

        n   
   
 
hk
hk
k
h
¨v 
  

+16
kk
k
k
hk
h
h
+20
kk
hh
h
h
+14
45
hh
h
k

£f

-16
kk
kk
k
h




 
 £f




-16
kk
kk
k
h
39
Monophones (2006) is a composition by Robin Hayward for horn in the 45-utonal tuning (with
normal third valve-slides tuned to 3/15 of the open horn) and 6-valve F tuba, treated as a
double tuba in F and C (see score example 2 above). Pitches are assigned to the horn and
tuba based on tetrachords arising from the perfect fourth relationship between the two
instruments.
Durations are assigned to the pitches according to which instrument they are played by, their
overtone number, and which chart (Bb, F or C) they occur in.
Table 12
Overtone:
Bb Horn
F Horn
F Tuba
C Tuba
1
h.
w
2,3
e.( e.( e.
h.
q.
h
4,5,6,7
x.( x.( x.
q.
e.
q
8+
y.( y.( y.
e.
x.
e
The intervals between horn and tuba are measured according to Euler’s gradus suavitatis
function outlined in section 1.2 above. Intervals with a consonance grade value ≤ 10 are
distinguished either by repetition or silence.
For example, the first interval in the score, 3/1, which has a consonance grade of 3, is played
3 times; the interval at the start of the second system, 16/3, having a consonance grade of 7,
is played 7 times. If the number of repetitions exceeds the duration assigned to the second
note of the interval, as in bars 45 - 46, the duration takes priority and the number of
repetitions is curtailed. If the second note is shorter than the first note, the interval is followed
by a silence lasting the difference between the two durations (see bars 62 - 65).
40
3. TUNEABLE INTERVALS BETWEEN THE TUBA AND HORN
3.1 Calculation and analysis of the available intervals
All possible intervals between the tuba and horn 45-utonal tunings (tables 5 and 6), have
been tabulated to find correspondences and near-correspondences with the set of tuneable
intervals. First, the tuba and horn pitches were calculated according to the following
conditions:
1.) It is assumed that a player can reach up to the 19th partial on any given length of tubing.
To prioritize the even-numbered partials when sorting the resulting lists, these overtones
are ordered {1,2,3,4,5,6,8,7,9,10,12,11,13,14,15,16,17,18,19}. (8 precedes 7 and 12
precedes 11.)
2.) The available undertone fundamentals resulting from all valve combinations are
determined as a set of integer proportions between 45 and 90 as discussed above.
3.) The set of overtones of each undertone are written as rational numbers
(overtone/undertone).
4.) The pitch of the open tube (with no valves depressed) is determined as a ratio in relation
to 1/1 (A4). The horn Bb fundamental is thus 2/15, and the tuba F fundamental is 1/10.
5.) Each available pitch is calculated by multiplying the (overtone/undertone) proportions
with the frequency ratio of the open tube fundamentals.
6.) The melodic distance from A4 is calculated for each pitch.
7.) The resulting pitches are sorted in rising order.
8.) Pitches outside the playable range of each instrument are eliminated.
In the case of the horn, most of the pedal tones are omitted (with the exception of the two
highest, Bb and A) as well as all pitches above F5 plus a comma. In the case of the tuba,
pitches above C5 plus a comma are omitted. Although the very high tuba pitches between F4
and C5 may be beyond the range of normal playing, they are included because of the
“double tuba” model discussed above, which is a perfect fourth below the horn. For an
experienced player the pitches are in fact playable, though the timbre in this extreme register
becomes increasingly thin and less typical of the normal tuba sound.
This results in 247 horn pitches ranging from A1 to F5, and 1088 tuba pitches from F0 to C5.
Many of these pitches are synonyms, which can be produced in several different overtoneundertone combinations and/or valve combinations. To simplify the process of calculating
tuneable intervals, the synonyms are eliminated. This leaves 178 distinct horn pitches and
563 distinct tuba pitches, allowing (178·563) = 100214 possible intervals.
An array TvH[i,j] is constructed, with rows for each tuba pitch Ti and columns for each horn
pitch Hj. This array is searched by a computer program in two different ways. The first
method calculates the ratios in lowest terms between tuba and horn, F(Hj)/F(Ti). Each ratio is
41
compared with the set of 183 tuneable interval ratios {TI}, and if there is a match it is
indicated in the array.
(6)
TvH[i,j] = F(Hj)/F(Ti) if F(Hj)/F(Ti) ∈ {TI} else TvH[i,j] = blank
In this case, the array contains 4782 tuneable intervals between tuba and horn.
The number of tuneable interval combinations is counted for each row and column of the
array. In the case of the horn, each pitch has at least 3 different corresponding tuba pitches.
In musical terms, every horn note can generate tuneable tuba melodies. Conversely,
however, 245 of the 563 tuba pitches have no corresponding horn notes, and 9 pitches have
only one.
This result suggests the need for a second calculation method, which looks for close
enharmonic approximations of tuneable intervals. Each interval between tuba and horn is
calculated as a melodic distance and is compared with the list of tuneable intervals
expressed as melodic distances. The results are compiled in five tables ranging from ± 0.5¢
maximum deviation up to ± 4.5¢ maximum deviation, at which tolerance all tuba pitches can
also be harmonized with two or more distinct horn pitches.
± 0.5¢
± 1.5¢
± 2.5¢
± 3.5¢
± 4.5¢
5334 tuneable intervals
8458 tuneable intervals
11860 tuneable intervals
16255 tuneable intervals
21016 tuneable intervals
These results are filtered into a single table using the following algorithm.
1.) Include all exact ratios.
2.) In a given row, if the number of horn pitches forming a tuneable interval is 0 or 1 (i.e., if
melodic movement is not possible in relation to a given tuba pitch), look for enharmonic
near-equivalents within a tolerance of ±0.5¢.
3.) If the row now has 2 or more possible horn pitches, indicate the maximum necessary
tolerance and proceed to the next row.
4.) If the number of possible horn pitches is still 0 or 1, proceed to the next tolerance range
of ±1.5¢.
5.) Repeat this process until each row has at least 2 possible horn pitches.
In the composite table, some rows contain only ratios, some contain combinations of cents
and ratios, and some contain only cents values. The intervals expressed as melodic
distances are enharmonic near-equivalents of tuneable intervals within an indicated
tolerance range. The resulting array may be ‘navigated’ by moving between rows and
columns, producing a mathematical model of tuneable interval counterpoint between tuba
and horn in which every tuba pitch and every horn pitch may be connected to other pitches
by a tuneable path.
Because of the size of this array, only a small portion of it is reproduced in this paper. It
exists as a searchable and sortable Excel spreadsheet, and may be obtained from the
authors or on the website www.plainsound.org. The table has 6728 possible intervals in total.
42
318 rows have at least 1 exact ratio; 69 rows have intervals within a tolerance of ±0.5¢; 155
rows have a tolerance of ±1.5¢; 17 rows have a tolerance of ±2.5¢; 2 rows have a tolerance
of ±3.5¢; 2 rows have a tolerance of ±4.5¢. Thus, 559 of 563 tuba pitches can be harmonized
with horn pitches falling within 2.5¢ of a tuneable interval, a highly acceptable tolerance for
acoustic instruments. An audible version of this table has been programmed in MAX/MSP,
producing available microtonal intervals between tuba and horn by clicking in a scrollable
array.
Table 13
-5186
v
v
v
v
Tuba v Horn ± up to 5¢
v
v
*
index
-3600
->
->
558
01|hh/hhhh
1 90
545
01|hh/hkhh
1 87
->
->
cents
->
1 20
-5186.31371
-5186
3 58
-5127.62219
-5128
526
01|hk/hhhh
1 86
9 172
-5107.6077
-5108
518
01|hh/khhh
1 84
3 56
-5066.87091
-5067
499
01|hk/hkhh
1 83
9 166
-5046.13732
-5046
480
01|kh/hhhh
1 82
9 164
-5025.1524
-5025
467
01|hh/hhkh
1 81
1 18
-5003.91
-5004
455
01|hk/khhh
1 80
9 160
-4982.40371
-4982
436
01|kh/hkhh
1 79
9 158
-4960.6269
-4961
424
01|kk/hhhh
1 78
3 52
-4938.57266
-4939
408
01|hk/hhkh
1 77
9 154
-4916.23385
-4916
394
01|kh/khhh
1 76
9 152
-4893.60301
-4894
378
01|hh/hhhk
1 75
3 50
-4870.67243
-4871
359
01|hk/hkkh
1 74
9 148
-4847.43404
-4847
340
01|kh/hhkh
1 73
9 146
-4823.87947
-4824
330
01|hh/hkhk
1 72
1 16
-4800
-4800
312
01|hk/hhhk
1 71
9 142
-4775.78654
-4776
299
01|kh/hkkh
1 70
9 140
-4751.22962
-4751
281
01|kk/hhkh
1 69
3 46
-4726.31935
-4726
268
01|hk/hkhk
1 68
9 136
-4701.04541
-4701
251
01|kh/hhhk
1 67
9 134
-4675.39703
-4675
237
01|hh/hhkk
1 66
3 44
-4649.36294
-4649
224
01|hk/khhk
1 65
9 130
-4622.93137
-4623
211
01|kh/hkhk
1 64
9 128
-4596.09
-4596
198
01|kk/hhhk
1 63
1 14
-4568.82591
-4569
182
01|hk/hhkk
1 62
9 124
-4541.12557
-4541
166
01|kh/khhk
1 61
9 122
-4512.9748
-4513
157
01|kk/hkhk
1 60
3 40
-4484.35871
-4484
142
01|hk/hkkk
1 59
9 118
-4455.26166
-4455
127
01|kh/hhkk
1 58
9 116
-4425.66719
-4426
112
01|kk/khhk
1 57
3 38
-4395.55802
-4396
101
01|hk/khkk
1 56
9 112
-4364.9159
-4365
88
01|kh/hkkk
1 55
9 110
77
01|kk/kkhk
1 54
1 12
-4333.72165
-4334
-4301.955
-4302
63
01|hk/kkkk
1 53
9 106
-4269.59454
-4270
51
01|kh/khkk
1 52
9 104
-4236.61766
-4237
38
01|kk/hkkk
1 51
3 34
-4203.00041
-4203
25
01|kh/kkkk
1 49
9 98
-4133.74181
-4134
13
01|kk/khkk
1 48
3 32
-4098.045
-4098
1
01|kk/kkkk
1 45
1 10
-3986.31371
-3986
546
02|hh/hkhh
2 87
3 29
-3927.62219
-3928
527
02|hk/hhhh
2 86
9 86
-3907.6077
-3908
519
02|hh/khhh
2 84
3 28
-3866.87091
-3867
500
02|hk/hkhh
2 83
9 83
-3846.13732
-3846
481
02|kh/hhhh
2 82
9 82
468
02|hh/hhkh
2 81
1 9
-3825.1524
-3825
-3803.91
-3804
456
02|hk/khhh
2 80
9 80
-3782.40371
-3782
437
02|kh/hkhh
2 79
9 79
-3760.6269
-3761
425
02|kk/hhhh
2 78
3 26
-3738.57266
-3739
409
02|hk/hhkh
2 77
9 77
-3716.23385
-3716
395
02|kh/khhh
2 76
9 76
-3693.60301
-3694
379
02|hh/hhhk
2 75
3 25
-3670.67243
-3671
360
02|hk/hkkh
2 74
9 74
-3647.43404
-3647
341
02|kh/hhkh
2 73
9 73
-3623.87947
-3624
331
02|hh/hkhk
2 72
1 8
-3600
-3600
313
02|hk/hhhk
2 71
9 71
-3575.78654
-3576
300
02|kh/hkkh
2 70
9 70
-3551.22962
-3551
282
02|kk/hhkh
2 69
3 23
-3526.31935
-3526
269
02|hk/hkhk
2 68
9 68
-3501.04541
-3501
252
02|kh/hhhk
2 67
9 67
-3475.39703
-3475
238
02|hh/hhkk
2 66
3 22
-3449.36294
-3449
225
02|hk/khhk
2 65
9 65
-3422.93137
-3423
212
02|kh/hkhk
2 64
9 64
-3396.09
-3396
199
02|kk/hhhk
2 63
1 7
-3368.82591
-3369
183
02|hk/hhkk
2 62
9 62
-3341.12557
-3341
167
02|kh/khhk
2 61
9 61
-3312.9748
-3313
158
02|kk/hkhk
2 60
3 20
-3284.35871
-3284
acc
o
≥u
{≥}v
§v
j
{m}
n
o
j
™n
®§v
Æv
g
{¨}®n
j
n
j
§o
±e
¨u
z
®n
™f
n
§n
µßn
{£}o
o
{©}v
≥u
Æv
§v
®o
n
{˝}o
™n
¨u
¶v
n
o
≥u
{≥}v
§v
j
{m}
n
o
j
™n
®§v
Æv
g
{¨}®n
j
n
j
§o
±e
¨u
z
®n
™f
n
§n
µßn
{£}o
o
note
F
F
F
F
G
G
G
G
G
G
G
G
A
A
A
A
A
A
B
A
A
B
B
B
B
B
C
C
C
C
C
C
D
D
D
D
D
D
E
F
F
F
F
G
G
G
G
G
G
G
G
A
A
A
A
A
A
B
A
A
B
B
B
B
B
C
C
± to ET
PB
midi
6728
v
acc
v
note
v
01|h/khk
01|h/kkk
02|k/hhh
02|k/hkh
02|k/khh
02|k/kkh
02|k/hhk
02|k/hkk
02|h/hhh
02|k/khk
02|h/hkh
02|h/khh
1
48
1
45
2
88
2
84
2
80
2
76
2
72
2
68
2
66
2
64
2
63
2
60
1
8
2
15
3
22
1
7
3
20
3
19
1
6
3
17
2
11
3
16
4
21
1
5
-3600
-3488.26871
-3449.36294
-3368.82591
-3284.35871
-3195.55802
-3101.955
-3003.00041
-2951.31794
-2898.045
-2870.78091
-2786.31371
-3600
-3488
-3449
-3369
-3284
-3196
-3102
-3003
-2951
-2898
-2871
-2786
n
f
®n
§n
o
Æv
n
¨u
®n
n
§n
A
B
B
B
C
C
D
D
E
E
E
o
F
± to ET
0
+12
-49
+31
+16
+4
-2
-3
-51
+2
+29
+14
v
PB
~B0,64
~B97,67
~B26,48
~B125,73
~B1,69
~B54,65
~B48,63
~B5,63
~B74,79
~B80,64
~B45,73
~B49,68
v
midi
33
34
35
35
36
37
38
39
39
40
40
41
v
6728
67
42
45
57
83
39
62
43
42
85
38
->
0
231
316
649
702
498
68
814
+14
~B49,68
17
14
ratio
5/2
8/3
x
20/7
3/1
x
10/3
x
x
15/4
x
4/1
-28
~B21,55
18
2
±4.5¢
x
1639¢
x
x
x
x
x
x
x
x
x
2341¢
0
-8
~B72,61
18
4
±1.5¢
x
x
x
x
x
x
x
2105¢
x
x
x
x
+33
~B77,74
18
14
ratio
7/3
x
x
8/3
14/5
x
28/9
x
x
7/2
x
x
-46
~B30,49
19
3
±2.5¢
x
x
x
x
x
x
x
2043¢
x
x
x
x
-25
~B122,55
19
8
±2.5¢
x
1537¢
x
1656¢
x
1830¢
x
x
x
x
x
x
-4
~B96,62
19
15
ratio
9/4
12/5
x
18/7
x
x
3/1
x
x
27/8
24/7
18/5
+18
~B81,69
19
7
ratio
x
x
x
x
8/3
x
x
x
x
10/3
x
x
+39
~B77,76
19
6
±4.5¢
1361¢
x
x
x
x
x
x
x
x
x
x
x
+61
~B84,51
20
6
x
x
x
x
13/4
x
x
-16
~B103,58
20
7
+6
~B6,66
20
9
+29
~B49,73
20
9
-47
~B105,48
21
-24
~B46,56
21
2
~B0,64
21
15
+24
~B96,71
21
10
+49
~B78,79
21
7
-26
~B74,55
22
6
-1
0
270
315
3
ratio
13/6
x
x
x
13/5
ratio
x
x
7/3
x
x
x
x
x
28/9
x
x
ratio
x
x
x
x
x
8/3
x
x
x
19/6
x
x
ratio
x
x
x
x
5/2
x
x
x
x
25/8
x
10/3
x
x
x
x
x
1652¢
x
x
x
x
x
±2.5¢
x
x
x
x
x
x
x
x
x
x
x
x
ratio
2/1
x
x
x
12/5
x
8/3
x
x
3/1
x
16/5
±3.5¢
x
x
x
1407¢
x
x
x
x
x
x
1905¢
x
ratio
x
x
x
x
7/3
x
x
x
x
x
x
28/9
ratio
±2.5¢
386
435
x
x
23/12
x
x
x
x
x
x
x
x
23/8
x
x
~B85,63
22
9
ratio
x
x
x
x
x
x
x
8/3
x
17/6
x
x
+25
~B112,71
22
3
±1.5¢
x
x
x
x
x
x
x
x
x
x
x
x
-49
~B26,48
23
18
ratio
11/6
x
2/1
x
11/5
x
x
x
8/3
11/4
x
x
+77
~B85,56
23
4
ratio
x
x
x
x
13/6
x
x
x
x
x
x
x
+4
~B32,65
23
6
ratio
x
x
x
x
x
x
x
x
x
8/3
x
x
+31
~B125,73
23
19
617
ratio
7/4
x
x
2/1
x
x
7/3
x
x
x
8/3
14/5
+59
3
645
±2.5¢
x
x
x
x
x
x
x
1538¢
x
x
x
±2.5¢
x
x
x
x
x
x
x
x
x
x
x
x
ratio
5/3
x
x
x
2/1
x
x
x
x
5/2
x
8/3
11
±2.5¢
x
967¢
x
1086¢
x
x
x
x
x
x
1584¢
x
2
±2.5¢
x
x
x
x
x
x
x
x
x
x
x
x
15
ratio
x
x
x
x
x
2/1
x
x
x
19/8
x
x
ratio
14/9
x
x
x
x
x
x
x
x
7/3
x
x
x
5/3
x
11/6
x
x
x
x
x
x
x
~B107,50
24
-13
~B109,59
24
2
+16
~B1,69
24
17
-55
~B40,78
24
-26
~B101,55
25
+4
~B54,65
25
+35
~B29,75
25
11
-34
~B27,53
26
12
-2
~B48,63
26
25
+30
~B93,73
26
+63
~B36,52
702
x
ratio
x
ratio
3/2
8/5
x
12/7
9/5
x
2/1
x
x
9/4
x
12/5
4
±1.5¢
x
781¢
x
x
x
x
x
x
x
x
x
x
27
7
ratio
13/9
x
x
x
x
x
x
x
x
13/6
x
x
ratio
x
x
x
x
x
2/1
x
x
x
x
884
~B5,63
27
15
x
x
+66
~B26,53
28
5
1052
ratio
x
x
x
14/9
x
x
x
x
x
x
x
+2
~B80,64
28
17
1088
ratio
4/3
x
16/11
x
8/5
x
x
x
x
2/1
x
x
+14
~B49,68
29
27
1200
ratio
5/4
4/3
x
10/7
3/2
x
5/3
x
x
15/8
x
2/1
-28
-3
~B21,55
30
3
±2.5¢
x
x
x
x
x
x
x
x
x
x
x
-8
~B72,61
30
5
±1.5¢
x
x
x
x
x
x
x
x
x
x
x
x
+33
~B77,74
30
23
ratio
7/6
x
x
4/3
7/5
x
14/9
x
x
7/4
x
x
-46
~B30,49
31
2
1340
±1.5¢
x
x
x
x
x
x
x
x
x
x
x
-25
~B122,55
31
3
1361
±1.5¢
x
x
x
x
x
x
x
x
x
x
x
x
-4
~B96,62
31
25
ratio
x
6/5
x
9/7
x
x
3/2
x
x
x
12/7
9/5
+18
~B81,69
31
16
ratio
x
x
x
x
4/3
x
x
x
x
5/3
x
x
+39
~B77,76
31
3
±3.5¢
x
x
x
x
x
x
x
x
x
x
x
x
+61
~B84,51
32
10
-16
~B103,58
32
14
+6
~B6,66
32
16
+29
~B49,73
32
12
-47
~B105,48
33
4
-24
~B46,56
33
2
~B0,64
33
36
+24
~B96,71
33
4
+49
~B78,79
33
13
-26
~B74,55
34
6
-1
~B85,63
34
+25
~B112,71
-49
~B26,48
+77
~B85,56
+4
1404
x
x
x
ratio
x
x
x
x
13/10
x
13/9
x
x
13/8
x
x
ratio
x
x
7/6
11/9
x
x
x
x
14/9
x
x
x
ratio
x
x
x
x
x
4/3
x
x
x
x
x
x
1515
ratio
x
x
x
x
5/4
x
x
x
x
x
x
5/3
1539
±1.5¢
1470
x
x
x
x
x
x
x
x
x
±1.5¢
x
x
x
x
x
x
x
x
x
x
x
x
ratio
1/1
x
x
8/7
6/5
x
4/3
x
16/11
3/2
x
8/5
±1.5¢
x
x
x
x
x
x
x
x
x
x
x
x
ratio
x
x
x
x
7/6
x
x
x
x
x
x
14/9
ratio
x
x
x
x
x
x
x
x
x
x
x
x
15
ratio
x
x
x
x
x
x
x
4/3
x
x
x
x
34
5
±1.5¢
x
x
x
x
x
x
x
x
x
x
x
35
31
ratio
x
x
1/1
x
x
x
11/9
x
4/3
11/8
x
x
35
6
ratio
x
x
x
x
x
x
x
x
x
x
x
13/9
~B32,65
35
15
ratio
x
x
x
x
x
x
x
x
x
4/3
x
x
+31
~B125,73
35
33
ratio
7/8
x
x
1/1
x
x
7/6
x
x
x
4/3
7/5
+59
~B107,50
36
6
-13
~B109,59
36
8
+16
~B1,69
36
32
0
1586
1635
1817
x
x
x
x
x
x
x
x
x
x
x
x
x
x
±2.5¢
x
x
x
x
x
x
x
x
x
x
x
x
ratio
5/6
x
x
x
1/1
x
x
x
x
5/4
x
4/3
±2.5¢
1902
x
x
43
3.2 Notation of the available intervals
To investigate tuneable intervals with acoustic brass instruments, the various tables of
pitches are written in staff notation. All 4782 exact tuneable ratios between tuba and horn,
which were generated in the first array discussed above, have been converted to information
compatible with Sibelius music notation software. The required Helmholtz-Ellis accidentals,
the note names, the cents deviations, as well as MIDI data (note numbers and 14-bit pitch
bend values) may be automatically calculated by transferring the pitch frequency ratios into a
spreadsheet file14 conceived and programmed by the composer Stefan Bartling. The
resulting text information is copied into his Sibelius plug-in “heHELIX”, which translates the
data into musical notation. The valve combination fingerings are added similarly, using his
plug-ins “heHOFI” and “heTUFI”. All of these tools are available from the authors.
Two examples of the results are given below – all possible occurrences of the intervals 13/10
and 10/13 (the inverted form simply implies that the horn is below the tuba), and all examples
of the interval 11/1 (note that this is one of the consonances greater than 3 octaves
mentioned in section 1.4 above).
Table 14 (overleaf)
14
“he_WRITE.xls”
44
13/10
1
Horn
(in F)

2
+16


o
kk
h
h
Horn
(in C)
Tuba


+61
Hn.
(C)
Tba.


11
n
hk
h
k

kk
h
h


™m

™ßn
kh
kh
h
h
12

kh
kh
h
h

o
13
kh
h
h
14



15


Hn.
(F)
Hn.
(C)

hk
h
k


n


™u
+50
Tba.

v
+4
kk
hh
k
k

20
¨©U
-62

o
+16
kh
k
k

¨©u

¨T
-17
21
kh
kk
k
k
hk
k
k

o

™n
+61
kh
kh
h
h
22

©v
-58


u
-12
23
kk
kh
k
k

´g
kk
h
k
©v
-55
kk
h
h
kk
h
h

´g

´™f
+66
kh
kh
h
h
24


-9

©o
-46
kk
kk
k
k

©v
-61
kk
h
h

+30



u
-16
kh
kh
h
h

o
+20
26


™n
+65
kk
hh
k
k

≠f
+13
kk
h
h
o
kh
h
k
©n
™£n


18
£f
hk
h
h
n
kk
hh
k
k
£o
-16
©o
0
+65

17
kk
kh
h
k
25

™n
kh
kh
h
h
kk
k
h
Æu
o
™n
+61




Æ©v
kk
kh
k
k
+21
19
Æ©v
kk
h
h
o


16
n
kh
kh
h
k


o
hk
h
h
kk
hh
k
k

+2


+20
o
™u
©o
§u
kk
hh
k
k

©o
-44
kh
k
h

™n
hh
hh
k
k

+50
kh
kh
h
h
©§v
+17
n
™£n

-28
o
®u

©§v
hk
h
h
+63
£f
+30

kk
h
k


9
+16
v
-16 kk
h
h
kk
hh
k
k
o
+18
8
+4
£o
™n
+63
7


o
kh
kh
h
h

-63
6
+18
kk
h
h
™m
©®v
-109


+48
©®v
ßo
+113
n



n
+48

ßo
+67

kh
kh
h
h
kk
hh
k
k
5
hk
h
k
™n

kh
kh
h
h
Hn.
(F)
+61
™n

+2
o
+63
™n
10


n
+2
hk
h
h
o

4
o
+16
kk
h
h
o

3
o
+18
kk
h
h

≠f

≠™e
+59
kh
kh
h
h
kk
kk
h
k
45
10/13
1
Horn
(in F)
Horn
(in C)
Tuba
2

0
Hn.
(C)
Tba.


u
n
hk
h
k

-18 hh
h
k

m
n





4
+4

kk
h
k


©p
©v
-46
-63
-42
-30
-59
hh
hh
k
h
9
+17

§u
kh
k
h
hk
h
k



-28
kh
kh
h
k
10

Æu
-9
kk
k
h

n
Æu
©o
Æ©v
§u
©§v

n
+2
hk
kh
h
h

-44
kh
kk
h
k

-55
kk
kh
h
k
§v
hh
hh
h
k
11

®v
-47
hh
h
h

-63
®u

-11
12

u
-16
kh
h
k

-109
kh
hk
k
h
13

¨T
-17

¨T
©®o
©n
¨©u
-93
kh
hk
k
k
-61
kk
kk
h
k
-62
hh
hh
k
k

kh
k
k
u


©®v
®v

kh
h
h


©§w
hh
hk
h
k

®u
kh
k
h

©o

§v
u
©n
hh
hh
h
h
+35
-14 hk
h
k
o

7



6
u
hk
h
h
n


o
v

5
+16
©o
8
Hn.
(F)
3

kh
kk
k
k

46
11/1
1
Horn
(in F)
2

Horn
(in C)

Tuba



+35
§v
kh
k
h
§v
Hn.
(C)

Tba.


n


¨ßu
+48
9
kh
k
k

¨ßu

ßn
+53
kk
h
k

ßn
+2

¨u
-3
kh
kk
k
k

ßo
+65
hk
h
h

n
kk
kh
k
k


+14

o
kk
kk
k
k

ßn
hh
h
k
ßn
ßf


o

ßn

®o
Æv
kh
kk
h
k
+47
+49
+4

11

ßn
kh
h
k


o
kh
kh
h
k
7
hk
h
h
Æßv
+16


o
+18
kk
k
h
ßo
§n
10

6
Æßv
+56


hh
hh
k
k
-49
5
kk
h
h
ߧn
®n
hh
kh
k
h

ßo
+67
kh
k
h


®§v
8


hk
h
k

4
ߧn
+82
+31
-16
Hn.
(F)

3
n
+2
-34
kk
kh
h
k
12

§v
+35
kh
hk
k
k
13


ßn
+51
kh
k
h
§v

n
-2
hk
h
k

ßn
kk
kk
h
k
14

n
+2
hk
h
k

n
0

n
hh
-4 hh
k
h

®§v
hh
-16 kh
k
h

n
hh
hk
h
k

®n
-49 hh
hh
k
k
47
3.3 Testing the tuneable intervals with tuba and horn
In order to test the full range between the tuba and horn, a book of music was prepared,
notating theoretically tuneable intervals between tuba and horn. It also includes some wider
tuneable intervals including 11/1 (shown above) and 13/1. A systematic assessment of larger
intervals is planned as part of our future work. To date, we have completed two working
sessions with Dianna Gaetjens playing horn and Robin Hayward playing tuba. Here are a
few general observations:
1.) The intervals empirically determined by earlier testing on string instruments are similarly
tuneable on brass instruments.
2.) The theoretical analysis of available intervals yields combinations which may be
accurately reproduced in practice. Unfamiliar valve combinations often required several
attempts to be tuned accurately. In general, the players managed to play the intervals
without having to substantially alter the pitches naturally occurring on their instruments.
3.) The very lowest tuba pedal tones are too indeterminate in pitch to produce tuneable
sounds.
4.) The lowest tuba pitch producing good pitch resolution is A0 (the lowest note of the
piano).
5.) Intervals with melodic distance less than 1200¢ are not tuneable in the very lowest
register. In the bass register (the lower of the two pitches above F2) these intervals
begin to demonstrate an audible periodic rumbling pattern which may be described as a
tuneable noise. In the tenor register (the lower of the two pitches above C3) they begin
to exhibit a clearly periodic composite sound, as was also found in tests with the same
intervals on cello. In the highest register, the intervals produce very strongly audible
combination tones reinforcing harmonics of their combined period. Once the horn
pitches rise above D5, they become spectrally weaker and do not tune as easily with the
tuba.
6.) In the low register, it is often easier to tune intervals when the horn is playing the lower
pitch, because the spectral balance of its tone favors higher partials (it produces a reedy
tone reminiscent of a bassoon). Conversely, the relatively strong fundamental of the tuba
aids production of difference tones when it takes the lower voice, and these are most
clearly evident in a higher register.
7.) As the intervals become melodically wider (> 2/1), the lower limits of tuneability also
descend in pitch.
8.) Many of the intervals wider than three octaves (> 8/1) are also tuneable and demand
further systematic consideration.
Based on these tests it is possible to conclude that the theoretical analysis of available
pitches produces results that are readily realizable by experienced musicians. These pitches
can therefore serve as a basis for microtonal composition using tuneable intervals.
48
3.4 Some final remarks about working with tuneable intervals on brass instruments
Tuneable intervals offer one possible conceptual framework for composing in Just Intonation,
allowing modulation between related harmonic and subharmonic series based on a set of
perceptible sounds. In light of this question of perceptibility, our analysis of the brass tunings
poses one difficulty: the pitches produced by the tuba’s valve combinations introduce
numerous fundamentals based on prime-number relationships higher than 23 (i.e., pitches
which may not be reached by tuneable interval steps).
In any music which is based on audible harmonic steps, the logical consequences would be
to either ignore these pitches as beings outside the limits of Fasslichkeit (Schoenberg’s term
for musical comprehensibility), to treat them as microtonal dissonances, or to consider
contextual redefinitions of these pitches based on an acoustically acceptable enharmonic
tolerance. Realizing subtle alterations of pitches is readily possible, since brass players are
accustomed to making slight adjustments to naturally occurring pitches with their lips, by finetuning the slide position in the case of the trombone or by moving their hand in and out of the
bell in the case of the horn.
Our current work includes developing tools for mapping limited subsets of 23-limit harmonic
space, which may be searched for enharmonic near-equivalents to the brass pitches falling
outside the perceptible harmonic space. In combination with the near-equivalents found in
the tuba-horn tuneable interval chart, it is possible to conceptualize various means to
harmonically employ the higher-prime fundamentals. It is hoped that these tools can serve as
a provisional basis for compositional investigations of the brass instruments’ microtonal
possibilities and how they may be combined with other acoustic instruments in order to make
music in Just Intonation.