A Method for Solmization of Melody
Yongwei Zhu
Institute for Infocomm
Research
[email protected]
Mohan Kankanhalli
National University of
Singapore
[email protected]
Abstract
This paper presents a novel method for automatic
solmization of melody, by which a melody (a sequence
of MIDI notes) can be transcribed to sol-fa syllables
(i.e. do re me fa sol la ti). Automatic solmization can
assist in music skill training, music notation and
content-based music retrieval.
The proposed method is based on an approach for
estimating the music scale of a melody. The key of the
major scale (“do”) is estimated using music scale
models. Due to the diversity of melody types, models
for both diatonic and pentatonic scales are employed
to avoid the possible key ambiguity for folk songs. The
decision of the key of a melody is based on the scale
estimation results for aggregating music notes, so that
the method can work for both short and long melodies.
Experiments have shown that the technique can
achieve 95% correct solmization of the melodies of
pop songs.
1. Introduction
Due to the advancement of digital technology,
digital music has become pervasive in human life,
which calls for innovative tools that can help people
effectively interact with the huge amount of music data
available today. An example is content-based music
retrieval, such as query-by-humming, which allows a
person to find a tune from a large database of music
[1]. Another example is automatic music notation, by
which a sequence of MIDI notes can be transformed to
a music notation that is appropriate for music reading
or learning.
Somization is a music notation system, where
musical notes are denoted using syllables. The
somization syllables that prevail Europe and many
other places in the modern world are the sol-fa
syllables, i.e. do re me fa sol la ti, [2,3,4]. This system
is widely adopted in music education (solfege and
aural training) and music notation for songs. For
instance, the Chinese cipher notation (Jian Pu or Dien
Pou) [5] that uses numbers from 1 to 7 to notate
0-7803-8603-5/04/$20.00 ©2004 IEEE.
Sheng Gao
Institute for Infocomm
Research
[email protected]
melodies of songs is directly derived from the sol-fa
syllables.
If a song or melody can be correctly and
ambiguously translated to the sol-fa syllables, it is then
possible to do music retrieval using the sol-fa syllables,
which could be more efficient and effective than by
using note pitch values. Furthermore, automatic
solmization of a melody could also assist a music
learner in the training of aural skills.
Although a music piece can be notated using staff
notation with any key, say C Major. However, an
arbitrary key could not reflect the structure of the
music, and could be awkward for practical music
performance. From our experience, the key signatures
encoded in MIDI files are often missing or incorrect.
We have not seen any existing method that can
automatically translate a melody into sol-fa syllables.
The sol-fa syllables are defined for diatonic scale,
where “do” corresponds to the first degree of the major
scale. Solmization of a melody is thus a problem of
finding the pitch that is associated with “do” in the
diatonic scale.
It has been shown in the work [6,7] that many songs
are composed using Major or Minor scales, such that
melody matching can be facilitated by estimation the
scale root. However, it does not guarantee a solution
for transcribing the melody into sol-fa syllables. The
diversity of melodies, such as scales types, usage of
accidental notes, and differences in length of songs,
pose challenges for automatic solmization of melody.
This paper presents a novel method to tackle this
problem.
This paper is organized as follows: section 2
presents relation of solmization and music scales, as
well a model of music scales; section 3 presents the
proposed method for solmization of melody based on
music scale estimation; section 4 presents the
experiments; and section 5 presents the conclusion.
2. Solmization and Scale Models
A music scale is a sequence of music notes within
an octave, from which a music piece is composed. In
the de facto tuning system, each octave has 12 equal
tempered notes. The diatonic scale, which dominated
western music for more than 500 years, is a scale with
7 notes (out of the 12 notes). Diatonic scale has 7
modes with each one using a different note as the
starting note. Major scale and minor scale are the 2
most widely used modes of diatonic scale. The diatonic
scale can be modeled as a wheel with 12 spokes, as
illustrated in figure 1. The darkened spokes correspond
to the notes in the diatonic scale (scale tones), and the
dimmed spokes correspond to the notes outside of the
scale or accidental notes. When forming a scale mode,
the major scale starts from note number 0, and the
minor scale starts from note number 9.
In the solmization system, the syllable “do”
corresponds to the first note (or degree) of the major
scale, and “la” corresponds to the first note of the
minor scale. In fact, all modes of the diatonic scale
share a same syllable for each note, as shown in figure
1.
Minor
11
10
Major
[ti]
[re]
7
[sol]
6
[mi]
10
11
Major
0
[do]
9
10
11
[ti]
9
1
[la]
0
[do]
1
[la]
8
[re]
7 [sol]
6
[mi]
5
4
3
2 8
2
7
[mi]
6
3
[fa]
4
5
(b) Hemitonic
Pentatonic Scale
Figure 2: Scale models for pentatonic scales
1
[la]
8
Minor
(a) Anhemitonic
Pentatonic Scale
0
[do]
9
diatonic scale. However, if using the pentatonic scale
to explain the melody, there is no ambiguity.
Besides major and minor pentatonic scale, there are
also other types of pentatonic scale, like the one shown
in figure 2(b). This pentatonic has semitone steps (thus
called hemitonic pentatonic scale), and is used in
traditional Japanese music. On the contrary, this type
of scale does not cause ambiguity with the diatonic
scale, where “do” in pentatonic scale only associates
with “do” in diatonic scale.
2
3
[fa]
4
5
Diatonic Scale
Figure 1: Scale model for diatonic scale
Thus as long as a melody is composed using
diatonic scale, the sol-fa syllables for any notes of a
melody can be solely determined, regardless which
(major or minor) mode is used. However, for certain
songs, especially folk songs, not all 7 notes in diatonic
scale are used, and as a result the song can be
associated to multiple modes of the diatonic scale,
which leads to ambiguity of sol-fa syllables for the
melody.
We have seen that such cases of ambiguity are
usually due to the use of pentatonic scale. The
pentatonic (scale with 5 notes) is an important type of
music scale beside diatonic scale in many eastern
Asian cultures [4]. The notes of typical pentatonic
scales are the subset of diatonic scale tones. Among
the various pentatonic scales, the major pentatonic and
minor pentatonic scales prevail over others. The model
for major and minor pentatonic scales is illustrated in
figure 2(a). It can be seen that major and minor
pentatonic scale is the diatonic scale with note number
5 and 11 omitted. That means syllable “fa” and “ti” do
not present in the songs of major and minor pentatonic
scales.
A song written in a major and minor pentatonic
scale, if explained using a diatonic scale, would have
ambiguity on the syllables: “do” in pentatonic scale
can be associated with “do”, “fa” or “sol” in the
Although most songs are composed with diatonic or
pentatonic scale, accidental notes do present in some
songs. From our observation and understanding, the
number of accidental notes in a song is usually small.
We also observed that short songs less likely have
accidental notes than long songs, while short songs are
more likely based on pentatonic scale.
Our proposed method for solmization of melody is
based on the use of diatonic scale and major/minor
pentatonic scale models. Possible accidental notes and
different length of melodies are considered.
3. Proposed Method for Solmization of
Melody
Solmization of a melody is equivalent to finding the
absolute note/pitch of the melody that corresponds to
syllable “do’. In the proposed approach, the note for
“do” is estimated by fitting the melody notes into the
scale models and then finding the best alignment of the
notes in the melody with the notes in the scale models.
The best alignment accounts for minimal number of
notes in the melody that reside outside of the scale
model. Since there are 12 notes in each octave, the
“do” could possibly correspond to any one of the 12
notes.
Section 3.1 presents estimation of “do” for a fixed
number of notes in a melody for one scale model;
section 3.2 presents how to decide “do” for melodies
with different length using both diatonic and
pentatonic scale models.
3.1. Melody Scale Estimation using One Scale
Model
The method for melody scale estimation using one
scale model takes 3 steps: (1) constructing note
histograms of the melody; (2) aligning the note
histogram to the scale model and computing the model
fitting error; and (3) deciding the right model
alignment that has minimal fitting error.
The note histogram is constructed by assigning the
notes into bins, where each bin corresponds to a note in
MIDI. Since there are 128 pitches for MIDI notes, each
note histogram has 128 bins.
The alignment of the note histogram with a scale
model is done by wrapping the bins around the scale
model, such that each bin will match to a spoke in the
model. It is obvious that notes that are just octaves
apart will be matched to a same spoke. Since a scale
model has 12 spokes, there are totally 12 possible
alignments. The model fitting error for a particular
alignment is computed by summing all the bins that are
matched to the dimmed spokes (accidental notes).
Table 1 shows the note histogram for the first 35 notes
for the melody of “Power of love”. The first row
contains the MIDI note number, and second row
contains number of notes in each bin. Table 2 shows
the diatonic scale model fitting errors of the 12
alignments. The first row holds the alignment numbers,
and the second row holds model fitting errors.
Table 1: Note histogram for “Power of love”
70
2
71
0
72
5
73
3
74
0
75
16
76
0
77
5
78
0
79
2
80
2
scale estimation is also inappropriate, since for some
melodies the key may be transposed in the middle, and
scale notes after the transposition could be treated as
accidental notes.
For the above issue, we propose to use aggregating
number of notes for scale estimation. And the final
result is based on the sequence of scale estimation
result for the 2 scale models. The method is described
as follows.
The melody is first separated into segments of equal
number of notes (e.g. 7 notes). The melody scale
estimation is then conducted for the melody by
aggregated number of segments. For example, the first
note histogram is constructed for the first segment, and
scale is estimated for the first note histogram. And the
second histogram is constructed for the first 2
segments, and scale is estimated. And so on.
In scale estimation, the scale model fitting error is
computed for each specific key with increasing number
segments. Thus for diatonic scale model, 12 fitting
error sequences are obtained. Similarly, 12 fitting error
sequences are obtained for pentatonic scale model.
Each model fitting error sequence monotonically
increases with the number of segments.
The final decision of scale is by choosing the
sequence that is consistently minimal for the diatonic
scale model. If more than one fitting error sequences
keep zero error, then the result of pentatonic scale
model is consulted, and the sequence with consistent
minimal value is then chosen.
4. Experiments and Discussion
Table 2: Model fitting errors for “Power …”
1
23
2
2
3
30
4
3
5
14
6
21
7
7
8
28
9
0
10
30
11
5
12
21
The alignment with minimal fitting error could be
chosen as the scale estimation result. In the above
example, alignment number 9 has a minimal error (0)
for diatonic scale. For this alignment, note 80 matches
with “do”.
3.2. Melody Solmization Using Diatonic and
Pentatonic Scale Models
As mentioned before, some melody based on
pentatonic scale may have ambiguity on diatonic scale.
So in our approach for melody solmization, both
diatonic and pentatonic scale models are employed.
The results of scale estimation from the 2 models are
combined to produce the final result.
It could be difficult to determine the best fixed
number of notes for scale estimation for all melodies,
since different melody progress in different ways. On
one hand, the predetermined number could be too
small, when many of the notes repeat a same pitch. On
the other hand, to use all the notes in a melody for
Figure 3: Scale model fitting errors for “Power
of love”
We have conducted experiments to evaluate the
proposed melody solmization method. In the
experiments 20 randomly chosen MIDI music songs,
including western and Asian songs, are used. The
solmization notes are first obtained manually from
musician experts or song books. The result of the
automatic solmization method is then evaluated with
the manual result.
Solmization of the song “Power of love” is given as
an example. Figure 3 shows the model fitting errors for
the 2 scale models. The fitting error sequence for each
specific key is plotted. It can be seen in figure 3(a) that
the sequence with consistent minimal value zero
should be taken as the correct key. Since there is only
one sequence chosen in diatonic scale, the results of
pentatonic can be ignored. However, it still can be seen
that the chosen sequence is also consistently minimal
in pentatonic scale in this case.
Figure 4 illustrates the solmization result for the
song “Power of love”.
pentatonic scale are all large. So the song should be
based on diatonic scale. However, neither of the 2
sequences is consistently minimal, thus lead to
ambiguity of 2 keys.
We found that the problem is due to the use of a
few accidental notes in the very beginning of the
melody.
For the cases of key ambiguity, the pitch range of
the melody could be further considered to pick the
appropriate key. In song books, usually, the lowest
pitch of the melody is not lower than “sol” in low
octave, and the highest note is not higher than “sol” in
the high octave. By using this rule, the song
“Yesterday” has a solely determined solmization,
which in fact is identical to the song book.
5. Conclusion
80
79
78
77
76
75
74
73
72
71
70
69
68
do
re
mi fa
sol
la
ti do
Figure 4: Solmization result for “Power of
love”
Figure 5 shows the scale model fitting results for a
Japanese song “Akatombo”. For diatonic scale model,
3 error sequences keep zero. While for pentatonic
scale, there is only 1 error sequence remaining zero,
which determines the right key of the melody.
This paper presented a novel method for
solmization of melody. This technique is based on the
approach for estimation of music scales of a melody.
The syllable “do” is associated with the first note of
major scale. Two types of scale (diatonic and
pentatonic) are employed to avoid possible scale
ambiguity. The scales are estimated by fitting the notes
into scale models, and finding the fitting error
sequence that is consistently minimal.
The experiments have shown the effectiveness of
the method, where 95% of the songs can be correctly
transcribed to sol-fa syllables.
10. References
[1] A. Ghias, J. Logan, and D. Chamberlin. “Query By
Humming”. Proceedings of ACM Multimedia 95, November
1995, pages 231-236.
[2] The Columbia Encyclopedia, Sixth Edition. Columbia
University Press 2003
[3] http://www.bartleby.com/65/gu/GuidodAr.html
[4] Karpinski, G.S., Lessons from the Past: Music Theory
Pdagogy and the Future, The Online Journal of the Society
for Music Theory, Volume 6, Number 3, August, 2000.
[5] http://www.wikipedia.org/wiki/Music_notation
[6] Pickens, J., "Key-specific Shrinkage Techniques for
Harmonic Models," Proceedings of ISMIR ’03 Conference,
Baltimore, MD, Oct. 26-30, 2003
Figure 5: Scale model fitting for “Akatombo”
In the experiments, 19 songs out of the 20 have the
correct scale root estimated. The song “Yesterday” has
2 fitting error sequences both with small values in
diatonic scale, and the respective error sequences in
[7] Zhu Y., Kankanhalli M. “Music Scale Modeling for
Melody Matching” Proc. Of ACM Multimedia 2003,
Berkeley, CA, Nov. 2-8, 2003.