H a s e g a w a
The Laws of Tonalia
Basic Theory on Scale and Harmony
− Ver.3 −
Kagemitsu Hasegawa
Music evolves, and continues to do so even now. So then, will the musical
evolution of the 21th century represent the creation of new, atonal musical
phraseologies, or a quest for microtone scales? This article is centered
around the existence of unexplored areas in the twelve-tone scale, in other
words the possibility of further sophistication and systematization of
composition techniques.
Well, the term “tonalia” in the title of this article refers to the range
of tonality. In exploring the structure of such an area, i.e. what scales
and harmonies exist, the realization of the following should be a prerequisite:
even if the scale and melody of a musical piece includes tonality, if an
atonal harmony is incorporated through composition or arrangement, the
work will become atonal. Similarly, even if the scale and melody of a musical
piece includes atonality, if a tonal harmony is incorporated through composition
or arrangement, the work will become tonal. From another perspective, it
can be said that scales and harmonies are equals.
◆Theory on Scale◆
1. Atonal scale
(1) Definition of atonal scale
A scale wherein an octave is formed by the repetitive use of a certain
tone row.
(2) Classification of atonal scale
The interval sequence in each scale / The formula showing the structure of the semitone number / Scales with the keynote of C
・Two-note scale／Diatonic scale
Augmented 4th, Augmented 4th／6×2／C, F#
・Three-note scale／Tritonic scale
Major 3rd, Major 3rd, Major 3rd／4×3／C, E, G#
・Four-note scale／Tetratonic scale
Minor 3rd, Minor 3rd, Minor 3rd, Minor 3rd／3×4／C, D#, F#, A
Minor 2nd, Perfect 4th, Minor 2nd, Perfect 4th／（1＋5）×2／C, C#, F#, G
Perfect 4th, Minor 2nd, Perfect 4th, Minor 2nd／（5＋1）×2／C, F, F#, B
Major 2nd, Major 3rd, Major 2nd, Major 3rd／（2＋4）×2／C, D, F#, G#
Major 3rd, Major 2nd, Major 3rd, Major 2nd／（4＋2）×2／C, E, F#, A#
・Six-note scale／Hexatonic scale
Major 2nd, Major 2nd, Major 2nd, Major 2nd, Major 2nd, Major 2nd／2×6／C, D, E, F#, G#, A#
Minor 2nd, Minor 3rd, Minor 2nd, Minor 3rd, Minor 2nd, Minor 3rd／（1＋3）×3／C, C#, E, F, G#, A
Minor 3rd, Minor 2nd, Minor 3rd, Minor 2nd, Minor 3rd, Minor 2nd／（3＋1）×3／C, D#, E, G, G#, B
Minor 2nd, Minor 2nd, Major 3rd, Minor 2nd, Minor 2nd, Major 3rd／（1＋1＋4）×2／C, C#, D, F#, G, G#
Minor 2nd, Major 3rd, Minor 2nd, Minor 2nd, Major 3rd, Minor 2nd／（1＋4＋1）×2／C, C#, F, F#, G, B
Major 3rd, Minor 2nd, Minor 2nd, Major 3rd, Minor 2nd, Minor 2nd／（4＋1＋1）×2／C, E, F, F#, A#, B
Minor 2nd, Major 2nd, Minor 3rd, Minor 2nd, Major 2nd, Minor 3rd／（1＋2＋3）×2／C, C#, D#, F#, G, A
Minor 2nd, Minor 3rd, Major 2nd, Minor 2nd, Minor 3rd, Major 2nd／（1＋3＋2）×2／C, C#, E, F#, G, A#
Major 2nd, Minor 2nd, Minor 3rd, Major 2nd, Minor 2nd, Minor 3rd／（2＋1＋3）×2／C, D, D#, F#, G#, A
Major 2nd, Minor 3rd, Minor 2nd, Major 2nd, Minor 3rd, Minor 2nd／（2＋3＋1）×2／C, D, F, F#, G#, B
Minor 3rd, Minor 2nd, Major 2nd, Minor 3rd, Minor 2nd, Major 2nd／（3＋1＋2）×2／C, D#, E, F#, A, A#
Minor 3rd, Major 2nd, Minor 2nd, Minor 3rd, Major 2nd, Minor 2nd／（3＋2＋1）×2／C, D#, F, F#, A, B
・Eight-note scale／Octotonic scale
Minor 2nd, Major 2nd, Minor 2nd, Major 2nd, Minor 2nd, Major 2nd, Minor 2nd, Major 2nd／（1＋2）×4／C, C#, D#, E, F#, G, A, A#
Major 2nd, Minor 2nd, Major 2nd, Minor 2nd, Major 2nd, Minor 2nd, Major 2nd, Minor 2nd／（2＋1）×4／C, D, D#, F, F#, G#, A, B
Minor 2nd, Minor 2nd, Major 2nd, Major 2nd, Minor 2nd, Minor 2nd, Major
2nd, Major 2nd／（1＋1＋2＋2）×2
／C, C#, D, E, F#, G, G#, A#
Minor 2nd, Major 2nd, Major 2nd, Minor 2nd, Minor 2nd, Major 2nd, Major 2nd, Minor 2nd／（1＋2＋2＋1）×2
／C, C#, D#, F, F#, G, A, B
Major 2nd, Minor 2nd, Minor 2nd, Major 2nd, Major 2nd, Minor 2nd, Minor
2nd, Major 2nd／（2＋1＋1＋2）×2
／C, D, D#, E, F#, G#, A, A#
Major 2nd, Major 2nd, Minor 2nd, Minor 2nd, Major 2nd, Major 2nd, Minor
2nd, Minor 2nd／（2＋2＋1＋1）×2
／C, D, E, F, F#, G#, A#, B
Minor 2nd, Minor 2nd, Minor 2nd, Minor 3rd, Minor 2nd, Minor 2nd, Minor 2nd, Minor 3rd／（1＋1＋1＋3）×2
／C, C#, D, D#, F#, G, G#, A
Minor 2nd, Minor 2nd, Minor 3rd, Minor 2nd, Minor 2nd, Minor 2nd, Minor
3rd, Minor 2nd／（1＋1＋3＋1）×2
／C, C#, D, F, F#, G, G#, B
Minor 2nd, Minor 3rd, Minor 2nd, Minor 2nd, Minor 2nd, Minor 3rd, Minor
2nd, Minor 2nd／（1＋3＋1＋1）×2
／C, C#, E, F, F#, G, A#, B
Minor 3rd, Minor 2nd, Minor 2nd, Minor 2nd, Minor 3rd, Minor 2nd, Minor
2nd, Minor 2nd／（3＋1＋1＋1）×2
／C, D#, E, F, F#, A, A#, B
・Nine-note scale／Nonatonic scale
Minor 2nd, Minor 2nd, Major 2nd, Minor 2nd, Minor 2nd, Major 2nd, Minor
2nd, Minor 2nd, Major 2nd／（1＋1＋2）×3
／C, C#, D, E, F, F#, G#, A, A#
Minor 2nd, Major 2nd, Minor 2nd, Minor 2nd, Major 2nd, Minor 2nd, Minor
2nd, Major 2nd, Minor 2nd／（1＋2＋1）×3
／C, C#, D#, E, F, G, G#, A, B
Major 2nd, Minor 2nd, Minor 2nd, Major 2nd, Minor 2nd, Minor 2nd, Major 2nd, Minor 2nd, Minor 2nd／（2＋1＋1）×3
／C, D, D#, E, F#, G, G#, A#, B
・Ten-note scale／Decatonic scale
Minor 2nd, Minor 2nd, Minor 2nd, Minor 2nd, Major 2nd, Minor 2nd, Minor 2nd, Minor 2nd, Minor 2nd, Major 2nd
／（1＋1＋1＋1＋2）×2／C, C#, D, D#, E, F#, G, G#, A, A#
Minor 2nd, Minor 2nd, Minor 2nd, Major 2nd, Minor 2nd, Minor 2nd, Minor 2nd, Minor 2nd, Major 2nd, Minor 2nd
／（1＋1＋1＋2＋1）×2／C, C#, D, D#, F, F#, G, G#, A, B
Minor 2nd, Minor 2nd, Major 2nd, Minor 2nd, Minor 2nd, Minor 2nd, Minor 2nd, Major 2nd, Minor 2nd, Minor 2nd
／（1＋1＋2＋1＋1）×2／C, C#, D, E, F, F#, G, G#, A#, B
Minor 2nd, Major 2nd, Minor 2nd, Minor 2nd, Minor 2nd, Minor 2nd, Major 2nd, Minor 2nd, Minor 2nd, Minor 2nd
／（1＋2＋1＋1＋1）×2／C, C#, D#, E, F, F#, G, A, A#, B
Major 2nd, Minor 2nd, Minor 2nd, Minor 2nd, Minor 2nd, Major 2nd, Minor 2nd, Minor 2nd, Minor 2nd, Minor 2nd
／（2＋1＋1＋1＋1）×2／C, D, D#, E, F, F#, G#, A, A#, B
・Twelve-note scale／Dodecatonic scale, Chromatic scale
Minor 2nd, Minor 2nd, Minor 2nd, Minor 2nd, Minor 2nd, Minor 2nd, Minor 2nd, Minor 2nd, Minor 2nd, Minor 2nd, Minor 2nd,
Minor 2nd／1×12／C, C#, D, D#, E, F, F#, G, G#, A, A#, B
The above 38 scales.
2. Tonal scale
A tonal scale refers to scales other than the 38 mentioned above. One
type of such tonal scale is the diatonic scale, which is, in keeping with
the primary law of tonalia, formed under three conditions: the degree of
consonance, non-permutation, and non-maldistribution.
It is assumed that these seven diatonic scales were not created out of
nowhere, but were developed over a long time from the three-tone scale
to the five-tone scale and finally to the seven-tone scale. The second law of tonalia
is that odd-numbered scales provide both stability and functionality. According
to this theory, as explained below, the seven-tone will be developed into
the nine-tone, and the nine-tone will eventually reach the eleven-tone.
・Three-note scale／Tritonic scale
C, E, G
・Five-note-note scale／Pentatonic scale
C, D, E, G, A
・Seven-note scale／Heptatonic scale／Diatonic scale
C, D, E, F, G, A, B
・Nine-note scale／Nonatonic scale
C, D, D#, E, F, G, G#, A, B
・Eleven-note scale／Undecatonic scale
C, D, D#, E, F, F#, G, G#, A, B
Of the three conditions for forming diatonic scales, non-maldistribution
means there is no bias in either the intervals or sequences as they relate
to the stability and functionality of the scales.
In particular, while diatonic scales are composed of five whole tones
and two semitones, those formed by, for example, one minor third, four
whole tones, and three semitones indicate the presence of maldistribution.
Furthermore, this implies that two semitones that are part of the diatonic
scale do not appear (A) consecutively or (B) in proximity to each other.
Example of A: semitone, semitone, whole tone, whole tone, whole tone, whole
tone, whole tone.
Example of B: semitone, whole tone, semitone, whole tone, whole tone, whole tone, whole tone.
With this in mind, presented below is the basis for the argument that
the above nine scales have evolved from diatonic scales: firstly, there
exist 180 (20 x 9) nine-tone scales formed on the twelve-tone scales. According
to the non-ubiquitous theory, the structure of nine-tone scales consists
of three whole tones and six semitones. There are sixteen of such scales,
as shown below.
After sifting them through the screen of diatonic scales, the only two
remaining scales are No. 6 and 11. Comparing these two scales shows that
the two tones added to the diatonic scales are C# and G# for No. 6, and
D# and G# for No. 11, where G# is shared between them. Now, to paraphrase
the three conditions of the degree of consonance, non-permutation, and
non-maldistribution into the tonal nine-tone scales, they share non-permutation
and non-maldistribution.
However, there is a major difference in the degree of consonance. Below
is a graph showing how the degree of dissonance changes depending on the
frequency ratio of the fundamental wave of the musical sound, drawing a
distinctive curve called a dissonance curve.
Atsushi Ogata (2007), “The Science of Temperament and Scale,”
P109, Kodansha (reproduction permission February 12th, 2016).
As this graph clearly shows, the degree of dissonance of C# is outstandingly
high. Therefore, from the context of the degree of consonance, which is
the first condition, it is probable that C# does not need to be used. When
actually played, the absence of C# accentuates the presence of C and the
tonality becomes clearer. From this, the tonal nine-tone scales, which
evolved from the diatonic scales, can be considered to consist of C, D,
D#, E, F, G, G#, A, and B, with the keynote of C.
Modes of nine-tone scale are indicated below.
The tonal eleven-note scales are formed by adding two tones from C#,
F#, or A# to the base of the tonal nine-tone scales. In other words, the
tonal eleven-note scales are formed by removing any one of C#, F#, or A#.
As previously mentioned, from the perspective of the degree of consonance,
C# should not be used. Thus, it can be logically concluded that the tonal
eleven-note scales will naturally be formed with C, D, D#, E, F, F#, G,
G#, A, and B, with the keynote of C.
◆Theory on Harmony◆
In the theory on scale, even in the theory on harmony, odd numbers similarly dominate like odd numbers that dominated the tonalia. My theory is that.
Also, in the tertian harmony of the seven-tone scales, the basic way
of forming the harmony is by building upwards from keynote two, which is
one and two notes away.
1. Harmony building by odd degrees
The third law of tonalia is that tonal harmonies build upwards by odd
degrees. This means that there are six harmonies on the seven-note scales:
the tertian harmony, fifth harmony, seventh harmony, ninth harmony, eleventh
harmony, and thirteenth harmony.
What is usually called the quartal harmony should actually be defined
correctly as a downwardly inverted form of the upwardly built fifth harmony,
or a shortened and inverted form of the upwardly built eleventh harmony.
6 harmonies of 4 voices with root note C
| Harmony | Constituent notes | Chord name | Abbreviations |
| Tertian harmony | C, E, G, B | C Major seventh | C Maj7 |
| Fifth harmony | C, G, D, A | C Triple fifths* | C Tri5 |
| Seventh harmony | C, B, A, G | C Later sevenths* | C Lat7 |
| Ninth harmony | C, D, E, F | C Previous ninths* | C Pre9 |
| Eleventh harmony | C, F, B, E | C Divided elevenths* | C Div11 |
| Thirteenth harmony | C, A, F, D | C Divided thirteenths* | C Div13 |
*Original terminology
2.Types of each harmony
The upwardly built fifth or more harmonies formed in the diatonic is
as follows.
Fifth harmony of 4 voices with root note C
| Constituent notes | Number of semitones | Chord name | Abbreviations |
| C, G, D, A | 7, 7, 7 | C Triple fifths | C Tri5 |
| D, A, E, B | 7, 7, 7 | D Triple fifths | D Tri5 |
| E, B, F, C | 7, 6, 7 | E Divided fifths | E Div5 |
| F, C, G, D | 7, 7, 7 | F Triple fifths | F Tri5 |
| G, D, A, E | 7, 7, 7 | G Triple fifths | G Tri5 |
| A, E, B, F | 7, 7, 6 | A Previous fifths | A Pre5 |
| B, F, C, G | 6, 7, 7 | B Later fifths | B Lat5 |
Seventh harmony of 4 voices with root note C
| Constituent notes | Number of semitones | Chord name | Abbreviations |
| C, B, A, G | 11, 10, 10 | C Later sevenths | C Lat7 |
| D, C, B, A | 10, 11, 10 | D Divided sevenths | D Div7 |
| E, D, C, B | 10, 10, 11 | E Previous sevenths | E Pre7 |
| F, E, D, C | 11, 10, 10 | F Later sevenths | F Lat7 |
| G, F, E, D | 10, 11, 10 | G Divided sevenths | G Div7 |
| A, G, F, E | 10, 10, 11 | A Previous sevenths | A Pre7 |
| B, A, G, F | 10, 10, 10 | B Triple sevenths | B Tri7 |
Ninth harmony of 4 voices with root note C
| Constituent notes | Number of semitones | Chord name | Abbreviations |
| C, D, E, F | 14, 14, 13 | C Previous ninths | C Pre9 |
| D, E, F, G | 14, 13, 14 | D Divided ninths | D Div9 |
| E, F, G, A | 13, 14, 14 | E Later ninths | E Lat9 |
| F, G, A, B | 14, 14, 14 | F Triple ninths | F Tri9 |
| G, A, B, C | 14, 14, 13 | G Previous ninths | G Pre9 |
| A, B, C, D | 14, 13, 14 | A Divided ninths | A Div9 |
| B, C, D, E | 13, 14, 14 | B Later ninths | B Lat9 |
Eleventh harmony of 4 voices with root note C
| Constituent notes | Number of semitones | Chord name | Abbreviations |
| C, F, B, E | 17, 18, 17 | C Divided elevenths | C Div11 |
| D, G, C, F | 17, 17, 17 | D Triple elevenths | D Tri11 |
| E, A, D, G | 17, 17, 17 | E Triple elevenths | E Tri11 |
| F, B, E, A | 18, 17, 17 | F Later elevenths | F Lat11 |
| G, C, F, B | 17, 17, 18 | G Previous elevenths | G Pre11 |
| A, D, G, C | 17, 17, 17 | A Triple elevenths | A Tri11 |
| B, E, A, D | 17, 17, 17 | B Triple elevenths | B Tri11 |
Thirteenth harmony of 4 voices with root note C
| Constituent notes | Number of semitones | Chord name | Abbreviations |
| C, A, F, D | 21, 20, 21 | C Divided thirteenths | C Div13 |
| D, B, G, E | 21, 20, 21 | D Divided thirteenths | D Div13 |
| E, C, A, F | 20, 21, 20 | E Middle thirteenth* | E Mid13 |
| F, D, B, G | 21, 21, 20 | F Previous thirteenth | F Pre13 |
| G, E, C, A | 21, 20, 21 | G Divided thirteenths | G Div13 |
| A, F, D, B | 20, 21, 21 | A Later thirteenths | A Lat13 |
| B, G, E, C | 20, 21, 20 | B Middle thirteenth | B Mid13 |
*Original terminology
3. Classification of chords
In a typical harmonic progression, the eleventh and thirteenth are used for chord tones, with the harmony laid out two octaves above them.
Now, chords are generally classified into consonance and dissonance.
In contrast, my theory first defines the concept of perfect consonance. what is more, after classifying into tonal chord and atonal chord, the perfect consonance is positioned as atonal chord. This is Tonalia's fourth law.
(1) Definition of atonal chord
The atonal chord is a chord that has three or more tones of the same
interval, and build upwardly.
Therefore, the tonal chords are everything else.
(2) Definition of perfect consonance
The perfect consonance is a chord that has three or more tones, and build upwardly by the interval of perfect 5th, perfect 11th, perfect 19th, perfect 25th, and etc.
Also in my theory, as mentioned above, the upwardly built chord by perfect
fourth must be positioned as shortened and inverted form of the upwardly
built perfect eleventh.
4. Conclusion
As mentioned at the beginning, there still remain undeveloped areas in
the twelve-tone scales. Those are not areas for atonal music as if the
game theory had been used, but ones for expansibility that represent possibilities
for tonalia.
What kind of music would a higher form of life create with the audible
range that is different from ours, on a planet with different gravity from
that of the earth? Certainly, somewhere in this broad universe, there must
exist some music that is beyond our imagination.
(Due to the restriction on symbol input, ♭ was not used.)
Published in February 16, 2016.
Revised in September 25, 2018.
Revised in April 17, 2020.
トーナリアの法則
音階と和声の基礎理論
− Ver.3 −
長谷川 景光
音楽は進化する。そして、現在も進化の途上にある。では、21世紀の音楽的進化とは、無調の新たな音楽語法の創造となるのか、はたまた微分音音階を希求することになるのか。
本論の中心にあるのは、12音階にはまだ未開拓の領域、すなわち作曲技法のさらなる高度化、高組織化の可能性が存在するということである。
さて、論題に掲げたトーナリアとは、有調の領域を意味している。その領域の構造、要するにどのような音階と和声が存在するのかに関しては、以下の認識が前提となる。
すなわち、その曲の音階、また旋律が有調性を有していたとしても、作曲、編曲によって無調性和声を組み込めば、その曲は無調化する。
また、その逆にその曲の音階、また旋律が無調性を有していたとしても、作曲、編曲によって有調性和声を組み込めば、その曲は有調化する。
観点を換えれば、音階と和声は対等であるということである。
◆音階理論◆
１．無調音階
（1）無調音階の定義
一定の音列を繰り返して、1オクターブを形成する音階。
（2）無調音階の分類
各音階における音程配列／半音数の構造を示す数式／Cを主音とする音階
・2音階／Diatonic scale
増4度、増4度／6×2／C, F#
・3 音階／Tritonic scale
長3度、長3度、長3度／4×3／C, E, G#
・4音階／Tetratonic scale
短3度、短3度、短3度、短3度／3×4／C, D#, F#, A
短2度、完全4度、短2度、完全4度／（1＋5）×2／C, C#, F#, G
完全4度、短2度、完全4度、短2度／（5＋1）×2／C, F, F#, B
長2度、長3度、長2度、長3度／（2＋4）×2／C, D, F#, G#
長3度、長2度、長3度、長2度／（4＋2）×2／C, E, F#, A#
・6音階／Hexatonic scale
長2度、長2度、長2度、長2度、長2度、長2度／2×6／C, D, E, F#, G#, A#
短2度、短3度、短2度、短3度、短2度、短3度／（1＋3）×3／C, C#, E, F, G#, A
短3度、短2度、短3度、短2度、短3度、短2度／（3＋1）×3／C, D#, E, G, G#, B
短2度、短2度、長3度、短2度、短2度、長3度／（1＋1＋4）×2／C, C#, D, F#, G, G#
短2度、長3度、短2度、短2度、長3度、短2度／（1＋4＋1）×2／C, C#, F, F#, G, B
長3度、短2度、短2度、長3度、短2度、短2度／（4＋1＋1）×2／C, E, F, F#, A#, B
短2度、長2度、短3度、短2度、長2度、短3度／（1＋2＋3）×2／C, C#, D#, F#, G, A
短2度、短3度、長2度、短2度、短3度、長2度／（1＋3＋2）×2／C, C#, E, F#, G, A#
長2度、短2度、短3度、長2度、短2度、短3度／（2＋1＋3）×2／C, D, D#, F#, G#, A
長2度、短3度、短2度、長2度、短3度、短2度／（2＋3＋1）×2／C, D, F, F#, G#, B
短3度、短2度、長2度、短3度、短2度、長2度／（3＋1＋2）×2／C, D#, E, F#, A, A#
短3度、長2度、短2度、短3度、長2度、短2度／（3＋2＋1）×2／C, D#, F, F#, A, B
・8音階／Octotonic scale
短2度、長2度、短2度、長2度、短2度、長2度、短2度、長2度／（1＋2）×4／C, C#, D#, E, F#, G, A, A#
長2度、短2度、長2度、短2度、長2度、短2度、長2度、短2度／（2＋1）×4／C, D, D#, F, F#, G#, A, B
短2度、短2度、長2度、長2度、短2度、短2度、長2度、長2度／（1＋1＋2＋2）×2／C, C#, D, E, F#, G, G#, A#
短2度、長2度、長2度、短2度、短2度、長2度、長2度、短2度／（1＋2＋2＋1）×2／C, C#, D#, F, F#, G, A, B
長2度、短2度、短2度、長2度、長2度、短2度、短2度、長2度／（2＋1＋1＋2）×2／C, D, D#, E, F#, G#, A, A#
長2度、長2度、短2度、短2度、長2度、長2度、短2度、短2度／（2＋2＋1＋1）×2／C, D, E, F, F#, G#, A#, B
短2度、短2度、短2度、短3度、短2度、短2度、短2度、短3度／（1＋1＋1＋3）×2／C, C#, D, D#, F#, G, G#, A
短2度、短2度、短3度、短2度、短2度、短2度、短3度、短2度／（1＋1＋3＋1）×2／C, C#, D, F, F#, G, G#, B
短2度、短3度、短2度、短2度、短2度、短3度、短2度、短2度／（1＋3＋1＋1）×2／C, C#, E, F, F#, G, A#, B
短3度、短2度、短2度、短2度、短3度、短2度、短2度、短2度／（3＋1＋1＋1）×2／C, D#, E, F, F#, A, A#, B
・9音階／Nonatonic scale
短2度、短2度、長2度、短2度、短2度、長2度、短2度、短2度、長2度／（1＋1＋2）×3／C, C#, D, E, F, F#, G#, A, A#
短2度、長2度、短2度、短2度、長2度、短2度、短2度、長2度、短2度／（1＋2＋1）×3／C, C#, D#, E, F, G, G#, A, B
長2度、短2度、短2度、長2度、短2度、短2度、長2度、短2度、短2度／（2＋1＋1）×3／C, D, D#, E, F#, G, G#, A#, B
・10音階／Decatonic scale
短2度、短2度、短2度、短2度、長2度、短2度、短2度、短2度、短2度、長2度／（1＋1＋1＋1＋2）×2／C, C#, D, D#, E, F#, G, G#, A, A#
短2度、短2度、短2度、長2度、短2度、短2度、短2度、短2度、長2度、短2度／（1＋1＋1＋2＋1）×2／C, C#, D, D#, F, F#, G, G#, A, B
短2度、短2度、長2度、短2度、短2度、短2度、短2度、長2度、短2度、短2度／（1＋1＋2＋1＋1）×2／C, C#, D, E, F, F#, G, G#, A#, B
短2度、長2度、短2度、短2度、短2度、短2度、長2度、短2度、短2度、短2度／（1＋2＋1＋1＋1）×2／C, C#, D#, E, F, F#, G, A, A#, B
長2度、短2度、短2度、短2度、短2度、長2度、短2度、短2度、短2度、短2度／（2＋1＋1＋1＋1）×2／C, D, D#, E, F, F#, G#, A, A#, B
・12音階／Dodecatonic scale, Chromatic scale
短2度、短2度、短2度、短2度、短2度、短2度、短2度、短2度、短2度、短2度、短2度、短2度／1×12／
C, C#, D, D#, E, F, F#, G, G#, A, A#, B
以上の38の音階。
２．有調音階
有調音階とは、上記38以外の音階である。その有調音階の1つが全音階であり、その全音階が形成される3条件は、協和度、非順列性、そして非偏在性であるというのが、トーナリアの第一法則である。
この全音階、すなわち7音階が突然発生したのではなく、長い時間をかけて3音階から5音階へ、そして5音階から7音階へと進化していったと考えられる。
このように、奇数音階は安定性と機能性を具備しているというのが、トーナリアの第二法則である。この理論に従えば、以下のとおり7音階は9音階に高度化し、9音階は最終的に11音階に到達することになる。
・3音階／Tritonic scale
C, E, G
・5音階／Pentatonic scale
C, D, E, G, A
・7音階／Heptatonic scale／Diatonic scale
C, D, E, F, G, A, B
・9音階／Nonatonic scale
C, D, D#, E, F, G, G#, A, B
・11音階／Undecatonic scale
C, D, D#, E, F, F#, G, G#, A, B
そして、全音階形成の3条件の1つである非偏在性とは、安定的、機能的音階には、音程及び配列の2つに偏りがないということである。
具体的に、全音階は5つの全音と2つの半音によって形成されているが、短3度が1つ、全音が4つ、半音が3つというような音階は、偏在性を有しているということを意味する。さらに、全音階を形成する2つの半音が連続したり（A）、接近する（B）ということがないことを意味している。
Aの例 半音、半音、全音、全音、全音、全音、全音。
Bの例 半音、全音、半音、全音、全音、全音、全音。
そこで、全音階から進化したのが上記の9音階であるという根拠を以下に示す。
まず、12音階上に形成される9音階は、20の9倍、180存在する。
そして、非偏在性理論に基づけば、9音階の構造は3つの全音と6つの半音によって形成される。そのような音階は、以下のとおり16存在する。
さらに、これを全音階というフィルターに通すと、適合する音階は6番と11番の2つのみとなる。この両者の音階を比較すると、全音階に加えられた2つの音は、6番がC#、G#、11番がD#、G#であり、G#は共通する。
そこで、前述の全音階が形成される3条件、すなわち協和度、非順列性、そして非偏在性を有調9音階にも敷衍して考察すると、共に非順列性、非偏在性については同様である。
しかし、協和度については大きな違いがある。
以下は、楽音の基本波の周波数比によって、不協和度がどのように変化するかを示すグラフであり、不協和曲線と言われる独特のカーブを描く。
小方厚(2007)『音律と音階の科学』講談社、109頁（転載許可2016.2.12）
このグラフを見れば一目瞭然であるが、突出して不協和度が高いのがC#である。
したがって、第一条件である協和度という観点から、C#を用いるべきではないことに蓋然性がある。実際に演奏してみると、C#を用いないと主音であるCの存在感が高まり調性が明確になる。
このことから、全音階の進化形である有調9音階はC、D、D#、E、F、G、G#、A、B（主音C）であると位置づけられる。
以下に9音階の旋法を示す。
次に、有調11音階であるが、この有調9音階をベースとして、C#、F#、A#の内の2音を加えることにより形成される。言い換えれば、C#、F#、A#の何れか1つを取り去ることにより有調11音階が形成されることになる。
前述のとおり、第一条件である協和度という観点からは、C#を用いるべきではないので、必然的に有調11音階がC、D、D#、E、F、F#、G、G#、A、B（主音C）であることが理論的に導き出される。
◆和声理論◆
音階理論において、トーナリアを支配していたのが奇数であったように、和声理論においても同様に奇数が支配している。というのが拙論である。
また、7音階における3度和声では、主音から1つ置きに上方堆積して和声を形成する。しかし、9音階における3度和声では、主音から2つ置き、1つ置き、2つ置きの順で上方堆積し和声が形成されるのが基本形である。
１．和声の奇数度堆積
根音に対し上方に奇数度堆積するのが有調和声であり、これがトーナリアの第三法則である。
したがって、7音階上には3度和声、5度和声、7度和声、9度和声、11度和声、13度和声の6和声が理論的に存在する。
通常、言われるところの4度和声は、正しくは上方堆積5度和声の下方転回形、または上方堆積11度和声の短縮転回形と位置づけなければならない。
根音をCとする四声の6和声
| 和声 | 構成音 | 和音名 | 略号表記 |
| 3度和声 | C, E, G, B | C Major seventh | C Maj7 |
| 5度和声 | C, G, D, A | C Triple fifths* | C Tri5 |
| 7度和声 | C, B, A, G | C Later sevenths* | C Lat7 |
| 9度和声 | C, D, E, F | C Previous ninths* | C Pre9 |
| 11度和声 | C, F, B, E | C Divided elevenths* | C Div11 |
| 13度和声 | C, A, F, D | C Divided thirteenths* | C Div13 |
*独自の術語
２．各和声の種類
ダイアトニックに形成される上方堆積の和声は以下のとおりである。
根音をCとする四声の3度和声
| 構成音 | 半音数 | 和音名 | 略号表記 |
| C, E, G, B | 4, 3, 4 | C Major seventh | C Maj7 |
| D, F, A, C | 3, 4, 3 | D Minor seventh | D m7 |
| E, G, B, D | 3, 4, 3 | E Minor seventh | E m7 |
| F, A, C, E | 4, 3, 4 | F Major seventh | F Maj7 |
| G, B, D, F | 4, 3, 3 | G Seventh | G 7 |
| A, C, E, G | 3, 4, 3 | A Minor seventh | A m7 |
| B, D, F, A | 3, 3, 4 | B Diminish seventh | B dim7 |
根音をCとする四声の5度和声
| 構成音 | 半音数 | 和音名 | 略号表記 |
| C, G, D, A | 7, 7, 7 | C Triple fifths | C Tri5 |
| D, A, E, B | 7, 7, 7 | D Triple fifths | D Tri5 |
| E, B, F, C | 7, 6, 7 | E Divided fifths | E Div5 |
| F, C, G, D | 7, 7, 7 | F Triple fifths | F Tri5 |
| G, D, A, E | 7, 7, 7 | G Triple fifths | G Tri5 |
| A, E, B, F | 7, 7, 6 | A Previous fifths | A Pre5 |
| B, F, C, G | 6, 7, 7 | B Later fifths | B Lat5 |
根音をCとする四声の7度和声
| 構成音 | 半音数 | 和音名 | 略号表記 |
| C, B, A, G | 11, 10, 10 | C Later sevenths | C Lat7 |
| D, C, B, A | 10, 11, 10 | D Divided sevenths | D Div7 |
| E, D, C, B | 10, 10, 11 | E Previous sevenths | E Pre7 |
| F, E, D, C | 11, 10, 10 | F Later sevenths | F Lat7 |
| G, F, E, D | 10, 11, 10 | G Divided sevenths | G Div7 |
| A, G, F, E | 10, 10, 11 | A Previous sevenths | A Pre7 |
| B, A, G, F | 10, 10, 10 | B Triple sevenths | B Tri7 |
根音をCとする四声の9度和声
| 構成音 | 半音数 | 和音名 | 略号表記 |
| C, D, E, F | 14, 14, 13 | C Previous ninths | C Pre9 |
| D, E, F, G | 14, 13, 14 | D Divided ninths | D Div9 |
| E, F, G, A | 13, 14, 14 | E Later ninths | E Lat9 |
| F, G, A, B | 14, 14, 14 | F Triple ninths | F Tri9 |
| G, A, B, C | 14, 14, 13 | G Previous ninths | G Pre9 |
| A, B, C, D | 14, 13, 14 | A Divided ninths | A Div9 |
| B, C, D, E | 13, 14, 14 | B Later ninths | B Lat9 |
根音をCとする四声の11度和声
| 構成音 | 半音数 | 和音名 | 略号表記 |
| C, F, B, E | 17, 18, 17 | C Divided elevenths | C Div11 |
| D, G, C, F | 17, 17, 17 | D Triple elevenths | D Tri11 |
| E, A, D, G | 17, 17, 17 | E Triple elevenths | E Tri11 |
| F, B, E, A | 18, 17, 17 | F Later elevenths | F Lat11 |
| G, C, F, B | 17, 17, 18 | G Previous elevenths | G Pre11 |
| A, D, G, C | 17, 17, 17 | A Triple elevenths | A Tri11 |
| B, E, A, D | 17, 17, 17 | B Triple elevenths | B Tri11 |
根音をCとする四声の13度和声
| 構成音 | 半音数 | 和音名 | 略号表記 |
| C, A, F, D | 21, 20, 21 | C Divided thirteenths | C Div13 |
| D, B, G, E | 21, 20, 21 | D Divided thirteenths | D Div13 |
| E, C, A, F | 20, 21, 20 | E Middle thirteenth* | E Mid13 |
| F, D, B, G | 21, 21, 20 | F Previous thirteenth | F Pre13 |
| G, E, C, A | 21, 20, 21 | G Divided thirteenths | G Div13 |
| A, F, D, B | 20, 21, 21 | A Later thirteenths | A Lat13 |
| B, G, E, C | 20, 21, 20 | B Middle thirteenth | B Mid13 |
*独自の術語
３．和音の分類
通常の和声進行において、コードトーンに11th、13th、が用いられ2オクターブ上に和声が展開される。前述のとおり、音階と和声は対等であるので、音階においても2オクターブどころか11オクターブ上に音階を展開することが可能である。
さて、和音は通常、協和音と不協和音に大別される。
これに対して拙論では、まず完全協和音という概念を規定している。さらに、有調和音と無調和音に分類した上で、完全協和音を無調和音に位置づけている。これが、トーナリアの第四法則である。
（1）無調和音の定義
無調和音とは、同じ音程が３音以上、上方堆積する和音である。
したがって、有調和音はそれ以外の全てである。
（2）完全協和音の定義
完全協和音とは、完全５度、完全11度、完全19度、完全25度などの音程によって３音以上、上方堆積する和音である。
なお、拙論では前述のとおり、完全４度の上方堆積和音は、完全11度の短縮転回形と位置づけなければならない。
４．おわりに
冒頭に述べたように、12音階にはまだ未開拓の領域がある。それは、ゲーム理論を用いたような無調音楽のための領域ではなく、トーナリアにおける可能性であり拡張性のための領域である。
地球とは異なる重力の星で、ヒトとは異なる可聴域を持つ高等生物が創り出す音楽とはどのようなものなのか。広い宇宙においては、我々の想像を絶するような音楽が必ずや存在するのであろう。
（注：記号入力の制約により♭を使用せず。）
2016年2月16日 公開
2018年9月25日 改訂
2020年4月17日 改訂
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