User:FloraC/Analysis on the 13-limit just intonation space: episode i

This page is a work in progress.

Preface

Just intonation is the repertoire of pitch materials that is the most concerning. In this essay, it refers to the infinite-dimensional harmonic space consisting of all rational numbers, sometimes dubbed as rational intonation.

It should take no effort to recognize the source of its significance, but making it clear will benefit. Just intonation is an elephant in the room – in that one cannot turn a deaf ear to it. Again, it matters to point it out that, analytically, inharmonicity is not the same as harmonicity, just like fractions are not the same as integers. Additions and multiplications in the harmonic space always result in something in the said space. In math, it is called a ring. The same cannot be said of arbitrary inharmonic spaces.

Of course, simply disregarding irrational intervals is not doing it just, as explained in my previous essay There Is Not a Third Side of the River, since the semioctave, for example, is a highly characteristic sound just as important as those rational intervals. These quadratic irrationals will be addressed in the later sections. It is an indispensible part of this analysis.

As for the artificial attempts where utterly irrelevant entities are forced together, like using π as frequency ratio of the sound, I have criticized in Fundamental Principles to Musical Sense and have no reason to discuss even further.

The 13-limit just intonation is of particular interest for multiple reasons. The structure is neat and cognitively accessible for homo sapiens, yet is sophisticated enough to breed rich harmonic gestures and tempering options. Each prime harmonic in this space has a relatively distinct identity, which unfolds fairly consistently, unlike higher primes.

This essay aims to be objective. Explanations on my favorite temperaments will not be present – at least not in sufficiency to single them out.

Now let us start with Pythagorean tuning aka 3-limit just intonation.

Chapter I. How Pythagoras Broke the Tuning

The Pythagorean tuning or 3-limit just intonation is the backbone of the interval space. My classification of Pythagorean intervals is exactly the same as Functional Just System (FJS), so I guess I should only cite it.

FJS technique #1: to convert from a Pythagorean ratio to an FJS representation

1. Factorize the ratio.

2. Initially ignore octaves (powers of two).

3. If the power of three is positive, move that many steps by fifths clockwise; if negative, move anticlockwise. Convert that number to an interval.

4. Adjust octaves as required.

Example: To convert 9/8 to the FJS, we factorize: 2^-3 3². We ignore the factor of two. The power of three is +2, so we move two fifths clockwise: C–G–D. We have a major second. No octave adjustment needs to be made. The answer is M2.

FJS Technique #2: to convert from an FJS representation of a Pythagorean ratio back to the ratio.

1. Initially ignore octaves.

2. Convert the interval to the number of steps by fifths, name it n.

3. Calculate red(3ⁿ)

4. Adjust octaves as required.

Example: To convert the FJS interval m3 to a Pythagorean ratio, we convert it first to -3 fifths: C–F–B♭–E♭. We now raise 3 to that power: 3^-3. This is 1/27. To bring this number between 1 (inclusive) and 2 (exclusive), we multiply by 32 to get the answer: 32/27.

and

This is pretty boring for now.

This tuning naturally gives us the 7-tone diatonic scale and the 12-tone chromatic scale. Based on the diatonic scale, each interval is assigned a diatonic degree by the mapping of 7et:

⟨7 11]

Each degree changes its size through rotation. As such, each interval is assigned major or minor based on their size. Meanwhile, the chromatic scale can be generated this way (discarding either of the tritones), and implies the mapping of 12et:

⟨12 19]

Together, as is found out by Mike Battaglia, the change of basis

[math]\displaystyle{ \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \rightarrow \begin{bmatrix} 7 & 11 \\ 12 & 19 \end{bmatrix} }[/math]

is helpful for Pythagorean interval classification. In particular, it is easy to identify the following tempered monzos:

[1 1⟩

is a diatonic semitone, in this case 256/243. A movement by this would change the diatonic degree as well as the chromatic degree.

[0 1⟩

is a chromatic semitone, in this case 2187/2048. A movement by this does not change the diatonic degree but only the chromatic degree.

[1 0⟩

is an enharmonic diesis, in this case the Pythagorean comma (531441/524288). A movement by this changes the diatonic degree but not the chromatic degree.

It is possible to build tonality using the Pythagorean tuning, but the major and minor categories of intervals here are only to be understood in combination with diatonic degrees for the purpose of marking generator steps i.e. fifth shifts, and are not to be confused with the major and minor tonality. An interval is perfect if its number of generator steps is -1, 0, or +1, major if it is +2 to +5, and minor if it is -5 to -2.

[Figure] Figure 1: generator step markers

Meantone tempers out 81/80, the syntonic comma, identifying 5/4 by 2 diatonic degrees and 4 generator steps. That is how 5/4 is called a major third.

5-limit just intonation can be analysed as the syntonic comma added to the Pythagorean tuning, or meantone with the syntonic comma recovered, represented by the following mapping:

[math]\displaystyle{ \begin{bmatrix} 7 & 11 & 16 \\ 12 & 19 & 28 \\ 0 & 0 & 1 \end{bmatrix} }[/math]

or in terms of generator steps:

[math]\displaystyle{ \begin{bmatrix} 1 & 1 & 0 \\ 0 & 1 & 4 \\ 0 & 0 & 1 \end{bmatrix} }[/math]

with the syntonic comma mapped to the tempered monzo of [0 0 -1⟩ in both cases. Since the first two entries are zero, this tempered monzo marks neither diatonic moves nor chromatic moves, but a commatic one, which only alters the color of the interval. A Pythagorean interval altered by the syntonic comma is dyed with the "color of 5". For example, 81/64 is M3, whereas 5/4 is M3⁵. While both are members of major thirds, 5/4 involves otonal-5.

Now these are some of the diatonic semitones:

[Table]

Each is separated by 81/80.

These are some of the chromatic semitones:

[Table]

Each is separated by 81/80.