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.