2.3-equivalent class and Pythagorean-commatic interval naming system
The 2.3-equivalent class characterizes a just interval by its deviation from Pythagorean tuning. If the Pythagorean chain of fifths is used to derive an interval's harmonic function, then the 2.3-equivalent class, being orthogonal thereto, characterizes an interval's harmonic "color". It is proposed as a replacement of harmonic class (HC) since it has been criticized that each HC lacks a consistent psychoacoustic quality.
This interval classification system takes full advantage of regular temperament theory. An interval's 2.3-equivalent class can be mathematically expressed as an interval's no-2 no-3 monzo. The notion finds applications in various notation systems that use the diatonic nominal-accidental chain including but not limited to FJS, HEJI, and color notation.
For example, 15/8, 5/4, 5/3 and 10/9 all have distinct harmonic functions, but it can be argued they have the same harmonic "color" of an otonal-5. In FJS they are denoted by the same superscript: M75, M35, M65, and M25, respectively. In HEJI they are notated by the same down-arrow on a certain root note; in color notation they share the same color name "yo".
From that basis we may derive the Pythagorean-commatic interval naming system, in which each interval's name consists of its abstract interval category prefixed by the commatic inflection from the Pythagorean reference point in that category. For example, 7/4 is the septimal minor seventh, since it is in the minor seventh category, and off from the Pythagorean minor seventh (16/9) by a septimal comma (64/63).
Intuitive as it is, these notes should be taken:
- Logically, there are two intervals for each name, residing on the opposite sides of the Pythagorean interval. The name should by default refer to the simpler one and the complex one should be further distinguished by retro. The septimal minor seventh refers to 7/4; retroseptimal minor seventh refers to 1024/567.
- The system does not state how one should conceive of the interval categories, so each interval may have multiple names. 7/4 is the septimal minor seventh as well as the garischismic double diminished octave, if one will call it.
Most traditional interval names loosely follow the spirit of the system, with some exceptions:
- Subdivided usage in terms of commas and intervals, and of upward and downward inflections, such as acute/grave for syntonic. It is not an actual problem so long as one memorizes all the details.
- Inflection on a non-Pythagorean interval, such as 55/32 being the keenanismic supermajor sixth, where the supermajor sixth is 12/7. It is not an actual problem since we can analyse the name in smaller parts.
- Septimal, undecimal, etc. These are not used to mean an inflection of the corresponding comma but the HC, and thus 49/48 is also marked septimal despite having a stack of two 7's in the numerator.
5-limit intervals are a particular mess being a mix of above, largely due to their historical significance. The corresponding HC is marked by classic(al) or synonymously pental, entangled with a finite set of intervals denominated as just. So 25/24 is the classic(al)/pental/just chromatic semitone. Alternative terms have been proposed to address that. Ptolemaic is used for inflection by the syntonic comma, so 25/24 is the double ptolemaic or diptolemaic chromatic semitone. Similarly, archytas is proposed for inflection by the septimal comma.