User:VectorGraphics/Equave tempering
Equave tempering refers to the idea that an interval of equivalence is mathematically identical to a comma being tempered out.
Explanation
A tempered interval is a single point in tempered space, but in JI interval space, it represents an "equivalence class" of intervals separated by some comma or combination of commas. For example, meantone can be analyzed as using the same interval to represent 5/4 and 81/64, or it can alternatively be analyzed in just intonation as treating 5/4 and 81/64 as equivalent. Thus, If you use both 5/4 and 81/64, but do not make a distinction between them, that is theoretically identical to meantone temperament. In other words, a regular temperament does not necessitate a regular tuning. This is similar to how a well temperament is often considered mathematically indistinguishable from the equal temperament it is based on, even though if you play in different keys, the precise tuning of each interval is different.
When you choose an interval of equivalence, it defines an equivalence class of just intervals in JI space that represents all intervals separated by the equave, and thus that belong to the same pitch class, and it collapses that equivalence class to a single point in a kind of "tempered interval space". When this is done with the octave 2/1, it collapses the 3D 5-limit lattice down to a 2D lattice that is similar to 3.5, but with the added qualifier that any interval may be replaced by its octave-reduction. Similarly, it collapses the 2D 3-limit lattice down to the 1D chain of fifths.
But this, as I established previously, is the exact same thing as a tempered interval! This means that one can just as easily say that an octave-equivalent system tempers out 2/1 as that a meantone system tempers out 81/80. Additionally, this allows the notion of equivalence to be formalized in the language of regular temperament theory, which is a concept that most usages of it currently lack. For example, this allows "octave-equivalent meantone" to be defined as the rank-1 temperament tempering out 81/80 and 2/1.
Diapasonic temperament
"Diapasonic" is the name Vector gives to the temperament tempering out 2/1, implying octave-equivalence. All octave-equivalent temperaments are expansions of the rank-0 temperament, though the canonical diapasonic temperament is a rank-1 structure generated by 3/2 (or, equivalently, 3/1), which all fifth-generated octave-equivalent temperaments are extensions of.
Diapasonic
Subgroup: 2.3
Comma list: 2/1
Mapping: [0 1]
- Mapping generators: ~3
Diapasonic meantone
Subgroup: 2.3.5 Comma list: 2/1, 81/80
Mapping: [0 1 4]
- Mapping generators: ~3
Diapasonic archy
Subgroup: 2.3.7 Comma list: 2/1, 64/63
Mapping: [0 1 -2]
- Mapping generators: ~3
Diapasonic porcupine
Subgroup: 2.3.5.11 Comma list: 2/1, 55/54, 100/99
Mapping: [0 3 5 4]
- Mapping generators: ~9/5
Classical diapasonic
Subgroup: 2.3.5
Comma list: 2/1
Mapping: [[0 1 0] [0 0 1]]
- Mapping generators: ~3, ~5
Diapasonic marvel
Subgroup: 2.3.5.7
Comma list: 2/1, 225/224
Mapping: [[0 1 0 2] [0 0 1 2]]
- Mapping generators: ~3, ~5
Septimal diapasonic
Subgroup: 2.3.5
Comma list: 2/1
Mapping: [[0 1 0] [0 0 1]]
- Mapping generators: ~3, ~5