User:VectorGraphics/Equave tempering

Equave tempering refers to the similarity between an equave and a comma being tempered out.

Similarities

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.

Differences

However, there is a slight difference in the way mappings with torsion are handled. Torsion happens when a comma being tempered out is a power of some other interval in the system that isn't explicitly tempered out.

• For a comma, the convention is to temper out any roots of the comma. For example, if 6561/6400 were tempered out, 81/80 would be.
• For an equave, the convention is to not temper out roots of the equave. For example, if 9/1 were treated as an equave, intervals separated by 3/1 would still be distinct, despite 9 being 3^2.

This difference makes itself apparent when one attempts to define an octave-equivalent equal temperament by tempering out the octave. If one were to temper out the octave normally, the temperament would collapse to 0edo, as the octave is by definition a power of the smallest step in the scale, so all the intervals involved would be tempered out. However, using the second convention, the temperament reduces to a finite set of notes, which is the expected behavior.

Diapasonic temperament

"Diapasonic" is the name Vector gives to the temperament tempering out 2/1, implying octave-equivalence so long as the octave is not divided equally. All octave-equivalent temperaments (with this caveat) 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 (again, with the caveat) 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