Tertian structure

Tertian structure is the relative spacing of the thirds in a tempered tuning system, such as an equal temperament. In general, it usually refers to the spacing of the 7-limit color-quality thirds 7/6, 6/5, 5/4, and 9/7, which can be expressed as the relative size of 25/24 and 36/35. Other thirds like 19/16, 19/15, or 11/9 may be invoked to describe more complex structures.

Common rank-3 temperaments can be expressed as tertian structures, and in fact, a tertian structure can always be thought of as a rank-3 temperament. These include Mint, Dicot.7, Keemic, and Mandos.

In just intonation, there is no rational relationship in size between 25/24 and 36/35. The ratio of their sizes is log_36/35(25/24) = 1.44908497.

All tertian structures are supported by 7edo, which collapses all distinctions between the four thirds in question down to a single interval, outlining the structure of diatonic harmony.

Mint

Ratio (36/35:25/24): 0:1

Comma: (36/35)^1 / (25/24)^0 = 36/35

Supported edos: 5, 9, 12, 14, 16, 21, 23, 28, 33

Mint tempers out 36/35 entirely, equating the pental and septimal third qualities to some tuning around 300 and 400 cents. Unusually enough, as a temperament structure in and of itself, this is best optimized with a flat fifth. The most prominent mint temperaments are dominant (meantone, supported by 5edo and 12edo), mavila (supported by 9edo, 16edo, and 23edo), whitewood (supported by 14edo, 21edo, and 28edo), and dimisept (supported by 12edo and 16edo alongside 28edo). Interestingly, mint temperaments overwhelmingly have flat fifths, except for dominant. This makes sense, as mint essentially equally spaces 3/2 between 5/4 and 7/4, and if both are kept just then 3/2 is set to about 678 cents.

Dicot

Ratio (36/35:25/24): 1:0

Comma: (25/24)^1 / (36/35)^0 = 25/24

Supported edos: 10, 13, 17, 20

Dicot is a much rarer archetype even in the small edos due to the fact that it can be described as a rank-2 temperament in a prime subgroup (although as a tertian structure 7 is given as an independent generator); additionally, its comma is the largest of any tertian structure discussed here. The temperaments supporting the main form of dicot, called dichotic, are 10, 13, 17, and 20. Unlike mint temperaments which prefer a flat fifth, dicot temperaments prefer a sharp fifth. Again, this makes sense if you examine the 5/4 eigenmonzo tuning, which jumps 3/2 all the way up to 773 cents; accurate tunings of dicot fall between the just tuning of 3/2 and that sharp tuning.

Keemic

Ratio (36/35:25/24): 1:1

Comma: (25/24)^1 / (36/35)^1 = 875/864

Supported edos: 8, 15, 19, 22, 26, 30, 34, 37, 41, 44, 45, 48, 56, 59, 60, 63...

Keemic is the first "reasonably accurate" tertian structure, with a 1:1 relationship between 25/24 and 36/35. It is the tertian structure for which color notation appears to have been designed, and which structurally relates the four 7-limit qualities by equal steps. There are three major keemic temperaments: porcupine (with a 11/10 generator, setting the diatonic major third to 9/7, and supported by 15edo, 22edo, 30edo, 37edo, 44edo, and 59edo), flattone (with a flat 3/2 generator, setting the diatonic major third to 5/4, and supported by 19edo, 26edo, and 45edo), and tetracot (which places the diatonic major third exactly in between 5/4 and 9/7, supporting Aberschismic and supported by 34edo, 41edo, 48edo, and 82edo).

Keemic is characterized by the narrowing of 25/24 and/or the widening of 36/35; eigenmonzo 54/49 (the distance between 7/6 and 9/7) and 3/2 leads to a 5/4 tuning of 379c; eigenmonzo 25/24 and 3/2 leads to a 9/7 tuning of 457c.

Mandos

Ratio (36/35:25/24): 1:2

Comma: (36/35)^2 / (25/24)^1 = 31104/30625

Supported edos: 24, 27, 31, 35, 38, 55, 58, 62, 69...

Mandos, instead of narrowing 25/24 and widening 36/35, widens 25/24 and narrows 36/35, making 25/24 twice as wide as 36/35. This allows (and in fact, structurally necessitates) a neutral third to be inserted between 5/4 and 6/5, which may be interpreted as 11/9. The overwhelming majority of the smaller mandos edos support mohajira, which adds meantone tempering and sets 5/4 to the diatonic major third and 9/7 to the semiaugmented third. These edos include 24, 31, 38, 55, 62, and 69, and mandos is tuned rather inaccurately in most of them save for 31edo, contributing to the interest in 31edo (at the intersection of septimal meantone and mandos/mohajira) as a system for extending classical meantone harmony. This can be understood by examining that meantone tempering usually flattens both 6/5 and 9/7, due to flattening the fifth; this makes mandos more justifiable as a tertian structure in that context. In most mohajira edos, the neutral third is 11/9. This is not supported by 69edo. Other mandos edos include 27edo, 58edo, and 35edo. 58edo is interesting in and of itself as a structure for interval categorization, which its mandos properties contribute to.

Myna is another rank-2 mandos temperament often confused with mandos as a whole. Myna further sets the 36/35 interval to 49/48. It is supported by 27edo (although 27edo sets the neutral third to 16/13 instead of 11/9) and 58edo, along with 31edo as previously mentioned.

TODO: rank-3 name for amity? also cover "oddball" structures like sixix, 3-2-3, 1-4-1, 1-5-1, etc