User:Eufalesio/12edo detemperament sequences

The chain of fifths is a very important framework historically. It's been in Western music THE way to think about everything all the way from plainchant to Renaissance meantone temperaments to the modern day – where the 12-pitch-class circle of fifths is taught; 12edo remains a massively over-represented tuning. It has a bit of a bad reputation in the xen circles, but the more I researched, the more I realized it is a paragon, and that its position nowadays is very much well earned.

12edo is an undisputed paragon of tonality among music theory of old and young. In the world of xenharmonic music, it is thus extremely desirable, if not necessary, to have a temperament supported by 12edo so that a simple workflow and notation are used. At least, RTT speaking. I will be neglecting Ringer scales and JI well temperaments as I don't care about them.

The easiest way to detemper 12edo is to observe (untemper) one or many more of its nullified commas; though observing a number of similar sized commas and tempering out their difference also works. This produces a rank-2 temperament. A rank-2 temperament can be "retempered" into a finer edo by adding another comma that 12edo doesn't temper out. All that I've explained here can be done with any rank temperament; rank-2s can be detempered into rank-3s, retempered into much finer edos.

The temperaments I'll be showing always have an whole or split octave period, and a whole or split fifth generator, and in any case, notation-wise from a D, the perfect fifth or fourth is NEVER spelt with an accidental.

7-limit

Let us take a look at some 7-limit temperaments through the patent valuation (direct approximation primes 2 through 7); which can be analyzed as tempering out 81/80, 64/63, 225/224; and by extension: 531441/524288, 126/125, 2048/2025, 36/35, 5120/5103... etc.

12

In 12 p.val, 5/4 is a major third, 7/4 is a minor seventh, subsequent temperaments will make modifications to those mappings.

• Observing 64/63 gives septimal meantone.
• Observing 81/80 gives pajara.
• Observing and tempering together 64/63 and 81/80 gives garibaldi.
• Observing 225/224 and tempering together 64/63 and 81/80 whilst tempering out 126/125 gives septimal diaschismic.
• Observing 64/63 and 81/80 whilst tempering out 531441/524288 gives compton.

These are rank-2 temperaments, which then can be retempered into finer edos by tempering out another comma not on the list;

• With septimal meantone:
  • Adding 126/125 or 1029/1024 or 1728/1715 gives 31edo.
  • Adding 15625/15552 or 3125/3072 or 49/48 gives 19edo.
  • Adding 16807/16384 gives 50edo.
• With pajara:
  • Adding 250/243 gives 22edo.
  • Adding 4375/4374 gives 34edo, through the 34d val.
  • Adding 686/675 gives 56edo.
• With garibaldi:
  • Adding 3125/3072 or 245/243 or 2401/2400 gives 41edo.
  • Adding 15625/15552 or 4375/4374 gives 53edo.
  • Adding the wizma or stearnsma gives 94edo.
• With septimal diaschismic:
  • Adding 245/243 or 4375/4374 gives 46edo.
  • Adding 1728/1715 gives 58edo.

• With compton:
  • Adding 15625/15552 or 4375/4374 or 2401/2400 gives 72edo.

Any of these rank-2s can be detempered into a rank-3 temperament which only tempers out one comma. The ones I deem most notable are:

• Marvel {225/224}
• Aberschismic {5120/5103}
• Orthoschismic {32805/32768}
• Garischismic {33554432/33480783}

12d

Taking the next best approximation of 7/4, a major sixth, however stupid that might sound... another family of temperaments can be derived. 12d can be analyzed as tempering out 81/80, 4375/4374, 2048/2025. This is a bad exotemperament because it implies 21/20, 28/27 are tempered out; but so are 32805/32768, and 875/864.

• Observing 2048/2045 gives flattone.
• Observing 81/80 gives srutal.
• Observing and tempering 2048/2405 and 81/80 together whilst tempering out 4375/4374 gives pontiac.

With flattone:

• Adding 50/49 gives 26edo.
• Adding 3125/3087 gives 45edo.

With srutal:

• Adding 126/125 or 245/243 gives 46edo.
• Adding 1728/1715 or 4000/3969 gives 80edo.

With pontiac:

• Adding 225/224 or 5120/5103 or 15625/15552 gives 53edo.
• Adding 1029/1024 or 3136/3125 gives 118edo.
• Adding 2401/2400 or wizma gives 171edo.
• Adding 16875/16807 gives 224edo.

Through this val, I deem the following rank-3 temperaments notable:

• Ragismic {4375/4374}
• Orthoschismic {32805/32768}

13-limit

Finding 13-limit 12edo detemperaments is harder, because 12edo notably does a horrible job of approximating primes 11 and 13, which means we have another vals to consider; which is to say, we don't take the direct approximation of some primes, but the next best. Here are 3 vals that I found to be worth of mention.

12e

Maps 11/8 to a fourth, 13/8 to a minor sixth. It is not the best val temperament wise, but since the corrections of p11 and p13 are in opposite directions to 5/4 and 7/4; but for this, the garibaldi extension cassandra is arguably the best pick, which is just a relabeling of the 3-limit, and can be easily detempered further into what I believe to be the BEST temperament (Ultimate)... see here for more info!

The path through cassandra into Ultimate is the easiest, as it keeps the chain of fifths, and a pythagorean-septimal comma (gary) with a hyper accurate zala, but adds an independent generator for 5 and 13, a "minicomma" that works as aberschisma ~ major minthma. Thus, the mappings of 5 and 13 can be thought as the same as cassandra, but altered by +1 / -1 minicomma. It can be detempered further into rank-4 olympic and finally, the goddamn-fucking-insanely accurate rank-5 temp Batshit AKA Insanismic for the prudes.

It can instead be detempered into hecate, which either cassandra or compton through the shit extension 12e & 72 can do. But I do not recommend going into hecate. It's decent, but it decouples the generic comma of cassandra from the chain of fifths and thus the garischisma is observed, and can't be detempered further into hyper-accurate temperaments.

12f

Maps 11/8 to a tritone, 13/8 to a major sixth. It is the best val temperament wise. All the corrections of primes 5 to 13 are in the same direction. There are two good paths to consider, through tridecimal compton and through diaschismic. Through tridecimal compton, the path ends at helios, yet another marvel temperament, which either way is very similar to compton still... so just use 72edo.

Through diaschismic, things get brighter. I don't really like diaschismic, but it can be detempered into the rank-3 46 & 58 & 270, which is more accurate than Ultimate, altering primes 5-13 by -2, -3, -5, -6 half-valinorsmas. It can be detempered into the ultra-accurate rank-4 {9801/9800, 123201/123200} by observing ibnsinmas, and ends at rank-5 chalmersic.

12de

Maps 7/4 to a major sixth, 11/8 to a fourth, 13/8 to a minor sixth. It is an exotemperament; but... if you can stomach the weirdly mapped 7/4, you can go places. The diaschisma is also tempered out but you go through srutal, and this can be dempered into the best 13-limit temperament: donar. It is much more accurate than Ultimate and 46 & 58 & 270. It alters primes 5-13 by -2, +8, +6, +5 half-valinorsmas. Donar then can be detempered into {9801/9800, 123201/123200} or {4375/4374, 4225/4224}, both to chalmersic.

Through the pontiac route you get Quadrapont, pontiac with 4th octaves, You either way can detemper into donar, or whatever the hell 12de & 53 & 171 is. That thing can only detemper into {4375/4374, 4225/4224} by observing the schisma.