User:Userminusone/Goldis comma

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The "goldis comma", or the golden diesis, is a 5 limit comma that is approximately 50.55 cents in size, which is the amount by which six classic augmented second intervals of 75/64 fall short of 8/3. Its ratio is 549755813888/533935546875, and its monzo is [39 -7 -12. It is the sum of the porcupine comma and the luna comma, the difference between the negri comma and the kwazy comma, and the difference between the passion comma and the semicomma. It is also the difference between 4 augmented commas and a pythagorean chromatic semitone, as well as the difference between a pythagorean whole tone and three negri commas.

Notes on naming

"Goldis" is a contraction of "Golden diesis". The diesis part represents the fact that this comma is close to the size of other diesis. The golden part represents the fact that the temperament tempering out this comma has a generator which is extremely close to logarithmic phi, or 1200/phi cents. As a result of this property, it is mostly tempered out by EDOs in the Fibonacci sequence. These EDOs are 13edo, 21edo, 34edo, 55edo, and 89edo. (144edo doesn't temper out this comma because 144edo is contorted in the 5 limit, meaning it has the same 5 limit patent val as 72edo)


The 5 limit parent temperament, Goldis, has a generator of approximately 458 cents. 5/4 is reached by -7 generators, and 3/2 is reached by +12 generators, making this a rather complex temperament. It is possible to use logarithmic phi (After octave reduction and octave inversion) as a generator for this temperament, but it isn't the most accurate option available. Valid generators for this temperament are between ~457.14 cents and ~458.63 cents. (With the exception of 13edo and 34edo)

Goldis pure fifths generator - 458.496250072 cents

(1200-1200/phi) - 458.3592135 cents

Perhaps the most accurate 7 limit extension of this temperament, which I call Semigoldis, splits the generator in half and maps one step to 8/7. Semigoldis tempers out the breedsma in addition to the goldis comma. The only downside is that this drastically increases the complexity. This temperament is supported by 21edo, 68edo, 89edo, 136edo, and 157edo.

Semigoldis pure fifths generator - 229.248125036 cents

(600-600/phi) - 229.17960675 cents

Curiously enough, Semigoldis naturally extends to the 11 limit by adding the one and only quartisma, which doesn't require the generator to be further split into any number of parts. In addition, 11-limit Semigoldis tempers out the valinorsma. High complexity is the downside for this temperament, as is the case with 7-limit Semigoldis. (The exception is 7/4, which is reached by only -1 generators). 5/4 is reached by -14 generators, 3/2 is reached by +24 generators, and 11/8 is reached by -29 generators. 89edo is a really good tuning for 11-limit Semigoldis, but all the EDOs that support 7-limit Semigoldis also support 11-limit Semigoldis.

11-limit Semigoldis pure 11/8s generator - 229.264898539 cents

temperament finder

Notes on generator ranges

It should be noted that there is an alternate 5/4, reached by +27 generators, which is more accurate than the -7 generator 5/4 whenever the generator is between ~458.63 cents and ~458.82 cents (or 13 steps of 34edo). Generators in this range generate Tetracot (which is contorted by order 3) rather than Goldis.

In addition, there is an alternate 3/2, reached by -22 generators, which is more accurate than the +12 generator 3/2 whenever the generator is between ~458.82 cents (or 13 steps of 34edo) and ~459.61 cents. Generators in this range generate Majvam (or 34&47) rather than Goldis. Majvam has the advantage of being more accurate than Goldis, but the disadvantage of being more complex than Goldis, which is probably why it isn't a very popular temperament.