User:Userminusone/Goldis comma
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.
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)
Temperaments
The 5 limit parent temperament, Goldis, has a generator of approximately 458 cents. The major third is 7 generators down, and the perfect fifth is 12 generators up, making this a rather complex temperament. It should be noted that there is an alternate major third 21 generators up which is the most accurate major third whenever the generator is between 458.6314 cents and 458.8235 cents (or 13 steps of 34edo). Generators in this range generate Tetracot (which is contorted by order 3) rather than Goldis.
Goldis pure fifths generator - 458.496250072 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