User:Userminusone/Goldis comma: Difference between revisions

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==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.

The 5 limit parent temperament, Goldis, has a generator of approximately 458 cents. The major third is reached by -7 generators, and the perfect fifth is reached by +12 generators, making this a rather complex temperament. It should be noted that there is an alternate major third, reached by +27 generators, which is more accurate than the -7 generator 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]].

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

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~~.

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

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

[http://x31eq.com/cgi-bin/uv.cgi?uvs=%5B39%2C-7%2C-12%3E&page=2&limit=5 temperament finder]

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 ==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.
+The 5 limit parent temperament, Goldis, has a generator of approximately 458 cents. The major third is reached by -7 generators, and the perfect fifth is reached by +12 generators, making this a rather complex temperament. It should be noted that there is an alternate major third, reached by +27 generators, which is more accurate than the -7 generator 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]].
+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
-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.
+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.
--limit semigoldis pure 11/8s generator - 229.264898539 cents
+-limit Semigoldis pure 11/8s generator - 229.264898539 cents
 [http://x31eq.com/cgi-bin/uv.cgi?uvs=%5B39%2C-7%2C-12%3E&page=2&limit=5 temperament finder]