User:Userminusone/Goldis comma: Difference between revisions

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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 {{monzo| 39 -7 -12 }}. It is the sum of the [[250/243|porcupine comma]] and the [[Luna family|luna comma]], the difference between the [[negri comma]] and the [[Very high accuracy temperaments|kwazy comma]], and the difference between the [[Passion|passion comma]] and the [[semicomma]].

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 {{monzo| 39 -7 -12 }}. It is the sum of the [[250/243|porcupine comma]] and the [[Luna family|luna comma]], the difference between the [[negri comma]] and the [[Very high accuracy temperaments|kwazy comma]], and the difference between the [[Passion|passion comma]] and the [[semicomma]]. It is also the difference between 4 [[128/125|augmented commas]] and a [[2187/2048|pythagorean chromatic semitone]], as well as the difference between a [[9/8|pythagorean whole tone]] and three [[negri comma|negri commas]].

==Notes on naming==

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==Temperaments==

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 ~~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~~.~~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. [[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 [[Golden ratio|logarithmic phi]] (After [[octave reduction]] and [[Octave complement|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.

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[http://x31eq.com/cgi-bin/uv.cgi?uvs=%5B39%2C-7%2C-12%3E&page=2&limit=5 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 [http://x31eq.com/cgi-bin/rt.cgi?ets=34%2647&limit=5 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.

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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 {{monzo| 39 -7 -12 }}. It is the sum of the [[250/243|porcupine comma]] and the [[Luna family|luna comma]], the difference between the [[negri comma]] and the [[Very high accuracy temperaments|kwazy comma]], and the difference between the [[Passion|passion comma]] and the [[semicomma]].
+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 {{monzo| 39 -7 -12 }}. It is the sum of the [[250/243|porcupine comma]] and the [[Luna family|luna comma]], the difference between the [[negri comma]] and the [[Very high accuracy temperaments|kwazy comma]], and the difference between the [[Passion|passion comma]] and the [[semicomma]]. It is also the difference between 4 [[128/125|augmented commas]] and a [[2187/2048|pythagorean chromatic semitone]], as well as the difference between a [[9/8|pythagorean whole tone]] and three [[negri comma|negri commas]].
 ==Notes on naming==
@@ Line 7: / Line 7: @@
 ==Temperaments==
-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 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 458.6314 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. [[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 [[Golden ratio|logarithmic phi]] (After [[octave reduction]] and [[Octave complement|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.
@@ Line 20: / Line 22: @@
 [http://x31eq.com/cgi-bin/uv.cgi?uvs=%5B39%2C-7%2C-12%3E&page=2&limit=5 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 [http://x31eq.com/cgi-bin/rt.cgi?ets=34%2647&limit=5 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.