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| | | The '''goldis comma''' ([[ratio]]: 549755813888/533935546875, {{monzo|legend=1| 39 -7 -12 }}) is a [[medium comma|medium]] [[5-limit]] [[comma]] which is the amount by which six classic augmented second intervals of [[75/64]] fall short of [[8/3]]. It is the sum of the [[250/243|porcupine comma]] (a.k.a. maximal diesis) 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 [[128/125|dieses]] 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]]. |
| The '''goldis comma''' 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]] (a.k.a. maximal diesis) and the [[Luna family|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 [[128/125|dieses]] 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]]. | |
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| "Goldis" is a contraction of "Golden diesis". The diesis part represents the fact that this comma is close to the size of a [[diesis]]. The golden part refers to that the temperament tempering out this comma has a generator which is extremely close to [[Golden ratio|logarithmic phi]], or 1200/phi cents. As a result of this property, it is mostly tempered out by edos in the Fibonacci sequence. These 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]], though the [[warts|144c val]] supports it.)
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| == Temperaments == | | == Temperaments == |
| Despite being a quarter-tone in size, due to its complexity, the damage is spread out, so that simple intervals of the [[5-limit]] tend to be tuned reasonably. Of the edos aforementioned, [[34edo]] is an especially good and tone-efficient tuning (also evidenced by being the largest "golden edo" appearing in the [[optimal ET sequence]]), [[55edo]] is good for combining it with an approximation of [[1/6-comma meantone]] that closes after 55 notes so that 5/4 is slightly more in tune, and [[89edo]] is an overlooked [[nestoria]] tuning supporting it (though it's very flat for a nestoria tuning).
| | [[Tempering out]] this comma leads to the [[goldis]] temperament. It is mostly [[support]]ed by [[edo]]s in the Fibonacci sequence. These are [[21edo]], [[34edo]], [[55edo]], and [[89edo]]. ([[144edo]] does not temper out this comma in the [[patent val]], though the [[warts|144c val]] supports it.) |
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| === Goldis ===
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| As the generator doesn't admit a plausible interpretation in the [[5-limit]], a number of extensions are possible. One possibility is to notice that the generator is close to [[49/32]], resulting in [[hemigoldis]], which splits the generator in half.
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| Subgroup: 2.3.5
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| Comma list: [[549755813888/533935546875]]
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| {{Mapping|legend=1| 1 9 -2 | 0 -12 7 }}
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| [[Optimal tuning]] ([[CWE]]): 2 = 1\1, ~131072/84375 = 741.381
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| {{Optimal ET sequence|legend=1| 13, 21, 34, 123, 157 }}
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| === Hemigoldis ===
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| Though fairly complex in the [[7-limit]], hemigoldis does a lot better in badness metrics than pure 5-limit goldis, and yet again has many possible extensions to other primes. For example, two periods minus six generators yields a "tetracot 2nd" which can be interpreted as ~[[21/19]] to add prime 19 or perhaps more accurately ~[[31/28]] to add prime 7, or even simply as ~[[32/29]] to add prime 29, though the other two have the benefit of clearly connecting to the 7-limit representation. Note that again [[89edo]] is a possible tuning for combining it with flat nestoria and not appearing in the optimal ET sequence.
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| Subgroup: 2.3.5.7
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| Comma list: [[549755813888/533935546875]], [[2401/2400]]
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| {{Mapping|legend=1| 1 21 -9 2 | 0 -24 14 1}}
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| [[Optimal tuning]] ([[CWE]]): 2 = 1\1, ~7/4 = 970.690
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| {{Optimal ET sequence|legend=1| 21, 47b, 68, 157 }}
| | == Etymology == |
| | This comma was named by [[Userminusone]] as a contraction of ''golden diesis''. The diesis part represents the fact that this comma is close to the size of a [[diesis (interval region)|diesis]]. The golden part refers to that the temperament tempering out this comma has a generator which is extremely close to [[golden ratio|logarithmic phi]], or 1200/phi cents. |
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| == See also == | | == See also == |
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| * [[Golden ratio]] | | * [[Golden ratio]] |
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| [[Category:Commas named for their regular temperament properties]] | | [[Category:Commas named for the generator of their temperament]] |
| | [[Category:Commas named after their interval size]] |