Goldis comma: Difference between revisions

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{{Infobox Interval
{{Infobox Interval
| Name = Goldis comma
| Name = Goldis comma
| Color name = Trisa-quadtrigu comma
| Ratio = 549755813888/533935546875
| Ratio = 549755813888/533935546875
| Comma = yes
| Comma = yes
}}
}}
 
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]].
 
"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]].)


== 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.)
 
=== Goldis ===
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.
 
Subgroup: 2.3.5
 
Comma list: [[549755813888/533935546875]]
 
{{Mapping|legend=1| 1 9 -2 | 0 -12 7 }}
 
[[Optimal tuning]] ([[CWE]]): 2 = 1\1, ~131072/84375 = 741.381
 
{{Optimal ET sequence|legend=1| 13, 21, 34, 123, 157 }}
 
=== Hemigoldis ===
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.
 
Subgroup: 2.3.5.7


Comma list: [[549755813888/533935546875]], [[2401/2400]]
== 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.  
{{Mapping|legend=1| 1 21 -9 2 | 0 -24 14 1}}
 
[[Optimal tuning]] ([[CWE]]): 2 = 1\1, ~7/4 = 970.690
 
{{Optimal ET sequence|legend=1| 21, 47b, 68, 157 }}


== See also ==
== See also ==
Line 42: Line 17:
* [[Acoustic phi]]
* [[Acoustic phi]]
* [[Golden ratio]]
* [[Golden ratio]]
[[Category:Commas named for the generator of their temperament]]
[[Category:Commas named after their interval size]]

Latest revision as of 16:16, 14 June 2025

Interval information
Ratio 549755813888/533935546875
Factorization 239 × 3-7 × 5-12
Monzo [39 -7 -12⟩
Size in cents 50.55043¢
Name Goldis comma
Color name Trisa-quadtrigu comma
FJS name [math]\displaystyle{ \text{8d5}_{5,5,5,5,5,5,5,5,5,5,5,5} }[/math]
Special properties reduced,
reduced subharmonic
Tenney height (log2 nd) 77.9579
Weil height (log2 max(n, d)) 78
Wilson height (sopfr(nd)) 159
Comma size medium
Open this interval in xen-calc

The goldis comma (ratio: 549755813888/533935546875, monzo: [39 -7 -12⟩) is a 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 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 dieses and a pythagorean chromatic semitone, as well as the difference between a pythagorean whole tone and three negri commas.

Temperaments

Tempering out this comma leads to the goldis temperament. It is mostly supported by edos in the Fibonacci sequence. These are 21edo, 34edo, 55edo, and 89edo. (144edo does not temper out this comma in the patent val, though the 144c val supports it.)

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. The golden part refers to that the temperament tempering out this comma has a generator which is extremely close to logarithmic phi, or 1200/phi cents.

See also

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