Goldis comma
| Interval information |
reduced subharmonic
(Shannon, [math]\sqrt{nd}[/math])
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 [39 -7 -12⟩. 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.
"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 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 144c val supports it.)
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).
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: [⟨1 9 -2], ⟨0 -12 7]]
Optimal tuning (CWE): 2 = 1\1, ~131072/84375 = 741.381
Optimal ET sequence: 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
Mapping: [⟨1 21 -9 2], ⟨0 -24 14 1]]
Optimal tuning (CWE): 2 = 1\1, ~7/4 = 970.690
Optimal ET sequence: 21, 47b, 68, 157