24576/24565: Difference between revisions
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== Temperaments == | == Temperaments == | ||
=== | === Mavka === | ||
By tempering out this comma in the full [[17-limit]], the rank-6 '''mavka temperament''' is defined. You may find a list of good equal temperaments supporting it below. | By tempering out this comma in the full [[17-limit]], the rank-6 '''mavka temperament''' is defined. You may find a list of good equal temperaments supporting it below. | ||
[[Subgroup]]: 2.3.5.7.11.13.17 | [[Subgroup]]: 2.3.5.7.11.13.17 | ||
[[Mapping]]: | [[Mapping]]:<br> | ||
[{{val| 1 0 1 0 0 0 4 }}<br> | |||
[ | {{val| 0 1 1 0 0 0 0 }}<br> | ||
{{val| 0 0 3 0 0 0 -1 }}<br> | |||
{{val| 0 0 0 1 0 0 0 }}<br> | |||
{{val| 0 0 0 0 1 0 0 }}<br> | |||
{{val| 0 0 0 0 0 1 0 }}] | |||
{{Val list|legend=1| 46, 58, 80, 103, 137, 149, 159, 171, 183, 217, 296, 320, 342f, 354, 400, 422, 525, 571, 581, 742, 764, 935, 1084, 1106, 1323, 1506, 3593g, 3947eg, 5053fgg, 6559defgg, 8065cdefggg, 10152cdeffgggg }}. | {{Val list|legend=1| 46, 58, 80, 103, 137, 149, 159, 171, 183, 217, 296, 320, 342f, 354, 400, 422, 525, 571, 581, 742, 764, 935, 1084, 1106, 1323, 1506, 3593g, 3947eg, 5053fgg, 6559defgg, 8065cdefggg, 10152cdeffgggg }}. | ||
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Subgroup: 2.75.85 | Subgroup: 2.75.85 | ||
Comma list: {{monzo| 13 1 -3 }} = 24576/24565 | |||
Sval mapping: [{{val| 1 2 5 }}], {{val| 0 3 1 }}] | |||
Sval mapping generators: ~2, ~85/32 | |||
Optimal GPV sequence: {{Val list| 5, 17, 22, 61, 83 }} | |||
=== Srutal archagall === | === Srutal archagall === | ||
Named because this lower-accuracy temperament is also an extension of (the 5-limit) [[srutal]] temperament that adds prime 17 (and which thereby is able to express the harmonics 75 and 85 in their appropriate prime subgroup). It achieves this by equating [[85/64]] with [[4/3]] by tempering their difference of S16 = [[256/255]]. Therefore it also tempers S17 = [[289/288]] and thus equates [[17/15]] with [[9/8]] due to tempering S16 | Named because this lower-accuracy temperament is also an extension of (the 5-limit) [[srutal]] temperament that adds prime 17 (and which thereby is able to express the harmonics 75 and 85 in their appropriate prime subgroup). It achieves this by equating [[85/64]] with [[4/3]] by tempering their difference of S16 = [[256/255]]. Therefore it also tempers S17 = [[289/288]] and thus equates [[17/15]] with [[9/8]] due to tempering S16 × S17. It could be described as the 10 & 12 temperament (with strong emphasis on [[12edo]] being the better tuning) on the following prior-discussed subgroup: | ||
Subgroup: 2.3.5.17 | Subgroup: 2.3.5.17 | ||
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Comma list: 256/255, 289/288 | Comma list: 256/255, 289/288 | ||
Sval mapping: [{{val| 2 0 11 5 }}], {{val| 0 1 -2 1 }}] | |||
Sval mapping generators: ~17/12, ~3 | |||
Optimal tuning (CTE): ~17/12 = 1\2, ~3/2 = 705.1272 | |||
Optimal GPV sequence: {{Val list| 10, 12, 22, 34, 80, 114, 194bc }} | |||
Badness: 0.00575 | |||
[[Category:Mavka]] | [[Category:Mavka]] | ||
Revision as of 04:32, 1 December 2022
| Interval information |
archagallisma
24576/24565, the mavka comma or archagallisma, is an unnoticeable 17-limit comma that represents the difference between two adjacent square superparticulars – 289/288 and 256/255, therefore, it is the amount by which a stack of three 17/16's fall short of a 6/5 minor third. It is also the amount by which a stack of two 128/85's octave-reduced exceeds 17/15 and the amount by which a stack of three 85//64's octave-reduced falls short of 75/64.
It can be factored into 4096/4095 × 4914/4913.
Temperaments
Mavka
By tempering out this comma in the full 17-limit, the rank-6 mavka temperament is defined. You may find a list of good equal temperaments supporting it below.
Subgroup: 2.3.5.7.11.13.17
Mapping:
[⟨1 0 1 0 0 0 4]
⟨0 1 1 0 0 0 0]
⟨0 0 3 0 0 0 -1]
⟨0 0 0 1 0 0 0]
⟨0 0 0 0 1 0 0]
⟨0 0 0 0 0 1 0]]
Archagall
By tempering it out in the 2.75.85 subgroup, we have three 85/64's up and one octave down as a 75/64 and we have two 128/85's up and one octave down as a 17/15 whole tone. This is a highly accurate temperament that could be considered to be encoding the "high accuracy logic" of superpyth and which is inescapably related to the 17L 5s scale form as it is the 17 & 22 temperament (or less accurately, the 5 & 17 temperament) in the following subgroup:
Subgroup: 2.75.85
Comma list: [13 1 -3⟩ = 24576/24565
Sval mapping: [⟨1 2 5]], ⟨0 3 1]]
Sval mapping generators: ~2, ~85/32
Optimal GPV sequence: Template:Val list
Srutal archagall
Named because this lower-accuracy temperament is also an extension of (the 5-limit) srutal temperament that adds prime 17 (and which thereby is able to express the harmonics 75 and 85 in their appropriate prime subgroup). It achieves this by equating 85/64 with 4/3 by tempering their difference of S16 = 256/255. Therefore it also tempers S17 = 289/288 and thus equates 17/15 with 9/8 due to tempering S16 × S17. It could be described as the 10 & 12 temperament (with strong emphasis on 12edo being the better tuning) on the following prior-discussed subgroup:
Subgroup: 2.3.5.17
Comma list: 256/255, 289/288
Sval mapping: [⟨2 0 11 5]], ⟨0 1 -2 1]]
Sval mapping generators: ~17/12, ~3
Optimal tuning (CTE): ~17/12 = 1\2, ~3/2 = 705.1272
Optimal GPV sequence: Template:Val list
Badness: 0.00575