4L 3s/Temperaments: Difference between revisions

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Created page with "== Myna (27&31) == == Kleismic (19&15, 2.3.5.7) == == Orgone (15&11, 2.7.11) == == Sixix (18&25) == Sixix can be viewed as a dual-fifth temperaments|dual-fifth temperame..."
 
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== Myna (27&31) ==
== Myna ==
== Kleismic (19&15, 2.3.5.7) ==
Subgroup: 2.3.5.7
== Orgone (15&11, 2.7.11) ==
 
== Sixix (18&25) ==
[[Comma list]]: 126/125, 1728/1715
[[Sixix]] can be viewed as a [[dual-fifth temperaments|dual-fifth temperament]], i.e. a temperament on the 2.3+.3-.5 "subgroup" (3+ = sharp 3, 3- = flat 3):
 
* It has both a sharp fifth and a flat fifth but no near-just 3/2.  
[[Mapping]]: [{{val| 1 9 9 8 }}, {{val| 0 -10 -9 -7 }}]
* Combining the sharp fifth and the flat fifth yields a good approximation of 9/8; two 9/8's make a 5/4, so it tempers out 81/80 in the underlying 2.9.5 subgroup.
 
* The chroma of sixix[7] is the difference between the sharp fifth and the flat fifth, and functions much like a(n untempered) comma in sixix harmony, giving two slightly different flavors of fifths, minor thirds, major thirds, etc, much like in [[porcupine]] harmony. Tempering out this comma leads to [[7edo]].
Mapping generators: ~2, ~5/3
 
[[POTE generator]]: ~6/5 = 310.146
 
[[Minimax tuning]]:
* 7- and [[9-odd-limit]]
: [{{monzo| 1 0 0 0 }}, {{monzo| 0 1 0 0 }}, {{monzo| 9/10 9/10 0 0 }}, {{monzo| 17/10 7/10 0 0 }}]
: [[Eigenmonzo]]s (unchanged-intervals): 2, 3
 
{{Optimal ET sequence|legend=1| 27, 31, 58, 89 }}
 
[[Badness]]: 0.0270
 
== Kleismic ==
Subgroup: 2.3.5
 
[[Comma list]]: 15625/15552
 
[[Mapping]]: [{{val| 1 0 1 }}, {{val|0 6 5 }}]
 
[[POTE generator]]: ~6/5 = 317.007
 
[[Tuning ranges]]:
* [[diamond monotone]] range: [300.000, 327.273] (4 to 11b)
* [[diamond tradeoff]] range: [315.641, 317.263]
* diamond monotone and tradeoff range: [315.641, 317.263]
 
{{Optimal ET sequence|legend=1| 15, 19, 34, 53, 458, 511c, …, 882c }}
 
== Orgone ==
{{main|Orgone}}
Commas:65536/65219
 
Subgroup: 2.7.11
 
POTE generator: ~77/64 = 323.372
 
Sval mapping: [<1 2 4|, <0 3 -2|]
 
EDOs: {{EDOs|7, 11, 15, 26, 37, 63, 89, 115, 141, 167, 308, 475bc, 783bc}}
 
== Dual-3 Sixix ==
Subgroup: 2.3⁻.9.5
 
Comma list: 81/80, {{monzo| 2 -3 0 1 }}
 
Mapping: [{{val| 1 2 4 4 }}, {{val| 0 -2 -3 -6 }}]
 
2.9.5 POTE generator: 335.8409
 
{{Optimal ET sequence|legend=1| 18, 25, 43 }}
== Sixix ==
 
Subgroup: 2.3.5
 
Comma list: 3125/2916
 
Mapping: [{{val| 1 3 4 }}, {{val| 0 -5 -6 }}]
 
POTE generator: ~6/5 = 338.365
 
{{Optimal ET sequence|legend=1| 7, 25, 32 }}
 
Badness: 0.1531
 
[[Category:Smitonic|T]]
[[Category:Lists of temperaments]]

Latest revision as of 04:47, 12 June 2025

Myna

Subgroup: 2.3.5.7

Comma list: 126/125, 1728/1715

Mapping: [⟨1 9 9 8], ⟨0 -10 -9 -7]]

Mapping generators: ~2, ~5/3

POTE generator: ~6/5 = 310.146

Minimax tuning:

[[1 0 0 0⟩, [0 1 0 0⟩, [9/10 9/10 0 0⟩, [17/10 7/10 0 0⟩]
Eigenmonzos (unchanged-intervals): 2, 3

Optimal ET sequence: 27, 31, 58, 89

Badness: 0.0270

Kleismic

Subgroup: 2.3.5

Comma list: 15625/15552

Mapping: [⟨1 0 1], ⟨0 6 5]]

POTE generator: ~6/5 = 317.007

Tuning ranges:

Optimal ET sequence: 15, 19, 34, 53, 458, 511c, …, 882c

Orgone

Main article: Orgone

Commas:65536/65219

Subgroup: 2.7.11

POTE generator: ~77/64 = 323.372

Sval mapping: [<1 2 4|, <0 3 -2|]

EDOs: 7, 11, 15, 26, 37, 63, 89, 115, 141, 167, 308, 475bc, 783bc

Dual-3 Sixix

Subgroup: 2.3⁻.9.5

Comma list: 81/80, [2 -3 0 1⟩

Mapping: [⟨1 2 4 4], ⟨0 -2 -3 -6]]

2.9.5 POTE generator: 335.8409

Optimal ET sequence: 18, 25, 43

Sixix

Subgroup: 2.3.5

Comma list: 3125/2916

Mapping: [⟨1 3 4], ⟨0 -5 -6]]

POTE generator: ~6/5 = 338.365

Optimal ET sequence: 7, 25, 32

Badness: 0.1531

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