Schismatic family: Difference between revisions
No edit summary |
|||
| Line 33: | Line 33: | ||
=== Overview to extensions === | === Overview to extensions === | ||
The second comma of the [[Normal lists #Normal interval list|normal comma list]] defines which 7-limit family member we are looking at. | The second comma of the [[Normal lists #Normal interval list|normal comma list]] defines which 7-limit family member we are looking at. | ||
* [[#Garibaldi]] adds [[garischisma|{{monzo| 25 -14 0 -1 }}]], | * [[#Garibaldi|Garibaldi]] adds [[garischisma|{{monzo| 25 -14 0 -1 }}]], | ||
* [[#Grackle]] adds {{monzo| -44 26 0 1 }}, | * [[#Grackle|Grackle]] adds {{monzo| -44 26 0 1 }}, | ||
* [[#Schism]] adds [[64/63|{{monzo| 6 -2 0 -1 }}]], | * [[#Schism|Schism]] adds [[64/63|{{monzo| 6 -2 0 -1 }}]], | ||
* [[#Pontiac]] adds {{monzo| -59 39 0 -1 }}. | * [[#Pontiac|Pontiac]] adds {{monzo| -59 39 0 -1 }}. | ||
Those all have a fifth as generator. | Those all have a fifth as generator. | ||
* [[#Bischismic]] adds {{monzo| -69 40 0 2 }} and has a fifth generator with a half-octave period. | * [[#Bischismic|Bischismic]] adds {{monzo| -69 40 0 2 }} and has a fifth generator with a half-octave period. | ||
* [[#Hemischis]] adds {{monzo| -34 25 0 -2 }} and has a hemififth generator. | * [[#Hemischis|Hemischis]] adds {{monzo| -34 25 0 -2 }} and has a hemififth generator. | ||
* [[#Guiron]] adds [[1029/1024|{{monzo| -10 1 0 3 }}]], with an ~8/7 generator, three of which give the fifth. | * [[Gamelismic clan #Guiron|Guiron]] adds [[1029/1024|{{monzo| -10 1 0 3 }}]], with an ~8/7 generator, three of which give the fifth. | ||
* [[#Term]] adds {{monzo| -94 54 0 3 }} with a 1/3 octave period. | * [[#Term|Term]] adds {{monzo| -94 54 0 3 }} with a 1/3 octave period. | ||
* [[#Sesquiquartififths]] adds {{monzo| -35 15 0 4 }} and slices the fifth in four. | * [[#Sesquiquartififths|Sesquiquartififths]] adds {{monzo| -35 15 0 4 }} and slices the fifth in four. | ||
Temperaments discussed elsewhere include | Temperaments discussed elsewhere include | ||
| Line 1,591: | Line 1,591: | ||
[[Optimal tuning]] ([[CTE]]): ~1536/1225 = 1\3, ~34300/19683 = 950.9654 | [[Optimal tuning]] ([[CTE]]): ~1536/1225 = 1\3, ~34300/19683 = 950.9654 | ||
{{Optimal ET sequence|legend=1| 24, 135, 159, 612ccdd }} | {{Optimal ET sequence|legend=1| 24, …, 111c, 135, 159, 612ccdd, 771ccdd }} | ||
[[Badness]]: 0. | [[Badness]]: 0.422026 | ||
=== 11-limit === | === 11-limit === | ||
| Line 1,602: | Line 1,602: | ||
Mapping: {{mapping| 3 0 45 -32 8 | 0 2 -16 17 1 }} | Mapping: {{mapping| 3 0 45 -32 8 | 0 2 -16 17 1 }} | ||
Optimal tuning (CTE): ~44/35 = 1\3, ~121/70 = 950. | Optimal tuning (CTE): ~44/35 = 1\3, ~121/70 = 950.9658 | ||
{{Optimal ET sequence | Optimal ET sequence: {{Optimal ET sequence| 24, …, 111c, 135, 159, 612ccdd, 771ccdd }} | ||
Badness: 0. | Badness: 0.101224 | ||
=== 13-limit === | |||
Subgroup: 2.3.5.7.11.13 | |||
Comma list: 364/363, 385/384, 676/675, 19712/19683 | |||
Mapping: {{mapping| 3 0 45 -32 8 42 | 0 2 -16 17 1 -13 }} | |||
Optimal tuning (CTE): ~44/35 = 1\3, ~26/15 = 950.9360 | |||
Optimal ET sequence: {{Optimal ET sequence| 24, …, 111cf, 135f, 159 }} | |||
Badness: 0.054894 | |||
== Sesquiquartififths == | == Sesquiquartififths == | ||
| Line 2,313: | Line 2,326: | ||
Badness: 0.030425 | Badness: 0.030425 | ||
== Nonant == | |||
The ''nonant'' temperament (36 & 135) has a period of 1/9 octave and tempers out the [[septimal ennealimma]], {{monzo| -11 -9 0 9 }}. | |||
[[Subgroup]]: 2.3.5.7 | |||
[[Comma list]]: 32805/32768, 40353607/40310784 | |||
{{Mapping|legend=1| 9 0 135 11 | 0 1 -8 1 }} | |||
: mapping generators: ~2592/2401, ~3 | |||
[[Optimal tuning]] ([[CTE]]): ~2592/2401 = 1\9, ~3/2 = 701.7232 | |||
{{Optimal ET sequence|legend=1| 36, 99c, 135, 171 }} | |||
[[Badness]]: 0.069896 | |||
=== 11-limit === | |||
Subgroup: 2.3.5.7.11 | |||
Comma list: 540/539, 32805/32768, 42875/42592 | |||
Mapping: {{mapping| 9 0 135 11 131 | 0 1 -8 1 -7 }} | |||
Optimal tuning (CTE): ~242/225 = 1\9, ~3/2 = 701.8398 | |||
Optimal ET sequence: {{Optimal ET sequence| 36, 99c, 135, 171, 477ce, 648cee }} | |||
Badness: 0.126910 | |||
=== 13-limit === | |||
Subgroup: 2.3.5.7.11.13 | |||
Comma list: 540/539, 729/728, 4096/4095, 16807/16731 | |||
Mapping: {{mapping| 9 0 135 11 131 -38 | 0 1 -8 1 -7 5 }} | |||
Optimal tuning (CTE): ~242/225 = 1\9, ~3/2 = 701.7998 | |||
Optimal ET sequence: {{Optimal ET sequence| 36, 99cf, 135, 171 }} | |||
Badness: 0.076195 | |||
== Tridecafifths == | == Tridecafifths == | ||
| Line 2,329: | Line 2,385: | ||
{{Optimal ET sequence|legend=1| 89, 200, 289 }} | {{Optimal ET sequence|legend=1| 89, 200, 289 }} | ||
[[Badness]]: 0. | [[Badness]]: 0.432580 | ||
=== 11-limit === | === 11-limit === | ||
| Line 2,340: | Line 2,396: | ||
Optimal tuning (CTE): ~2 = 1\1, ~33/32 = 53.9744 | Optimal tuning (CTE): ~2 = 1\1, ~33/32 = 53.9744 | ||
{{Optimal ET sequence | Optimal ET sequence: {{Optimal ET sequence| 89, 200, 289 }} | ||
Badness: 0. | Badness: 0.127820 | ||
[[Category:Temperament families]] | [[Category:Temperament families]] | ||