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Gammic family: Difference between revisions

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=== Overview to extensions ===
=== Overview to extensions ===
==== 7-limit extensions ====
Because 171 is such a strong [[7-limit]] system, it is well motivated to extend gammic to the 7-limit. This we may do by adding [[4375/4374]] to the comma list. 96 gammic generators finally reach 7, which is a long way to go compared to the 39 generator steps of [[pontiac]]. If someone wants to make the trip, a 103-note generator chain is possible.
Because 171 is such a strong [[7-limit]] system, it is well motivated to extend gammic to the 7-limit. This we may do by adding [[4375/4374]] to the comma list. 96 gammic generators finally reach 7, which is a long way to go compared to the 39 generator steps of [[pontiac]]. If someone wants to make the trip, a 103-note generator chain is possible.
==== Subgroup extensions ====
Gammic also naturally extends with the [[17/1|17th harmonic]], as is given in [[#Subgroup extensions_2|#Subgroup extensions]].


== Septimal gammic ==
== Septimal gammic ==
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Comma list: 243/242, 4375/4356, 100352/99825
Comma list: 243/242, 4375/4356, 100352/99825


Mapping: {{mapping| 1 1 2 0 2 | 0 20 11 96 50 }}
{{Mapping|legend=0| 1 1 2 0 2 | 0 20 11 96 50 }}


Optimal tunings:  
Optimal tunings:  
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Comma list: 243/242, 364/363, 625/624, 2200/2197
Comma list: 243/242, 364/363, 625/624, 2200/2197


Mapping: {{mapping| 1 1 2 0 2 3 | 0 20 11 96 50 24 }}
{{Mapping|legend=0| 1 1 2 0 2 3 | 0 20 11 96 50 24 }}


Optimal tunings:  
Optimal tunings:  
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Comma list: 243/242, 364/363, 375/374, 595/594, 2200/2197
Comma list: 243/242, 364/363, 375/374, 595/594, 2200/2197


Mapping: {{mapping| 1 1 2 0 2 3 4 | 0 20 11 96 50 24 3 }}
{{Mapping|legend=0| 1 1 2 0 2 3 4 | 0 20 11 96 50 24 3 }}


Optimal tunings:  
Optimal tunings:  
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Comma list: 225/224, 243/242, 215622/214375
Comma list: 225/224, 243/242, 215622/214375


Mapping: {{mapping| 1 1 2 1 2 | 0 20 11 62 50 }}
{{Mapping|legend=0| 1 1 2 1 2 | 0 20 11 62 50 }}


Optimal tunings:  
Optimal tunings:  
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Comma list: 225/224, 243/242, 351/350, 1188/1183
Comma list: 225/224, 243/242, 351/350, 1188/1183


Mapping: {{mapping| 1 1 2 1 2 3 | 0 20 11 62 50 24 }}
{{Mapping|legend=0| 1 1 2 1 2 3 | 0 20 11 62 50 24 }}


Optimal tunings:  
Optimal tunings:  
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Comma list: 225/224, 243/242, 351/350, 375/374, 1188/1183
Comma list: 225/224, 243/242, 351/350, 375/374, 1188/1183


Mapping: {{mapping| 1 1 2 1 2 3 4 | 0 20 11 62 50 24 3 }}
{{Mapping|legend=0| 1 1 2 1 2 3 4 | 0 20 11 62 50 24 3 }}


Optimal tunings:  
Optimal tunings:  
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== Neptune ==
== Neptune ==
A more interesting extension is to neptune, which divides an octave plus a gammic generator in half, to get a 10/7 generator. Neptune adds [[2401/2400]] to the gammic comma, and may be described as the 68&171 temperament. The generator chain goes merrily on, stacking one 10/7 over another, until after eighteen generator steps 6/5 (up nine octaves) is reached. Then in succession we get 12/7, the neutral third, 7/4 and 5/4. Two neutral thirds then gives a fifth, and these intervals with their inverses are the full set of [[7-odd-limit]] consonances. [[171edo]] makes a good tuning, and we can also choose to make any of the consonances besides 7/5 and 10/7 just, including the fifth, which gives a tuning extending [[Carlos Gamma]].  
A more interesting extension is to neptune, which divides an octave plus a gammic generator in half, to get a 10/7 generator. Neptune adds [[2401/2400]] to the gammic comma, and may be described as the {{nowrap| 68 & 171 }} temperament. The generator chain goes merrily on, stacking one 10/7 over another, until after eighteen generator steps [[6/5]] (up nine octaves) is reached. Then in succession we get [[12/7]], the neutral third, [[7/4]] and [[5/4]]. Two neutral thirds then gives a fifth, and these intervals with their inverses are the full set of [[7-odd-limit]] consonances. [[171edo]] makes a good tuning, and we can also choose to make any of the consonances besides 7/5 and 10/7 just, including the fifth, which gives a tuning extending [[Carlos Gamma]].  


Adding 385/384 or 1375/1372 to the list of commas allows for an extension to the [[11-limit]], where (7/5)<sup>3</sup> equates to 11/4.  
Adding 385/384 or 1375/1372 to the list of commas allows for an extension to the [[11-limit]], where (7/5)<sup>3</sup> equates to 11/4.  
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Comma list: 1225/1224, 2401/2400, 24576/24565
Comma list: 1225/1224, 2401/2400, 24576/24565


Subgroup-val mapping: {{mapping| 1 -19 -9 -8 1 | 0 40 22 21 6 }}
{{Mapping|legend=2| 1 -19 -9 -8 1 | 0 40 22 21 6 }}


Optimal tunings:  
Optimal tunings:  
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Comma list: 868/867, 961/960, 1225/1224, 2401/2400
Comma list: 868/867, 961/960, 1225/1224, 2401/2400


Subgroup-val mapping: {{mapping| 1 -19 -9 -8 1 -11 | 0 40 22 21 6 31 }}
{{Mapping|legend=2| 1 -19 -9 -8 1 -11 | 0 40 22 21 6 31 }}


Optimal tunings:  
Optimal tunings:  
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Comma list: 385/384, 1375/1372, 78408/78125
Comma list: 385/384, 1375/1372, 78408/78125


Mapping: {{mapping| 1 -19 -9 -8 5 | 0 40 22 21 -3 }}
{{Mapping|legend=0| 1 -19 -9 -8 5 | 0 40 22 21 -3 }}


Optimal tunings:  
Optimal tunings:  
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Comma list: 385/384, 625/624, 1188/1183, 1375/1372
Comma list: 385/384, 625/624, 1188/1183, 1375/1372


Mapping: {{mapping| 1 -19 -9 -8 5 -21 | 0 40 22 21 -3 48 }}
{{Mapping|legend=0| 1 -19 -9 -8 5 -21 | 0 40 22 21 -3 48 }}


Optimal tunings:  
Optimal tunings:  
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Comma list: 385/384, 561/560, 625/624, 715/714, 1188/1183
Comma list: 385/384, 561/560, 625/624, 715/714, 1188/1183


Mapping: {{mapping| 1 -19 -9 -8 5 -21 1 | 0 40 22 21 -3 48 6 }}
{{Mapping|legend=0| 1 -19 -9 -8 5 -21 1 | 0 40 22 21 -3 48 6 }}


Optimal tunings:  
Optimal tunings:  
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Comma list: 243/242, 441/440, 9765625/9732096
Comma list: 243/242, 441/440, 9765625/9732096


Mapping: {{mapping| 1 -19 -9 -8 -48 | 0 40 22 21 100 }}
{{Mapping|legend=0| 1 -19 -9 -8 -48 | 0 40 22 21 100 }}


Optimal tunings:  
Optimal tunings:  
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Comma list: 243/242, 441/440, 625/624, 2200/2197
Comma list: 243/242, 441/440, 625/624, 2200/2197


Mapping: {{mapping| 1 -19 -9 -8 -48 -21 | 0 40 22 21 100 48 }}
{{Mapping|legend=0| 1 -19 -9 -8 -48 -21 | 0 40 22 21 100 48 }}


Optimal tunings:  
Optimal tunings:  
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Comma list: 243/242, 375/374, 441/440, 625/624, 2200/2197
Comma list: 243/242, 375/374, 441/440, 625/624, 2200/2197


Mapping: {{mapping| 1 -19 -9 -8 -48 -21 1 | 0 40 22 21 100 48 6 }}
{{Mapping|legend=0| 1 -19 -9 -8 -48 -21 1 | 0 40 22 21 100 48 6 }}


Optimal tunings:  
Optimal tunings:  
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Comma list: 2401/2400, 9801/9800, 9453125/9437184
Comma list: 2401/2400, 9801/9800, 9453125/9437184


Mapping: {{mapping| 2 2 4 5 8 | 0 40 22 21 -37 }}
{{Mapping|legend=0| 2 2 4 5 8 | 0 40 22 21 -37 }}
: mapping generators: ~99/70, ~99/98
: mapping generators: ~99/70, ~99/98


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Comma list: 15625/15606, 24576/24565
Comma list: 15625/15606, 24576/24565


Subgroup-val mapping: {{mapping| 1 1 2 4 | 0 20 11 3 }}
{{Mapping|legend=2| 1 1 2 4 | 0 20 11 3 }}
: mapping generators: ~2, ~51/50
: mapping generators: ~2, ~51/50


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Badness (Sintel): 0.320
Badness (Sintel): 0.320


[[Category:Gammic family| ]] <!-- main article -->
[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Gammic family| ]] <!-- main article -->
[[Category:Catalogs of rank-2 temperaments]]
[[Category:Rank 2]]
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