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Gammic family: Difference between revisions

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Switch to Sintel's badness, WE & CWE tunings
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: mapping generators: ~2, ~1990656/1953125
: mapping generators: ~2, ~1990656/1953125


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~1990656/1953125 = 35.0964{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0419{{c}}, ~1990656/1953125 = 35.0977{{c}}
: [[error map]]: {{val| +0.042 +0.399 -0.156 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~1990656/1953125 = 35.0981{{c}}
: error map: {{val| 0.000 +0.008 -0.234 }}


{{Optimal ET sequence|legend=1| 34, 103, 137, 171, 547, 718, 889, 1607 }}
{{Optimal ET sequence|legend=1| 34, 103, 137, 171, 547, 718, 889, 1607 }}


[[Badness]] (Smith): 0.087752
[[Badness]] (Sintel): 2.06


=== 2.3.5.17 subgroup ===
=== 2.3.5.17 subgroup ===
The interval of 3 generators represents one-third of [[6/5]], which is very close to [[17/16]], with the comma between 6/5 and (17/16)<sup>3</sup> being [[24576/24565]] = {{S|16/S17}}. This then naturally interprets the generator as [[51/50]] with two generators representing [[25/24]], tempering out [[15625/15606]] = S49×S50<sup>2</sup>.
The interval of 3 generators represents 1/3 of [[6/5]], which is very close to [[17/16]], with the comma between 6/5 and (17/16)<sup>3</sup> being [[24576/24565]] ([[S-expression|S16/S17]]). This then naturally interprets the generator as [[51/50]] with two generators representing [[25/24]], tempering out [[15625/15606]] (S49⋅S50<sup>2</sup>).


[[Subgroup]]: 2.3.5.17
Subgroup: 2.3.5.17


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


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


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~51/50 = 35.1011{{c}}
Optimal tunings:
* WE: ~2 = 1199.9899{{c}}, ~51/50 = 35.1008{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~51/50 = 35.1008{{c}}


{{Optimal ET sequence|legend=0| 34, 103, 137, 171, 376, 547, 2564g, 3111cg, 3658cgg }}
{{Optimal ET sequence|legend=0| 34, 103, 137, 171, 376, 547 }}


[[Badness]] (Sintel): 0.320
Badness (Sintel): 0.320


== Septimal gammic ==
== Septimal gammic ==
Line 41: Line 47:
{{Mapping|legend=1| 1 1 2 0 | 0 20 11 96 }}
{{Mapping|legend=1| 1 1 2 0 | 0 20 11 96 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~234375/229376 = 35.0904{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0712{{c}}, ~234375/229376 = 35.0924{{c}}
: [[error map]]: {{val| +0.071 -0.035 -0.154 +0.049 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~234375/229376 = 35.0913{{c}}
: error map: {{val| 0.000 -0.130 -0.310 -0.065 }}


{{Optimal ET sequence|legend=1| 34d, 171, 205, 1402, 1573, 1744, 1915 }}
{{Optimal ET sequence|legend=1| 34d, …, 137d, 171, 1402, 1573, 1744, 1915, 2086c, …, 2599c, 5369bccd }}


[[Badness]] (Smith): 0.047362
[[Badness]] (Sintel): 1.20


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


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~45/44 = 35.089{{c}}
Optimal tunings:
* WE: ~2 = 1199.8949{{c}}, ~45/44 = 35.0855{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~45/44 = 35.0872{{c}}


{{Optimal ET sequence|legend=0| 34d, 137d, 171 }}
{{Optimal ET sequence|legend=0| 34d, …, 137d, 171 }}


Badness (Smith): 0.097061
Badness (Sintel): 3.21


=== 13-limit ===
=== 13-limit ===
Line 67: Line 79:
Mapping: {{mapping| 1 1 2 0 2 3 | 0 20 11 96 50 24 }}
Mapping: {{mapping| 1 1 2 0 2 3 | 0 20 11 96 50 24 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~45/44 = 35.091{{c}}
Optimal tunings:
* WE: ~2 = 1199.8098{{c}}, ~45/44 = 35.0855{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~45/44 = 35.0888{{c}}


{{Optimal ET sequence|legend=0| 34d, 137d, 171 }}
{{Optimal ET sequence|legend=0| 34d, 137d, 171 }}


Badness (Smith): 0.047822
Badness (Sintel): 1.98


=== 17-limit ===
=== 17-limit ===
Line 80: Line 94:
Mapping: {{mapping| 1 1 2 0 2 3 4 | 0 20 11 96 50 24 3 }}
Mapping: {{mapping| 1 1 2 0 2 3 4 | 0 20 11 96 50 24 3 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~45/44 = 35.090{{c}}
Optimal tunings:
* WE: ~2 = 1199.8393{{c}}, ~45/44 = 35.0851{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~45/44 = 35.0882{{c}}


{{Optimal ET sequence|legend=0| 34d, 137d, 171 }}
{{Optimal ET sequence|legend=0| 34d, 137d, 171 }}


Badness (Smith): 0.031466
Badness (Sintel): 1.60


== Gammy ==
== Gammy ==
Line 91: Line 107:
[[Comma list]]: 225/224, 94143178827/91913281250
[[Comma list]]: 225/224, 94143178827/91913281250


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


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~1990656/1953125 = 34.984{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.5055{{c}}, ~1990656/1953125 = 34.9984{{c}}
: [[error map]]: {{val| +0.506 -1.482 -0.321 +1.577 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~1990656/1953125 = 34.9947{{c}}
: error map: {{val| 0.000 -2.060 -1.372 +0.848 }}


{{Optimal ET sequence|legend=1| 34d, 69d, 103, 240, 343b }}
{{Optimal ET sequence|legend=1| 34d, 69d, 103, 240, 343b }}


[[Badness]] (Smith): 0.230839
[[Badness]] (Sintel): 5.84


=== 11-limit ===
=== 11-limit ===
Line 106: Line 126:
Mapping: {{mapping| 1 1 2 1 2 | 0 20 11 62 50 }}
Mapping: {{mapping| 1 1 2 1 2 | 0 20 11 62 50 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~45/44 = 34.985{{c}}
Optimal tunings:
* WE: ~2 = 1200.5129{{c}}, ~45/44 = 34.9999{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~45/44 = 34.9967{{c}}


{{Optimal ET sequence|legend=0| 34d, 69de, 103, 240, 343be }}
{{Optimal ET sequence|legend=0| 34d, 69de, 103, 240, 343be }}


Badness (Smith): 0.065326
Badness (Sintel): 2.16


=== 13-limit ===
=== 13-limit ===
Line 119: Line 141:
Mapping: {{mapping| 1 1 2 1 2 3 | 0 20 11 62 50 24 }}
Mapping: {{mapping| 1 1 2 1 2 3 | 0 20 11 62 50 24 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~45/44 = 34.988{{c}}
Optimal tunings:
* WE: ~2 = 1200.4356{{c}}, ~45/44 = 35.0008{{c}}
* CWE: ~2 = 1200.000{{c}}, ~45/44 = 34.9975{{c}}


{{Optimal ET sequence|legend=0| 34d, 69de, 103, 240, 343be }}
{{Optimal ET sequence|legend=0| 34d, 69de, 103, 240, 343be }}


Badness (Smith): 0.033418
Badness (Sintel): 1.38


=== 17-limit ===
=== 17-limit ===
Line 132: Line 156:
Mapping: {{mapping| 1 1 2 1 2 3 4 | 0 20 11 62 50 24 3 }}
Mapping: {{mapping| 1 1 2 1 2 3 4 | 0 20 11 62 50 24 3 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~45/44 = 34.997{{c}}
Optimal tunings:
* WE: ~2 = 1200.2936{{c}}, ~45/44 = 35.0057{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~45/44 = 35.0021{{c}}


{{Optimal ET sequence|legend=0| 34d, 69de, 103, 137, 240 }}
{{Optimal ET sequence|legend=0| 34d, 69de, 103, 137, 240 }}


Badness (Smith): 0.025030
Badness (Sintel): 1.28


== Neptune ==
== Neptune ==
Line 150: Line 176:
[[Comma list]]: 2401/2400, 48828125/48771072
[[Comma list]]: 2401/2400, 48828125/48771072


{{Mapping|legend=1| 1 21 13 13 | 0 -40 -22 -21 }}
{{Mapping|legend=1| 1 -19 -9 -8 | 0 40 22 21 }}
: mapping generators: 2, ~7/5
: mapping generators: 2, ~10/7


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~7/5 = 582.452{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0660{{c}}, ~10/7 = 617.5815{{c}}
: [[error map]]: {{val| +0.066 +0.053 -0.114 -0.141 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 617.5489{{c}}
: error map: {{val| 0.000 +0.000 -0.238 -0.299 }}


{{Optimal ET sequence|legend=1| 35, 68, 103, 171, 1094, 1265, 1436, 1607, 1778 }}
{{Optimal ET sequence|legend=1| 35, 68, 103, 171, 1094, 1265, 1436, 1607, 1778, 1949d, 3727cdd, 5676ccddd }}


[[Badness]] (Smith): 0.023427
[[Badness]] (Sintel): 0.593


==== 2.3.5.7.17 subgroup ====
==== 2.3.5.7.17 subgroup ====
Extending 2.3.5.17 gammic via neptune, we find that both 2401/2400 ({{S|49}}) and 2500/2499 (S50) are tempered out; their product, 1225/1224 (S35) is therefore also tempered out.
Extending 2.3.5.17 gammic via neptune, we find that both 2401/2400 ({{S|49}}) and 2500/2499 (S50) are tempered out; their product, 1225/1224 (S35) is therefore also tempered out.


[[Subgroup]]: 2.3.5.7.17
Subgroup: 2.3.5.7.17


[[Comma list]]: 1225/1224, 2401/2400, 24576/24565
Comma list: 1225/1224, 2401/2400, 24576/24565


{{Mapping|legend=1| 1 21 13 13 7 | 0 -40 -22 -21 -6 }}
Subgroup-val mapping: {{mapping| 1 -19 -9 -8 1 | 0 40 22 21 6 }}
: mapping generators: ~2, ~7/5


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, 7/5 = 582.450{{c}}
Optimal tunings:
* WE: ~2 = 1200.0136{{c}}, ~10/7 = 617.5572{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 617.5504{{c}}


{{Optimal ET sequence|legend=0| 35, 68, 103, 171, 581, 752, 923, 1094 }}
{{Optimal ET sequence|legend=0| 35, 68, 103, 171, 581, 752, 923, 1094 }}


[[Badness]] (Sintel): 0.404
Badness (Sintel): 0.404


==== 2.3.5.7.17.31 subgroup ====
==== 2.3.5.7.17.31 subgroup ====
Since neptune splits the interval of [[5/3]] into two, we can accurately map each part to [[40/31]]~[[31/24]] by tempering out [[961/960]] (S31). This is especially natural, as combined with tempering out 1225/1224 (S35) and 24576/24565 (S16/S17), we can map (17/16)<sup>2</sup> (6 gammic generators) to [[35/31]]. This also gives us its complement with respect to [[5/4]], the interval of 5 gammic generators representing a quarter of a perfect fifth, as [[31/28]].
Since neptune splits the interval of [[5/3]] into two, we can accurately map each part to [[40/31]]~[[31/24]] by tempering out [[961/960]] (S31). This is especially natural, as combined with tempering out 1225/1224 (S35) and 24576/24565 (S16/S17), we can map (17/16)<sup>2</sup> (6 gammic generators) to [[35/31]]. This also gives us its complement with respect to [[5/4]], the interval of 5 gammic generators representing a quarter of a perfect fifth, as [[31/28]].


[[Subgroup]]: 2.3.5.7.17.31
Subgroup: 2.3.5.7.17.31


[[Comma list]]: 868/867, 961/960, 1225/1224, 2401/2400
Comma list: 868/867, 961/960, 1225/1224, 2401/2400


{{Mapping|legend=1| 1 21 13 13 7 20 | 0 -40 -22 -21 -6 -31 }}
Subgroup-val mapping: {{mapping| 1 -19 -9 -8 1 -11 | 0 40 22 21 6 31 }}
: mapping generators: ~2, ~7/5


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, 7/5 = 582.451{{c}}
Optimal tunings:
* WE: ~2 = 1200.0519{{c}}, ~10/7 = 617.5760{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 617.5501{{c}}


{{Optimal ET sequence|legend=0| 35, 68, 103, 171, 752k, 923k, 1094k, 1265kk }}
{{Optimal ET sequence|legend=0| 35, 68, 103, 171, 752k, 923k }}


[[Badness]] (Sintel): 0.393
Badness (Sintel): 0.393


=== 11-limit ===
=== 11-limit ===
Line 196: Line 228:
Comma list: 385/384, 1375/1372, 78408/78125
Comma list: 385/384, 1375/1372, 78408/78125


Mapping: {{mapping| 1 21 13 13 2 | 0 -40 -22 -21 3 }}
Mapping: {{mapping| 1 -19 -9 -8 5 | 0 40 22 21 -3 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~7/5 = 582.475{{c}}
Optimal tunings:
* WE: ~2 = 1200.4655{{c}}, ~10/7 = 617.7648{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 617.5317{{c}}


{{Optimal ET sequence|legend=0| 35, 68, 103, 171e, 274e, 445ee }}
{{Optimal ET sequence|legend=0| 35, 68, 103, 171e, 274e, 445ee }}


Badness (Smith): 0.063602
Badness (Sintel): 2.10


==== 13-limit ====
==== 13-limit ====
Line 209: Line 243:
Comma list: 385/384, 625/624, 1188/1183, 1375/1372
Comma list: 385/384, 625/624, 1188/1183, 1375/1372


Mapping: {{mapping| 1 21 13 13 2 27 | 0 -40 -22 -21 3 -48 }}
Mapping: {{mapping| 1 -19 -9 -8 5 -21 | 0 40 22 21 -3 48 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~7/5 = 582.480{{c}}
Optimal tunings:
* WE: ~2 = 1200.4067{{c}}, ~10/7 = 617.7290{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 617.5257{{c}}


{{Optimal ET sequence|legend=0| 35f, 68, 103, 171e, 274e }}
{{Optimal ET sequence|legend=0| 35f, 68, 103, 171e, 274e }}


Badness (Smith): 0.037156
Badness (Sintel): 1.54


==== 17-limit ====
==== 17-limit ====
Line 222: Line 258:
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 21 13 13 2 27 7 | 0 -40 -22 -21 3 -48 -6 }}
Mapping: {{mapping| 1 -19 -9 -8 5 -21 1 | 0 40 22 21 -3 48 6 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~7/5 = 582.475{{c}}
Optimal tunings:
* WE: ~2 = 1200.2971{{c}}, ~10/7 = 617.6784{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 617.5291{{c}}


{{Optimal ET sequence|legend=0| 35f, 68, 103, 171e, 274e, 445ee }}
{{Optimal ET sequence|legend=0| 35f, 68, 103, 171e, 274e }}


Badness (Smith): 0.025909
Badness (Sintel): 1.32


=== Salacia ===
=== Salacia ===
Line 235: Line 273:
Comma list: 243/242, 441/440, 9765625/9732096
Comma list: 243/242, 441/440, 9765625/9732096


Mapping: {{mapping| 1 21 13 13 52 | 0 -40 -22 -21 -100 }}
Mapping: {{mapping| 1 -19 -9 -8 -48 | 0 40 22 21 100 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~7/5 = 582.478{{c}}
Optimal tunings:
* WE: ~2 = 1200.2180{{c}}, ~10/7 = 617.6341{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 617.5253{{c}}


{{Optimal ET sequence|legend=0| 68e, 103, 171, 274, 719be, 993bcde, 1267bbcde }}
{{Optimal ET sequence|legend=0| 68e, 103, 171, 274 }}


Badness (Smith): 0.069721
Badness (Sintel): 2.30


==== 13-limit ====
==== 13-limit ====
Line 248: Line 288:
Comma list: 243/242, 441/440, 625/624, 2200/2197
Comma list: 243/242, 441/440, 625/624, 2200/2197


Mapping: {{mapping| 1 21 13 13 52 27 | 0 -40 -22 -21 -100 -48 }}
Mapping: {{mapping| 1 -19 -9 -8 -48 -21 | 0 40 22 21 100 48 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~7/5 = 582.477{{c}}
Optimal tunings:
* WE: ~2 = 1200.1492{{c}}, ~10/7 = 617.5993{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 617.5249{{c}}


{{Optimal ET sequence|legend=0| 68e, 103, 171, 274, 719be, 993bcde }}
{{Optimal ET sequence|legend=0| 68e, 103, 171, 274 }}


Badness (Smith): 0.034977
Badness (Sintel): 1.45


==== 17-limit ====
==== 17-limit ====
Line 261: Line 303:
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 21 13 13 52 27 7 | 0 -40 -22 -21 -100 -48 -6 }}
Mapping: {{mapping| 1 -19 -9 -8 -48 -21 1 | 0 40 22 21 100 48 6 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~7/5 = 582.475{{c}}
Optimal tunings:
* WE: ~2 = 1200.0872{{c}}, ~10/7 = 617.5702{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 617.5264{{c}}


{{Optimal ET sequence|legend=0| 68e, 103, 171, 274, 445e, 719be, 1164bcdeef }}
{{Optimal ET sequence|legend=0| 68e, 103, 171, 274, 445e }}


Badness (Smith): 0.024577
Badness (Sintel): 1.25


=== Poseidon ===
=== Poseidon ===
Line 277: Line 321:
: mapping generators: ~99/70, ~99/98
: mapping generators: ~99/70, ~99/98


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~99/98 = 17.545{{c}}
Optimal tunings:  
* WE: ~99/70 = 600.0509{{c}}, ~99/98 = 17.5466{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~99/98 = 17.5458{{c}}


{{Optimal ET sequence|legend=0| 68, 206b, 274, 342 }}
{{Optimal ET sequence|legend=0| 68, 206b, 274, 342, 2804cdee, 3146cdee, …, 5198bccdddeeee }}


Badness (Smith): 0.041727
Badness (Sintel): 1.38


[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Gammic family| ]] <!-- main article -->
[[Category:Gammic family| ]] <!-- main article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Gammic_family"