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Breedsmic–syntonic equivalence continuum: Difference between revisions

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{{Mathematical interest}}
The '''breedsmic–syntonic equivalence continuum''' is a [[equivalence continuum|continuum]] of 7-limit temperament families which equate a number of [[2401/2400|breedsmas (2401/2400)]] with a [[81/80|syntonic comma (81/80)]]. This continuum is theoretically interesting in that these are all 7-limit temperament families supported by [[Meantone family#Squares|squares]] temperament. In addition, 81/80 and 2401/2400 are the smallest 5-limit and 7-limit [[superparticular]] intervals to be tempered out by [[31edo]].
The '''breedsmic–syntonic equivalence continuum''' is a [[equivalence continuum|continuum]] of 7-limit temperament families which equate a number of [[2401/2400|breedsmas (2401/2400)]] with a [[81/80|syntonic comma (81/80)]]. This continuum is theoretically interesting in that these are all 7-limit temperament families supported by [[Meantone family#Squares|squares]] temperament. In addition, 81/80 and 2401/2400 are the smallest 5-limit and 7-limit [[superparticular]] intervals to be tempered out by [[31edo]].


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|-
|-
| −4
| −4
| 217 & 31 & 14c
| 217 & 31 & 14c
|  
|  
| {{monzo|-24 0 -9 16}}
| {{Monzo| -24 0 -9 16 }}
|-
|-
| −3
| −3
| 159 & 31 & 14c
| 159 & 31 & 14c
|  
|  
| {{monzo|-19 1 -7 12}}
| {{Monzo| -19 1 -7 12 }}
|-
|-
| −2
| −2
| 87 & 31 & 14c
| 87 & 31 & 14c
| 51883209/51200000
| 51883209/51200000
| {{monzo|-14 2 -5 8}}
| {{Monzo| -14 2 -5 8 }}
|-
|-
| −1
| −1
| [[Squalentine]]
| [[Squalentine]]
| [[64827/64000]]
| [[64827/64000]]
| {{monzo|-9 3 -3 4}}
| {{Monzo| -9 3 -3 4 }}
|-
|-
| 0
| 0
| [[Didymus rank three family|Didymus]]
| [[Didymus]]
| [[81/80]]
| [[81/80]]
| {{monzo|-4 4 -1}}
| {{Monzo| -4 4 -1 }}
|-
|-
| 1
| 1
| [[Nuwell family|Nuwell]]
| [[Nuwell]]
| [[2430/2401]]
| [[2430/2401]]
| {{monzo|1 5 1 -4}}
| {{Monzo| 1 5 1 -4 }}
|-
|-
| 2
| 2
| 14c & 31 & 80
| 14c & 31 & 80
| 5832000/5764801
| 5832000/5764801
| {{monzo|6 6 3 -8}}
| {{Monzo| 6 6 3 -8 }}
|-
|-
| 3
| 3
| 14c & 31 & 152
| 14c & 31 & 152
| 13996800000/13841287201
| 13996800000/13841287201
| {{monzo|11 7 5 -12}}
| {{Monzo| 11 7 5 -12 }}
|-
|-
| 4
| 4
| 14c & 31 & 224
| 14c & 31 & 224
|  
|  
| {{monzo|16 8 7 -16}}
| {{Monzo| 16 8 7 -16 }}
|-
|-
| 5
| 5
| 265 & 31 & 282
| 265 & 31 & 282
|  
|  
| {{monzo|21 9 9 -20}}
| {{Monzo| 21 9 9 -20 }}
|-
|-
| 6
| 6
| 14c & 31 & 323
| 14c & 31 & 323
|  
|  
| {{monzo|26 10 11 -24}}
| {{Monzo| 26 10 11 -24 }}
|-
|-
| 7
| 7
| 17c & 395 & 364
| 17c & 395 & 364
|  
|  
| {{monzo|31 11 13 -28}}
| {{Monzo| 31 11 13 -28 }}
|-
|-
| 8
| 8
| 14c & 31 & 422
| 14c & 31 & 422
|  
|  
| {{monzo|36 12 15 -32}}
| {{Monzo| 36 12 15 -32 }}
|-
|-
| …
| …
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|-
|-
| 30
| 30
| 1677 & 6691 & 41854
| 1677 & 6691 & 41854
|  
|  
| {{monzo|146 34 59 -120}}
| {{Monzo| 146 34 59 -120 }}
|-
|-
| …
| …
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| [[Breedsmic temperaments|Breedsmic]]
| [[Breedsmic temperaments|Breedsmic]]
| [[2401/2400]]
| [[2401/2400]]
| {{monzo| -5 -1 -2 4}}
| {{Monzo| -5 -1 -2 4 }}
|}
|}


Examples of temperaments with fractional values of ''n'':
Examples of temperaments with fractional values of ''n'':
* 34p & 31 & 14c ({{nowrap|''n'' {{=}} −{{frac|1|2}} {{=}} −0.5}})
* 34 & 31 & 14c ({{nowrap|''n'' {{=}} −{{frac|1|2}} {{=}} −0.5}})
* [[Subgroup temperaments#Skwares 3|Skwares]] ({{nowrap|''n'' {{=}} {{frac|1|2}} {{=}} 0.5}})
* [[Subgroup temperaments#Skwares 3|Skwares]] ({{nowrap|''n'' {{=}} {{frac|1|2}} {{=}} 0.5}})


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Comma list: {{monzo| 146 34 59 -120 }}
Comma list: {{monzo| 146 34 59 -120 }}


POTE generators: 1901.9549, -775.6679
POTE generators: ~3 = 1901.9549, 775.6679


Mapping: [{{val| 1 0 26 14 }}, {{val| 0 1 34 17 }}, {{val| 0 0 120 59 }}]
Mapping: [{{val| 1 0 26 14 }}, {{val| 0 1 34 17 }}, {{val| 0 0 -120 -59 }}]


{{Optimal ET sequence|legend=1| 1677, 5014, 6691, 41854, 43531, 46868, 48545, 50222, 55236, 97090 }}
{{Optimal ET sequence|legend=1| 1677, 5014, 6691, 41854, 43531, 46868, 48545, 50222, 55236, 97090 }}


[http://x31eq.com/cgi-bin/rt.cgi?ets=1677_6691_41854&limit=7 The temperament finder - 7-limit 1677 & 6691 & 41854]
Badness (Sintel): 475.7


== 34p & 31 & 14c ==
== 34 & 31 & 14c ==
Comma list: {{monzo| -13 7 -4 4 }} = 5250987/5120000
Comma list: {{monzo| -13 7 -4 4 }} = 5250987/5120000


POTE generators: -425.3382, ~5 = 2785.5671
POTE generators: ~80/63 = 425.3382, ~5 = 2785.5671


Mapping: [{{val| 1 3 0 -2 }}, {{val| 0 4 0 -7 }}, {{val| 0 0 1 1 }}]
Mapping: [{{val| 1 3 0 -2 }}, {{val| 0 -4 0 7 }}, {{val| 0 0 1 1 }}]


{{Optimal ET sequence|legend=1| 3, 14c, 17c, 17d, 20c, 31, 34, 45, 62, 65 }}
{{Optimal ET sequence|legend=1| 3, 14c, 17c, 17d, 20c, 31, 34, 45, 62, 65 }}


[http://x31eq.com/cgi-bin/rt.cgi?ets=34p_31_14c&limit=7 The temperament finder - 7-limit 34p & 31 & 14c]
Badness (Sintel): 19.984


[[Category:Squares]]
[[Category:Squares]]
[[Category:Equivalence continua]]
[[Category:Equivalence continua]]
Retrieved from "https://en.xen.wiki/w/Breedsmic–syntonic_equivalence_continuum"