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Breedsmic–syntonic equivalence continuum: Difference between revisions

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The '''breedsmic-syntonic equivalence continuum''' is a 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]].
{{Mathematical interest}}


All temperaments in the continuum satisfy (2401/2400)<sup>''n''</sup> ~ 81/80. Varying ''n'' results in different temperament families listed in the table below. It converges to [[Breedsmic temperaments|breedsmic]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all [[7-limit]] temperament families supported by squares (due to it being the unique rank-2 temperament that tempers both commas and thus tempers all combinations of them). The just value of ''n'' is approximately 29.82025..., and temperaments having ''n'' near this value tend to be the most accurate ones.
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]].
 
All temperaments in the continuum satisfy {{nowrap|(2401/2400)<sup>''n''</sup> ~ 81/80}}. Varying ''n'' results in different temperament families listed in the table below. It converges to [[Breedsmic temperaments|breedsmic]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all [[7-limit]] temperament families supported by squares (due to it being the unique rank-2 temperament that tempers both commas and thus tempers all combinations of them). The just value of ''n'' is approximately 29.820259, and temperaments having ''n'' near this value tend to be the most accurate ones.


{| class="wikitable center-1 center-2"
{| class="wikitable center-1 center-2"
|+ Temperament families in the continuum
|+ style="font-size: 105%;" | Temperament families in the continuum
|-
|-
! rowspan="2" | ''n''
! rowspan="2" | ''n''
Line 13: Line 15:
! Monzo
! Monzo
|-
|-
| -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 }}
|-
|-
| …
| …
Line 86: Line 88:
| 1677 & 6691 & 41854
| 1677 & 6691 & 41854
|  
|  
| {{monzo|146 34 59 -120}}
| {{Monzo| 146 34 59 -120 }}
|-
|-
| …
| …
Line 96: Line 98:
| [[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 (''n'' = -1/2 = -0.5)
* 34 & 31 & 14c ({{nowrap|''n'' {{=}} −{{frac|1|2}} {{=}} −0.5}})
* [[Subgroup temperaments#Skwares 3|Skwares]] (''n'' = 1/2 = 0.5)
* [[Subgroup temperaments#Skwares 3|Skwares]] ({{nowrap|''n'' {{=}} {{frac|1|2}} {{=}} 0.5}})


== 1677 & 6691 & 41854 ==
== 1677 & 6691 & 41854 ==
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]]
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