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 temperament|Squalentine]]
| [[Squalentine]]
| [[64827/64000]]
| [[64827/64000]]
| {{monzo|-9 3 -3 4}}
| {{Monzo| -9 3 -3 4 }}
|-
|-
| 0
| 0
| [[Meantone family|Meantone]]
| [[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 }}]


{{Val list|legend=1| 1677, 5014, 6691, 41854, 43531, 46868, 48545, 50222, 55236, 97090 }}
Mapping: [{{val| 1 0 26 14 }}, {{val| 0 1 34 17 }}, {{val| 0 0 -120 -59 }}]


[http://x31eq.com/cgi-bin/rt.cgi?ets=1677_6691_41854&limit=7 The temperament finder - 7-limit 1677 & 6691 & 41854]
{{Optimal ET sequence|legend=1| 1677, 5014, 6691, 41854, 43531, 46868, 48545, 50222, 55236, 97090 }}


== 34p & 31 & 14c ==
Badness (Sintel): 475.7


== 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 }}]


{{Val list|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]]

Latest revision as of 12:39, 9 January 2026

This page presents a topic of primarily mathematical interest.

While it is derived from sound mathematical principles, its applications in terms of utility for actual music may be limited, highly contrived, or as yet unknown.

The breedsmic–syntonic equivalence continuum is a continuum of 7-limit temperament families which equate a number of breedsmas (2401/2400) with a syntonic comma (81/80). This continuum is theoretically interesting in that these are all 7-limit temperament families supported by 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 (2401/2400)n ~ 81/80. Varying n results in different temperament families listed in the table below. It converges to 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.

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

Examples of temperaments with fractional values of n:

  • 34 & 31 & 14c (n = −1⁄2 = −0.5)
  • Skwares (n = 1⁄2 = 0.5)

1677 & 6691 & 41854

Comma list: [146 34 59 -120⟩

POTE generators: ~3 = 1901.9549, 775.6679

Mapping: [⟨1 0 26 14], ⟨0 1 34 17], ⟨0 0 -120 -59]]

Optimal ET sequence: 1677, 5014, 6691, 41854, 43531, 46868, 48545, 50222, 55236, 97090

Badness (Sintel): 475.7

34 & 31 & 14c

Comma list: [-13 7 -4 4⟩ = 5250987/5120000

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

Mapping: [⟨1 3 0 -2], ⟨0 -4 0 7], ⟨0 0 1 1]]

Optimal ET sequence: 3, 14c, 17c, 17d, 20c, 31, 34, 45, 62, 65

Badness (Sintel): 19.984

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