Breedsmic–syntonic equivalence continuum: Difference between revisions
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The ''' | 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]]. | ||
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. | 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 13: | ||
! Monzo | ! Monzo | ||
|- | |- | ||
| | | −4 | ||
| 217 & 31 & 14c | | 217 & 31 & 14c | ||
| | | | ||
| {{monzo|-24 0 -9 16}} | | {{monzo|-24 0 -9 16}} | ||
|- | |- | ||
| | | −3 | ||
| 159 & 31 & 14c | | 159 & 31 & 14c | ||
| | | | ||
| {{monzo|-19 1 -7 12}} | | {{monzo|-19 1 -7 12}} | ||
|- | |- | ||
| | | −2 | ||
| 87 & 31 & 14c | | 87 & 31 & 14c | ||
| 51883209/51200000 | | 51883209/51200000 | ||
| {{monzo|-14 2 -5 8}} | | {{monzo|-14 2 -5 8}} | ||
|- | |- | ||
| | | −1 | ||
| [[Squalentine]] | | [[Squalentine]] | ||
| [[64827/64000]] | | [[64827/64000]] | ||
| Line 44: | Line 44: | ||
|- | |- | ||
| 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 84: | Line 84: | ||
|- | |- | ||
| 30 | | 30 | ||
| 1677 & 6691 & 41854 | | 1677 & 6691 & 41854 | ||
| | | | ||
| {{monzo|146 34 59 -120}} | | {{monzo|146 34 59 -120}} | ||
| Line 100: | Line 100: | ||
Examples of temperaments with fractional values of ''n'': | Examples of temperaments with fractional values of ''n'': | ||
* 34p & 31 & 14c (''n'' = | * 34p & 31 & 14c ({{nowrap|''n'' {{=}} −{{frac|1|2}} {{=}} −0.5}}) | ||
* [[Subgroup temperaments#Skwares 3|Skwares]] (''n'' = 1 | * [[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 }} | ||
| Line 116: | Line 115: | ||
== 34p & 31 & 14c == | == 34p & 31 & 14c == | ||
Comma list: {{monzo| -13 7 -4 4 }} = 5250987/5120000 | Comma list: {{monzo| -13 7 -4 4 }} = 5250987/5120000 | ||