Marvel–syntonic equivalence continuum: Difference between revisions
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The '''marvel–syntonic equivalence continuum''' is a [[equivalence continuum|continuum]] of 7-limit rank-3 temperament families which equate a number of [[225/224|marvel commas (225/224)]] with a [[81/80|syntonic comma (81/80)]]. This continuum is theoretically interesting in that these are all 7-limit rank-3 temperament families supported by [[septimal meantone]] temperament. | The '''marvel–syntonic equivalence continuum''' is a [[equivalence continuum|continuum]] of 7-limit rank-3 temperament families which equate a number of [[225/224|marvel commas (225/224)]] with a [[81/80|syntonic comma (81/80)]]. This continuum is theoretically interesting in that these are all 7-limit rank-3 temperament families supported by [[septimal meantone]] temperament. | ||
All temperaments in the continuum satisfy {{nowrap|( | All temperaments in the continuum satisfy {{nowrap|(225/224)<sup>''n''</sup> ~ 81/80}}. Varying ''n'' results in different temperament families listed in the table below. It converges to [[marvel]] 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 septimal meantone (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 2.788851, and temperaments having ''n'' near this value will be more accurate. | ||
{| class="wikitable center-1 center-2" | {| class="wikitable center-1 center-2" | ||
| Line 29: | Line 29: | ||
| [[3136/3125]] | | [[3136/3125]] | ||
| {{Monzo| 6 0 -5 2 }} | | {{Monzo| 6 0 -5 2 }} | ||
|- | |- | ||
| 3 | | 3 | ||
| Line 39: | Line 34: | ||
| [[703125/702464]] | | [[703125/702464]] | ||
| {{Monzo| -11 2 7 -3 }} | | {{Monzo| -11 2 7 -3 }} | ||
|- | |||
|4 | |||
| {{Nowrap| 31 & 19 & 224 }} | |||
|[[158203125/157351936]] | |||
|{{Monzo| -16 4 9 -4}} | |||
|- | |||
|5 | |||
| {{Nowrap| 31 & 19 & 265 }} | |||
| <abbr title="35595703125/35246833664">(22 digits)</abbr> | |||
|{{Monzo| -21 6 11 -5}} | |||
|- | |||
|... | |||
|... | |||
|... | |||
|... | |||
|- | |- | ||
| ∞ | | ∞ | ||
| Line 45: | Line 55: | ||
| {{Monzo| -5 2 2 -1 }} | | {{Monzo| -5 2 2 -1 }} | ||
|} | |} | ||
{| class="wikitable" | |||
|+Temperaments with fractional n | |||
!n | |||
!Temperament | |||
!Comma | |||
|- | |||
|3/2 = 1.5 | |||
|{{Nowrap|31 & 19 & 95}} | |||
|{{Monzo|7 2 -8 3}} | |||
|- | |||
|7/3 = 2.{{Overline|3}} | |||
|{{Nowrap|31 & 19 & 369}} | |||
|{{Monzo|23 -2 -17 7}} | |||
|- | |||
|5/2 = 2.5 | |||
|{{Nowrap|270 & 31 & 19}} | |||
|{{Monzo|17 -2 -12 5}} | |||
|- | |||
|8/3 = 2.{{Overline|6}} | |||
|{{Nowrap|441 & 31 & 19}} | |||
|{{Monzo|18 -4 -19 8}} | |||
|- | |||
|11/4 = 2.75 | |||
|{{Nowrap| 1205 & 612 & 1236 }} | |||
|{{Monzo| 39 -6 -26 11 }} | |||
|- | |||
|14/5 = 2.8 | |||
|{{Nowrap|3125 & 1547 & 783}} | |||
|{{Monzo|50 -8 -33 14}} | |||
|- | |||
|7/2 = 3.5 | |||
|{{Nowrap|31 & 19 & 395}} | |||
|{{Monzo|-27 6 16 -7}} | |||
|} | |||
[[Category:Equivalence continua]] | [[Category:Equivalence continua]] | ||