Schismic–commatic equivalence continuum: Difference between revisions
ArrowHead294 (talk | contribs) m →Sextile |
m add (81/80)^k ~ 128/125 as an alternative choice of coordinates unifying both continuums |
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* [[Quindromeda]] ({{nowrap|''n'' {{=}} 5}}) does not split the octave but splits the fourth in five, as 5 is coprime with 12. | * [[Quindromeda]] ({{nowrap|''n'' {{=}} 5}}) does not split the octave but splits the fourth in five, as 5 is coprime with 12. | ||
{| class="wikitable center-1" | Alternatively, because the [[duodene|5-limit otonal detemper]] of 12edo is a 4x3 rectangle (known as the [[duodene]]), we may be interested in expressing the continuum in terms of the boundary commas of this detemper, that is, as ([[81/80]])<sup>''k''</sup> ~ ([[128/125]]). This corresponds to these commas' structural significance via 128/125 being entirely in the 2.5 subgroup while 81/80 explains 5 in the simplest way relative to the 3-limit. This choice of coordinates has the advantage of finding all temperaments discussed in a relatively intuitive way and is noted for its nontrivial relation to the other better-motivated coordinates discussed; specifically, it's related to the inverted continuum by a translation followed by a flip. Its JIP is at 1.90915584... which is approximated closely by the microtemperament atomic at 21/11 = 1.90909... so that the main continuum can be seen as taking successive mediants towards 2/1 (schismic) starting from 1/1 (diaschismic). | ||
{| class="wikitable center-1 center-2" | |||
|+ style="font-size: 105%;" | Temperaments with integer ''n'' | |+ style="font-size: 105%;" | Temperaments with integer ''n'' | ||
|- | |- | ||
! rowspan="2" | ''n'' | ! rowspan="2" | ''n'' | ||
! rowspan="2" | ''k'' | |||
! rowspan="2" | Temperament | ! rowspan="2" | Temperament | ||
! colspan="2" | Comma | ! colspan="2" | Comma | ||
| Line 21: | Line 24: | ||
|- | |- | ||
| -1 | | -1 | ||
| 5/2 | |||
| [[Gracecordial]] | | [[Gracecordial]] | ||
| (22 digits) | | (22 digits) | ||
| Line 26: | Line 30: | ||
|- | |- | ||
| 0 | | 0 | ||
| 3/1 | |||
| [[Compton]] | | [[Compton]] | ||
| [[531441/524288]] | | [[531441/524288]] | ||
| Line 31: | Line 36: | ||
|- | |- | ||
| 1 | | 1 | ||
| ∞ | |||
| [[Meantone]] | | [[Meantone]] | ||
| [[81/80]] | | [[81/80]] | ||
| Line 36: | Line 42: | ||
|- | |- | ||
| 2 | | 2 | ||
| 1 | |||
| [[Diaschismic]] | | [[Diaschismic]] | ||
| [[2048/2025]] | | [[2048/2025]] | ||
| Line 41: | Line 48: | ||
|- | |- | ||
| 3 | | 3 | ||
| 3/2 | |||
| [[Misty]] | | [[Misty]] | ||
| [[67108864/66430125]] | | [[67108864/66430125]] | ||
| Line 46: | Line 54: | ||
|- | |- | ||
| 4 | | 4 | ||
| 5/3 | |||
| [[Undim]] | | [[Undim]] | ||
| (26 digits) | | (26 digits) | ||
| Line 51: | Line 60: | ||
|- | |- | ||
| 5 | | 5 | ||
| 7/4 | |||
| [[Quindromeda]] | | [[Quindromeda]] | ||
| (34 digits) | | (34 digits) | ||
| Line 56: | Line 66: | ||
|- | |- | ||
| 6 | | 6 | ||
| 9/5 | |||
| [[Sextile]] | | [[Sextile]] | ||
| (44 digits) | | (44 digits) | ||
| Line 61: | Line 72: | ||
|- | |- | ||
| 7 | | 7 | ||
| 11/6 | |||
| [[Heptacot]] | | [[Heptacot]] | ||
| (52 digits) | | (52 digits) | ||
| Line 66: | Line 78: | ||
|- | |- | ||
| 8 | | 8 | ||
| 13/7 | |||
| [[World calendar]] | | [[World calendar]] | ||
| (62 digits) | | (62 digits) | ||
| Line 71: | Line 84: | ||
|- | |- | ||
| 9 | | 9 | ||
| 15/8 | |||
| Quinbisa-tritrigu (12&441) | | Quinbisa-tritrigu (12&441) | ||
| (70 digits) | | (70 digits) | ||
| Line 76: | Line 90: | ||
|- | |- | ||
| 10 | | 10 | ||
| 17/9 | |||
| Lesa-quinbigu (12&494) | | Lesa-quinbigu (12&494) | ||
| (80 digits) | | (80 digits) | ||
| Line 81: | Line 96: | ||
|- | |- | ||
| 11 | | 11 | ||
| 19/10 | |||
| Quadtrisa-legu (12&559) | | Quadtrisa-legu (12&559) | ||
| (88 digits) | | (88 digits) | ||
| Line 86: | Line 102: | ||
|- | |- | ||
| 12 | | 12 | ||
| 21/11 | |||
| [[Atomic]] | | [[Atomic]] | ||
| (98 digits) | | (98 digits) | ||
| Line 91: | Line 108: | ||
|- | |- | ||
| 13 | | 13 | ||
| 23/12 | |||
| Quintrila-theyo (12&677) | | Quintrila-theyo (12&677) | ||
| (106 digits) | | (106 digits) | ||
| Line 101: | Line 119: | ||
|- | |- | ||
| ∞ | | ∞ | ||
| 2 | |||
| [[Schismic]] | | [[Schismic]] | ||
| [[32805/32768]] | | [[32805/32768]] | ||
| Line 108: | Line 127: | ||
We may invert the continuum by setting ''m'' such that {{nowrap|1/''m'' + 1/''n'' {{=}} 1}}. This may be called the ''syntonic-Pythagorean equivalence continuum'', which is essentially the same thing. The just value of ''m'' is 1.0908441588… The [[syntonic comma]] is way larger but much simpler than the schisma. As such, this continuum does not contain as many [[microtemperament]]s, but has more useful lower-complexity temperaments. | We may invert the continuum by setting ''m'' such that {{nowrap|1/''m'' + 1/''n'' {{=}} 1}}. This may be called the ''syntonic-Pythagorean equivalence continuum'', which is essentially the same thing. The just value of ''m'' is 1.0908441588… The [[syntonic comma]] is way larger but much simpler than the schisma. As such, this continuum does not contain as many [[microtemperament]]s, but has more useful lower-complexity temperaments. | ||
{| class="wikitable center-1" | {| class="wikitable center-1 center-2" | ||
|+ style="font-size: 105%;" | Temperaments with integer ''m'' | |+ style="font-size: 105%;" | Temperaments with integer ''m'' | ||
|- | |- | ||
! rowspan="2" | ''m'' | ! rowspan="2" | ''m'' | ||
! rowspan="2" | ''k'' | |||
! rowspan="2" | Temperament | ! rowspan="2" | Temperament | ||
! colspan="2" | Comma | ! colspan="2" | Comma | ||
| Line 119: | Line 139: | ||
|- | |- | ||
| -1 | | -1 | ||
| 4 | |||
| [[Python]] | | [[Python]] | ||
| [[43046721/41943040]] | | [[43046721/41943040]] | ||
| Line 124: | Line 145: | ||
|- | |- | ||
| 0 | | 0 | ||
| 3 | |||
| [[Compton family|Compton]] | | [[Compton family|Compton]] | ||
| [[531441/524288]] | | [[531441/524288]] | ||
| Line 129: | Line 151: | ||
|- | |- | ||
| 1 | | 1 | ||
| 2 | |||
| [[Schismic]] | | [[Schismic]] | ||
| [[32805/32768]] | | [[32805/32768]] | ||
| Line 134: | Line 157: | ||
|- | |- | ||
| 2 | | 2 | ||
| 1 | |||
| [[Diaschismic family|Diaschismic]] | | [[Diaschismic family|Diaschismic]] | ||
| [[2048/2025]] | | [[2048/2025]] | ||
| Line 139: | Line 163: | ||
|- | |- | ||
| 3 | | 3 | ||
| 0 | |||
| [[Augmented]] | | [[Augmented]] | ||
| [[128/125]] | | [[128/125]] | ||
| Line 144: | Line 169: | ||
|- | |- | ||
| 4 | | 4 | ||
| -1 | |||
| [[Diminished]] | | [[Diminished]] | ||
| [[648/625]] | | [[648/625]] | ||
| Line 149: | Line 175: | ||
|- | |- | ||
| 5 | | 5 | ||
| -2 | |||
| [[Ripple]] | | [[Ripple]] | ||
| [[6561/6250]] | | [[6561/6250]] | ||
| Line 154: | Line 181: | ||
|- | |- | ||
| 6 | | 6 | ||
| -3 | |||
| [[Wronecki]] | | [[Wronecki]] | ||
| [[531441/500000]] | | [[531441/500000]] | ||
| Line 163: | Line 191: | ||
| … | | … | ||
|- | |- | ||
| ∞ | |||
| ∞ | | ∞ | ||
| [[Meantone]] | | [[Meantone]] | ||
| Line 172: | Line 201: | ||
|+ style="font-size: 105%;" | Temperaments with fractional ''n'' and ''m'' | |+ style="font-size: 105%;" | Temperaments with fractional ''n'' and ''m'' | ||
|- | |- | ||
! ''n'' !! ''m'' !! Temperament !! Comma | ! ''n'' !! ''m'' !! ''k'' !! Temperament !! Comma | ||
|- | |- | ||
| 5/3 = 1.{{overline|6}} || 5/2 = 2.5 || [[Passion]] || {{monzo| 18 -4 -5 }} | | 5/3 = 1.{{overline|6}} || 5/2 = 2.5 || 1/2 || [[Passion]] || {{monzo| 18 -4 -5 }} | ||
|- | |- | ||
| 5/2 = 2.5 || 5/3 = 1.{{overline|6}} || [[Quintaleap]] || {{monzo| 37 -16 -5 }} | | 5/2 = 2.5 || 5/3 = 1.{{overline|6}} || 4/3 || [[Quintaleap]] || {{monzo| 37 -16 -5 }} | ||
|} | |} | ||