Sensamagic–gariboh equivalence continuum: Difference between revisions
Created page with "The '''sensamagic-gariboh equivalence continuum''' is a continuum of 3.5.7 subgroup temperaments which equate a number of 245/243|sensamagic co..." |
ArrowHead294 (talk | contribs) mNo edit summary |
||
| (20 intermediate revisions by 2 users not shown) | |||
| Line 1: | Line 1: | ||
The ''' | The '''sensamagic–gariboh equivalence continuum''' is a [[equivalence continuum|continuum]] of [[3.5.7 subgroup]] temperaments which equate a number of [[245/243|sensamagic commas (245/243)]] with the [[3125/3087|greater BP diesis (3125/3087)]]. | ||
All temperaments in the continuum satisfy (245/243)<sup>''n''</sup> ~ 3125/3087. Varying ''n'' results in different temperaments listed in the table below. It converges to [[ | All temperaments in the continuum satisfy {{nowrap|(245/243)<sup>''n''</sup> ~ 3125/3087}}. Varying ''n'' results in different temperaments listed in the table below. It converges to [[Bohlen–Pierce–Stearns]] as ''n'' approaches infinity and [[Sirius]] as ''n'' approaches 0. If we allow non-integer and infinite ''n'', the continuum describes the set of all 3.5.7 subgroup temperaments supported by [[13edt]] (due to it being the unique equal temperament that tempers both commas and thus tempers all combinations of them); additionally, if prime 2 is added, this describes the set of all rank-3 7-limit temperaments supported by [[bohpier]]. Due to 13edt, the equal-tempered [[Bohlen–Pierce scale]], being a very good representation of the subgroup, and many named temperaments within the subgroup being supported by 13edt, this continuum is structurally important. The just value of ''n'' is 1.4926…, and temperaments near this tend to be the most accurate ones. | ||
An alternative definition of this equivalence continuum satisfies (245/243)<sup>''k''</sup> ~ 7<sup>13</sup>/3<sup>23</sup> = 96889010407/94143178827, which sets ''k'' = 5 | An alternative definition of this equivalence continuum satisfies {{nowrap|(245/243)<sup>''k''</sup> ~ 7<sup>13</sup>/3<sup>23</sup>}} = 96889010407/94143178827, which sets {{nowrap|''k'' {{=}} 5 − ''n''}}. This definition might be advantageous as ''k'' gives the order of harmonic 5 in the corresponding comma, and equals the number of generators to obtain a harmonic 7 in the MOS scale. | ||
{| class="wikitable center-1 center-2" | {| class="wikitable center-1 center-2" | ||
|+Temperaments in the continuum | |+ style="font-size: 105%;" | Temperaments in the continuum | ||
|- | |- | ||
! rowspan="2" | ''n'' | ! rowspan="2" | ''n'' | ||
| Line 16: | Line 16: | ||
! Monzo (3.5.7 subgroup) | ! Monzo (3.5.7 subgroup) | ||
|- | |- | ||
| | | −2 | ||
| 7 | | 7 | ||
| Muphrid ([https://sintel.pythonanywhere.com/result?subgroup=3.5.7&reduce=on&weights=weil&target=&edos=&commas=546875%2F531441&submit_comma=submit | | Muphrid ([https://sintel.pythonanywhere.com/result?subgroup=3.5.7&reduce=on&weights=weil&target=&edos=&commas=546875%2F531441&submit_comma=submit b13 & b2d]) | ||
| 546875/531441 | | 546875/531441 | ||
| {{monzo| -12 7 1 }} | | {{monzo| -12 7 1 }} | ||
|- | |- | ||
| | | −1 | ||
| 6 | | 6 | ||
| [[No-twos subgroup temperaments#Arcturus|Arcturus]] | | [[No-twos subgroup temperaments#Arcturus|Arcturus]] | ||
| Line 42: | Line 42: | ||
| 2 | | 2 | ||
| 3 | | 3 | ||
| Procyon | | [[Procyon]] | ||
| 823543/820125 | | 823543/820125 | ||
| {{monzo| -8 -3 7 }} | | {{monzo| -8 -3 7 }} | ||
| Line 48: | Line 48: | ||
| 3 | | 3 | ||
| 2 | | 2 | ||
| Betelgeuse ([https://sintel.pythonanywhere.com/result?subgroup=3.5.7&reduce=on&weights=weil&target=&edos=&commas=40353607%2F39858075&submit_comma=submit | | Betelgeuse ([https://sintel.pythonanywhere.com/result?subgroup=3.5.7&reduce=on&weights=weil&target=&edos=&commas=40353607%2F39858075&submit_comma=submit b13 & b148]) | ||
| 40353607/39858075 | | 40353607/39858075 | ||
| {{monzo| -13 -2 9 }} | | {{monzo| -13 -2 9 }} | ||
| Line 54: | Line 54: | ||
| 4 | | 4 | ||
| 1 | | 1 | ||
| Pollux ([https://sintel.pythonanywhere.com/result?subgroup=3.5.7&reduce=on&weights=weil&target=&edos=&commas=1977326743%2F1937102445&submit_comma=submit | | Pollux ([https://sintel.pythonanywhere.com/result?subgroup=3.5.7&reduce=on&weights=weil&target=&edos=&commas=1977326743%2F1937102445&submit_comma=submit b13 & b139]) | ||
| 1977326743/1937102445 | | 1977326743/1937102445 | ||
| {{monzo| -18 -1 11 }} | | {{monzo| -18 -1 11 }} | ||
| Line 69: | Line 69: | ||
| | | | ||
|- | |- | ||
| ∞ | |||
| ∞ | | ∞ | ||
| [[BPS]] | | [[BPS]] | ||
| Line 78: | Line 79: | ||
{| class="wikitable center-1 center-2" | {| class="wikitable center-1 center-2" | ||
|+Temperaments in the continuum | |+ style="font-size: 105%;" | Temperaments in the continuum | ||
|- | |- | ||
! rowspan="2" | ''m'' | ! rowspan="2" | ''m'' | ||
| Line 85: | Line 86: | ||
|- | |- | ||
! Ratio | ! Ratio | ||
! Monzo ( | ! Monzo (3.5.7 subgroup) | ||
|- | |- | ||
| | | −1 | ||
| Miaplacidus | | [[Miaplacidus]] | ||
| 5859375/5764801 | | 5859375/5764801 | ||
| {{monzo| 1 9 -8 }} | | {{monzo| 1 9 -8 }} | ||
| Line 98: | Line 99: | ||
|- | |- | ||
| 1 | | 1 | ||
| [[ | | [[Bohlen–Pierce–Stearns]] | ||
| [[245/243]] | | [[245/243]] | ||
| {{monzo| -5 1 2 }} | | {{monzo| -5 1 2 }} | ||
|- | |- | ||
| 2 | | 2 | ||
| Procyon ([https://sintel.pythonanywhere.com/result?subgroup=3.5.7&reduce=on&weights=weil&target=&edos=&commas=823543%2F820125&submit_comma=submit | | Procyon ([https://sintel.pythonanywhere.com/result?subgroup=3.5.7&reduce=on&weights=weil&target=&edos=&commas=823543%2F820125&submit_comma=submit b13 & b144]) | ||
| 823543/820125 | | 823543/820125 | ||
| {{monzo| -8 -3 7 }} | | {{monzo| -8 -3 7 }} | ||
| Line 113: | Line 114: | ||
|- | |- | ||
| 4 | | 4 | ||
| [https://sintel.pythonanywhere.com/result?subgroup=3.5.7&reduce=on&weights=weil&target=&edos=&commas=%5B14+11+-17%5D&submit_comma=submit | | [https://sintel.pythonanywhere.com/result?subgroup=3.5.7&reduce=on&weights=weil&target=&edos=&commas=%5B14+11+-17%5D&submit_comma=submit b13 & b307] | ||
| 233543408203125/232630513987207 | | 233543408203125/232630513987207 | ||
| {{monzo| 14 11 -17 }} | | {{monzo| 14 11 -17 }} | ||
| Line 128: | Line 129: | ||
|} | |} | ||
[[ | == Projective tuning space == | ||
[[Category: | Below is a depiction of the temperaments of this continuum in 3.5.7 [[projective tuning space]]. | ||
{{center|<div style{{=}}"display: inline-grid; margin-right: 25px;"> | |||
{{(!}} class{{=}}"wikitable" | |||
{{!-}} | |||
{{!}} [[File:357ptslines1n.png|320px]] | |||
{{!-}} | |||
{{!}} Labelled by name | |||
{{!)}} | |||
</div><div style{{=}}"display: inline-grid;"> | |||
{{(!}} class{{=}}"wikitable" | |||
{{!-}} | |||
{{!}} [[File:357ptslines1c.png|320px]] | |||
{{!-}} | |||
{{!}} Labelled by comma | |||
{{!)}} | |||
</div>}} | |||
[[Category:Bohlen–Pierce]] | |||
[[Category:Equivalence continua]] | [[Category:Equivalence continua]] | ||