Sensamagic–gariboh equivalence continuum: Difference between revisions
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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 b13 & b2d]) | | Muphrid ([https://sintel.pythonanywhere.com/result?subgroup=3.5.7&reduce=on&weights=weil&target=&edos=&commas=546875%2F531441&submit_comma=submit b13 & b2d]) | ||
| Line 22: | Line 22: | ||
| {{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 79: | 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 88: | Line 88: | ||
! Monzo (3.5.7 subgroup) | ! Monzo (3.5.7 subgroup) | ||
|- | |- | ||
| | | −1 | ||
| Miaplacidus | | [[Miaplacidus]] | ||
| 5859375/5764801 | | 5859375/5764801 | ||
| {{monzo| 1 9 -8 }} | | {{monzo| 1 9 -8 }} | ||
| Line 99: | Line 99: | ||
|- | |- | ||
| 1 | | 1 | ||
| [[ | | [[Bohlen–Pierce–Stearns]] | ||
| [[245/243]] | | [[245/243]] | ||
| {{monzo| -5 1 2 }} | | {{monzo| -5 1 2 }} | ||
| Line 129: | 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]] | ||
Latest revision as of 13:04, 10 March 2025
The sensamagic–gariboh equivalence continuum is a continuum of 3.5.7 subgroup temperaments which equate a number of sensamagic commas (245/243) with the greater BP diesis (3125/3087).
All temperaments in the continuum satisfy (245/243)n ~ 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)k ~ 713/323 = 96889010407/94143178827, which sets 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.
| n | k | Temperament | Comma | |
|---|---|---|---|---|
| Ratio | Monzo (3.5.7 subgroup) | |||
| −2 | 7 | Muphrid (b13 & b2d) | 546875/531441 | [-12 7 1⟩ |
| −1 | 6 | Arcturus | 15625/15309 | [-7 6 -1⟩ |
| 0 | 5 | Sirius | 3125/3087 | [-2 5 -3⟩ |
| 1 | 4 | Canopus | 16875/16807 | [3 4 -5⟩ |
| 2 | 3 | Procyon | 823543/820125 | [-8 -3 7⟩ |
| 3 | 2 | Betelgeuse (b13 & b148) | 40353607/39858075 | [-13 -2 9⟩ |
| 4 | 1 | Pollux (b13 & b139) | 1977326743/1937102445 | [-18 -1 11⟩ |
| 5 | 0 | 13-7-commatic | 96889010407/94143178827 | [-23 0 13⟩ |
| … | … | … | ||
| ∞ | ∞ | BPS | 245/243 | [-5 1 2⟩ |
We may invert the continuum by setting m such that 1/m + 1/n = 1. This may be called the canopus-sirius equivalence continuum, as temperaments satisfy (16875/16807)m ~ 3125/3087. The just value of m is 3.0300…, and temperaments close to this value are the most accurate (note how accurate the Izar temperament then must be!)
| m | Temperament | Comma | |
|---|---|---|---|
| Ratio | Monzo (3.5.7 subgroup) | ||
| −1 | Miaplacidus | 5859375/5764801 | [1 9 -8⟩ |
| 0 | Sirius | 3125/3087 | [-2 5 -3⟩ |
| 1 | Bohlen–Pierce–Stearns | 245/243 | [-5 1 2⟩ |
| 2 | Procyon (b13 & b144) | 823543/820125 | [-8 -3 7⟩ |
| 3 | Izar | 13841287201/13839609375 | [-11 -7 12⟩ |
| 4 | b13 & b307 | 233543408203125/232630513987207 | [14 11 -17⟩ |
| … | … | … | |
| ∞ | Canopus | 16875/16807 | [3 4 -5⟩ |
Projective tuning space
Below is a depiction of the temperaments of this continuum in 3.5.7 projective tuning space.
|
| Labelled by name |
|
| Labelled by comma |