Syntonic–limmic equivalence continuum: Difference between revisions
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The '''syntonic-diatonic equivalence continuum''' is a [[equivalence continuum|continuum]] of [[regular temperament|temperaments]] which equate a number of [[81/80|syntonic commas (81/80)]] with the [[256/243|limma (256/243)]]. This continuum is theoretically interesting in that these are all [[5-limit]] temperaments [[support]]ed by [[5edo]]. | The '''syntonic-diatonic equivalence continuum''' is a [[equivalence continuum|continuum]] of [[regular temperament|temperaments]] which equate a number of [[81/80|syntonic commas (81/80)]] with the [[256/243|limma (256/243)]]. This continuum is theoretically interesting in that these are all [[5-limit]] temperaments [[support]]ed by [[5edo]]. | ||
All temperaments in the continuum satisfy (81/80)<sup>''n''</sup> ~ 256/243. Varying ''n'' results in different temperaments listed in the table below. It converges to [[meantone]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all 5-limit temperaments supported by 5edo due to it being the unique equal temperament that [[tempering out|tempers out]] both commas and thus tempers out all combinations of them. The just value of ''n'' is 4.1952…, and temperaments near this tend to be the most accurate ones. | All temperaments in the continuum satisfy {{nowrap|(81/80)<sup>''n''</sup> ~ 256/243}}. Varying ''n'' results in different temperaments listed in the table below. It converges to [[meantone]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all 5-limit temperaments supported by 5edo due to it being the unique equal temperament that [[tempering out|tempers out]] both commas and thus tempers out all combinations of them. The just value of ''n'' is 4.1952…, and temperaments near this tend to be the most accurate ones. | ||
256/243 is the characteristic [[3-limit]] comma tempered out in 5edo, and has many advantages as a target. In each case, ''n'' equals the order of [[5/1|harmonic 5]] in the corresponding comma, and equals the number of generators to obtain a harmonic 3 in the generator chain. For example: | 256/243 is the characteristic [[3-limit]] comma tempered out in 5edo, and has many advantages as a target. In each case, ''n'' equals the order of [[5/1|harmonic 5]] in the corresponding comma, and equals the number of generators to obtain a harmonic 3 in the generator chain. For example: | ||
* Superpyth (''n'' = 1) is generated by a fifth; | * Superpyth {{nowrap|(''n'' {{=}} 1)}} is generated by a fifth; | ||
* Immunity (''n'' = 2) splits its twelfth in two; | * Immunity {{nowrap|(''n'' {{=}} 2)}} splits its twelfth in two; | ||
* Rodan (''n'' = 3) splits its fifth in three; | * Rodan {{nowrap|(''n'' {{=}} 3)}} splits its fifth in three; | ||
* Etc. | * Etc. | ||
At {{nowrap|''n'' {{=}} 5}}, the corresponding temperament splits the ''octave'' into five instead, as after a stack of five syntonic commas, both the orders of 3 and 5 are multiples of 5 again. | |||
* 16/15 is the classic diatonic semitone, notable in the 5-limit as the difference between 4/3 and 5/4, so this shifted continuum could also logically be termed the "syntonic-diatonic equivalence continuum". This means that at ''k'' = 0, 4/3 and 5/4 are mapped to the same interval while 81/80 becomes independent of 16/15 (meaning 81/80 may or may not be tempered) because the relation becomes (81/80)<sup>0</sup> ~ 1/1 ~ 16/15. | |||
* ''k'' = 1 and upwards (up to a point) represent temperaments with (the potential for) reasonably good accuracy as equating at least one 81/80 with 16/15 seems like a good lower bound for a temperament intended to model JI. A good upper bound might be rodan (''k'' = 4), with the only exception being meantone (''n'' = ''k'' = | If we let {{nowrap|''k'' {{=}} ''n'' + 1}} so that {{nowrap|''k'' {{=}} 0}} means {{nowrap|''n'' {{=}} −1}}, {{nowrap|''k'' {{=}} 1}} means {{nowrap|''n'' {{=}} 0}}, etc. then the continuum corresponds to {{nowrap|(81/80)<sup>''k''</sup> {{=}} 16/15}}. Some prefer this way of conceptualising it because: | ||
* 16/15 is the classic diatonic semitone, notable in the 5-limit as the difference between 4/3 and 5/4, so this shifted continuum could also logically be termed the "syntonic-diatonic equivalence continuum". This means that at {{nowrap|''k'' {{=}} 0}}, 4/3 and 5/4 are mapped to the same interval while 81/80 becomes independent of 16/15 (meaning 81/80 may or may not be tempered) because the relation becomes {{nowrap|(81/80)<sup>0</sup> ~ 1/1 ~ 16/15}}. | |||
* {{nowrap|''k'' {{=}} 1}} and upwards (up to a point) represent temperaments with (the potential for) reasonably good accuracy as equating at least one 81/80 with 16/15 seems like a good lower bound for a temperament intended to model JI. A good upper bound might be rodan {{nowrap|(''k'' {{=}} 4)}}, with the only exception being meantone {{nowrap|(''n'' {{=}} ''k'' {{=}} ∞)}}. (Temperaments corresponding to {{nowrap|''k'' {{=}} 0, −1, −2...}} are comparatively low-accuracy to the point of developing various intriguing structures and consequences.) | |||
* 16/15 is the simplest ratio to be tempered in the continuum. | * 16/15 is the simplest ratio to be tempered in the continuum. | ||
{| class="wikitable center-1 center-2" | {| class="wikitable center-1 center-2" | ||
|+ Temperaments with integer ''n'' | |+ style="font-size: 105%;" | Temperaments with integer ''n'' | ||
|- | |- | ||
! rowspan="2" | ''k'' | ! rowspan="2" | ''k'' | ||
| Line 26: | Line 27: | ||
! Monzo | ! Monzo | ||
|- | |- | ||
| | | −3 | ||
| | | −4 | ||
| Laquadgu (5 & 28) | | Laquadgu (5 & 28) | ||
| [[177147/160000]] | | [[177147/160000]] | ||
| {{monzo| -8 11 -4 }} | | {{monzo| -8 11 -4 }} | ||
|- | |- | ||
| | | −2 | ||
| | | −3 | ||
| [[Gamelismic clan #Gorgo|Laconic]] | | [[Gamelismic clan #Gorgo|Laconic]] | ||
| [[2187/2000]] | | [[2187/2000]] | ||
| {{monzo| -4 7 -3 }} | | {{monzo| -4 7 -3 }} | ||
|- | |- | ||
| | | −1 | ||
| | | −2 | ||
| [[Bug]] | | [[Bug]] | ||
| [[27/25]] | | [[27/25]] | ||
| Line 45: | Line 46: | ||
|- | |- | ||
| 0 | | 0 | ||
| | | −1 | ||
| [[Father]] | | [[Father]] | ||
| [[16/15]] | | [[16/15]] | ||
| Line 107: | Line 108: | ||
{| class="wikitable center-1" | {| class="wikitable center-1" | ||
|+ Temperaments with integer ''m'' | |+ style="font-size: 105%;" | Temperaments with integer ''m'' | ||
|- | |- | ||
! rowspan="2" | ''m'' | ! rowspan="2" | ''m'' | ||
| Line 116: | Line 117: | ||
! Monzo | ! Monzo | ||
|- | |- | ||
| | | −1 | ||
| [[Ultrapyth]] | | [[Ultrapyth]] | ||
| [[5242880/4782969]] | | [[5242880/4782969]] | ||
| Line 153: | Line 154: | ||
{| class="wikitable" | {| class="wikitable" | ||
|+ Temperaments with fractional ''n'' and ''m'' | |+ style="font-size: 105%;" | Temperaments with fractional ''n'' and ''m'' | ||
|- | |- | ||
! ''n'' !! ''m'' !! Temperament !! Comma | ! ''n'' !! ''m'' !! Temperament !! Comma | ||
|- | |- | ||
| | | −3/2 = −1.5 || 3/5 = 0.6 || [[University]] || {{monzo| 4 2 -3 }} | ||
|- | |- | ||
| | | −1/2 = −0.5 || 1/3 = 0.{{overline|3}} || [[Uncle]] || {{monzo| 12 -6 -1 }} | ||
|- | |- | ||
| 5/2 = 2.5 || 5/3 = 1.{{overline|6}} || [[Counterpental]] || {{monzo| 36 -30 5 }} | | 5/2 = 2.5 || 5/3 = 1.{{overline|6}} || [[Counterpental]] || {{monzo| 36 -30 5 }} | ||