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Syntonic–limmic equivalence continuum: Difference between revisions

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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 tempers both commas and thus tempers 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 (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 tempers both commas and thus tempers 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]]. In each case, we notice that ''n'' equals the order of harmonic 5 in the corresponding comma, and equals the number of generators to obtain a harmonic 3 in the MOS scale. However, if we let ''k'' = ''n'' + 1 (meaning ''n'' = ''k'' - 1) so that ''k'' = 0 means ''n'' = -1, ''k'' = 1 means ''n'' = 0, etc. then the continuum corresponds to (81/80)<sup>''k''</sup> = 16/15, which might be a preferred way of conceptualising it because:
256/243 is the characteristic [[3-limit]] comma tempered out in [[5edo]]. In each case, we notice that ''n'' equals the order of harmonic 5 in the corresponding comma, and equals the number of generators to obtain a harmonic 3 in the generator chain.  
 
However, if we let ''k'' = ''n'' + 1 (meaning ''n'' = ''k'' - 1) so that ''k'' = 0 means ''n'' = -1, ''k'' = 1 means ''n'' = 0, etc. then the continuum corresponds to (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 ''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.
* 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'' = (unsigned) infinity). (Temperaments corresponding to ''k'' = 0, -1, -2 are comparatively low-accuracy to the point of developing various intriguing structures and consequences.)
* ''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'' = (unsigned) infinity). (Temperaments corresponding to ''k'' = 0, -1, -2 are comparatively low-accuracy to the point of developing various intriguing structures and consequences.)
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{| class="wikitable center-1 center-2"
{| class="wikitable center-1 center-2"
|+ Temperaments in the continuum
|+ Temperaments with integer ''n''
|-
|-
! rowspan="2" | ''k'' = ''n'' + 1
! rowspan="2" | ''k''
! rowspan="2" | ''n'' = ''k'' − 1
! rowspan="2" | ''n''
! rowspan="2" | Temperament
! rowspan="2" | Temperament
! colspan="2" | Comma
! colspan="2" | Comma
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| -3
| -3
| -4
| -4
| [[Laquadgu]]
| Laquadgu (5 & 28)
| [[177147/160000]]
| [[177147/160000]]
| {{monzo| -8 11 -4 }}
| {{monzo| -8 11 -4 }}
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| 3
| 3
| 2
| 2
| [[Immunity family|Immunity]]
| [[Immunity]]
| [[1638400/1594323]]
| [[1638400/1594323]]
| {{monzo| 16 -13 2 }}
| {{monzo| 16 -13 2 }}
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|}
|}


Examples of temperaments with fractional values of ''n'':
We may invert the continuum by setting ''m'' such that 1/''m'' + 1/''n'' = 1. This may be called the ''superpyth-diatonic equivalence continuum'', which is essentially the same thing. The just value of ''m'' is 1.3130…
* [[University temperament|University]] (''n'' = -1.5)
 
* [[Uncle]] (''n'' = -0.5)
{| class="wikitable center-1"
* [[Ultrapyth]] (''n'' = 0.5)
|+ Temperaments with integer ''m''
* 5 &amp; 56 (''n'' = 1.5)
|-
* Counterpental (''n'' = 2.5)
! rowspan="2" | ''m''
* [[Septiquarter]] (''n'' = 3.5)
! rowspan="2" | Temperament
* 559 &amp; 2513 (''n'' = 4.2)
! colspan="2" | Comma
* 5 &amp; 118 (''n'' = 4.5)
|-
* 5 &amp; 137 (''n'' = 5.5)
! Ratio
! Monzo
|-
| -1
| [[Ultrapyth]]
| [[5242880/4782969]]
| {{monzo| 20 -14 1 }}
|-
| 0
| [[Blackwood]]
| [[256/243]]
| {{monzo| 8 -5 }}
|-
| 1
| [[Meantone]]
| [[81/80]]
| {{monzo| -4 4 -1 }}
|-
| 2
| [[Immunity]]
| [[1638400/1594323]]
| {{monzo| 16 -13 2 }}
|-
| 3
| 5 & 56
|
| {{monzo| 28 -22 3 }}
|-
| …
| …
| …
| …
|-
| ∞
| [[Superpyth]]
| [[20480/19683]]
| {{monzo| 12 -9 1 }}
|}
 
{| class="wikitable"
|+ Temperaments with fractional ''n'' and ''m''
|-
! Temperament !! ''n'' !! ''m''
|-
| [[University]] || -3/2 = -1.5 || 3/5 = 0.6
|-
| [[Uncle]] || -1/2 = -0.5 || 1/3 = 0.{{overline|3}}
|-
| [[Counterpental]] || 5/2 = 2.5 || 5/3 = 1.{{overline|6}}
|-
| [[Septiquarter]] || 7/2 = 3.5 || 7/5 = 1.4
|-
| 559 &amp; 2513 || 21/5 = 4.2 || 21/16 = 1.3125
|-
| 5 &amp; 118 || 9/2 = 4.5 || 9/7 = 1.{{overline|285714}}
|-
| 5 &amp; 137 || 11/2 = 5.5 || 11/9 = 1.{{overline|2}}
|}


== Hemiseven ==
== Hemiseven ==
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