Syntonic–diatonic equivalence continuum: Difference between revisions

* Superpyth ({{nowrap| ''n'' {{=}} 1 }}) is generated by a fifth;

* Immunity ({{nowrap| ''n'' {{=}} 2 }}) splits its twelfth in two;

* Rodan ({{nowrap| ''n'' {{=}} 3 }}) splits its fifth in three;

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.  

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)^''k'' {{=}} 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 out) because the relation becomes {{nowrap| (81/80)⁰ ~ 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 out in the continuum.  

| {{Monzo| -8 11 -4 }}

| [[Laconic]]

| {{Monzo| -4 7 -3 }}

| {{Monzo| 0 3 -2 }}

| {{Monzo| 4 -1 -1 }}

| {{Monzo| 8 -5 }}

| {{Monzo| 12 -9 1 }}

| {{Monzo| 16 -13 2 }}

| {{Monzo| 20 -17 3 }}

| {{Monzo| 24 -21 4 }}

| {{Monzo| -28 25 -5 }}

| {{Monzo| -32 29 -6 }}

| {{Monzo| -4 4 -1 }}

We may invert the continuum by setting ''m'' such that {{nowrap| 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…. The [[superpyth comma]] is both larger and more complex than the syntonic comma. As such, this continuum does not contain as many useful temperaments, but still interesting nonetheless.  

| {{Monzo| 20 -14 1 }}

| {{Monzo| 8 -5 }}

| {{Monzo| -4 4 -1 }}

| {{Monzo| 16 -13 2 }}

| {{Monzo| 28 -22 3 }}

| {{Monzo| 12 -9 1 }}

| −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 }}

| 1/3 = 0.{{overline|3}} || −1/2 = −0.5 || [[Dirt]] || {{Monzo| 28 -19 1 }}

| 5/2 = 2.5 || 5/3 = 1.{{overline|6}} || [[Counterpental]] || {{Monzo| 36 -30 5 }}

| 7/2 = 3.5 || 7/5 = 1.4 || [[Septiquarter]] || {{Monzo| 44 -38 7 }}

| 21/5 = 4.2 || 21/16 = 1.3125 || 559 & 2513 || {{Monzo| -124 109 -21 }}

| 9/2 = 4.5 || 9/7 = 1.{{overline|285714}} || 5 & 118 || {{Monzo| -52 46 -9 }}

| 11/2 = 5.5 || 11/9 = 1.{{overline|2}} || 5 & 137 || {{Monzo| -60 54 -11 }}

In the 5-limit, superpyth tempers out [[20480/19683]]. It has a fifth generator of {{nowrap| ~3/2 {{=}} ~710{{c}} }} and ~5/4 is found at +9 generator steps, as an augmented second (C–D#). It corresponds to {{nowrap| ''n'' {{=}} 1 }}, meaning that the syntonic comma is equated with the diatonic semitone.  

* [[WE]]: ~2 = 1197.6520{{c}}, ~3/2 = 708.6882{{c}}

: [[error map]]: {{val| -2.348 +4.385 -1.076 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 709.8213{{c}}

: error map: {{val| 0.000 +7.866 +2.078 }}

[[Badness]] (Sintel): 3.17

* [[WE]]: ~2 = 1189.7544{{c}}, ~3/2 = 724.6670{{c}}

: [[error map]]: {{val| -10.246 +12.466 +4.210 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 731.7318{{c}}

[[Badness]] (Sintel): 6.33

The 5-limit version of ultrapyth tempers out the [[ultrapyth comma]]. It is generated by a perfect fifth. The interval class of 5 is found at +14 fifths as a double-augmented unison (C–Cx). It corresponds to {{nowrap| ''m'' {{=}} -1 }} and {{nowrap| ''n'' {{=}} 1/2 }}.  

* [[WE]]: ~2 = 1196.4357{{c}}, ~3/2 = 711.7085{{c}}

: [[error map]]: {{val| -3.564 +6.189 -1.009 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 713.5968{{c}}

: error map: {{val| 0.000 +11.642 +4.041 }}

[[Badness]] (Sintel): 18.7

* [[WE]]: ~2 = 1195.8566{{c}}, ~3/2 = 713.0611{{c}}

: [[error map]]: {{val| -4.143 +6.963 -0.863 }}

* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 715.3406{{c}}

[[Badness]] (Sintel): 55.3

The 5-limit version of rodan tempers out the [[rodan comma]], which is the difference between a stack of three [[729/640|retroptolemaic whole tones (729/640)]] and a perfect fifth (3/2). The only 7-limit extension that makes any sense to use is to add the gamelisma to the comma list, whereby the generator represents [[8/7]]. It corresponds to {{nowrap| ''n'' {{=}} 3 }}.  

* [[WE]]: ~2 = 1199.5618{{c}}, ~729/640 = 234.4424{{c}}

: [[error map]]: {{val| -0.438 +0.934 -0.355 }}

* [[CWE]]: ~2 = 1200.000{{c}}, ~729/640 = 234.4999{{c}}

: error map: {{val| 0.000 +1.545 +0.185 }}

[[Badness]] (Sintel): 3.95

* [[WE]]: ~2 = 1203.1925{{c}}, ~10/9 = 228.0305{{c}}

: [[error map]]: {{val| +3.193 -14.671 +13.092 }}

* [[CWE]]: ~2 = 1200.000{{c}}, ~10/9 = 228.0128{{c}}

: error map: {{val| 0.000 -17.917 +9.776 }}

[[Badness]] (Sintel): 3.80

Named by [[John Moriarty]], university is the {{nowrap| 5 & 6b }} temperament, and tempers out [[144/125]], the triptolemaic diminished third. It corresponds to {{nowrap| ''n'' {{=}} −3/2 }} and {{nowrap| ''m'' {{=}} 3/5 }}. In this temperament, two instances of [[6/5]] make a [[5/4]], and three make a [[3/2]]. Equating 6/5 with [[8/7]] (which makes sense since it is already very flat in the most accurate tunings of this temperament) leads to [[Gamelismic clan #Gidorah|gidorah]], and 6/5 with [[7/6]] leads to [[Mint temperaments #Penta|penta]].

* [[WE]]: ~2 = 1186.1969{{c}}, ~6/5 = 232.7334{{c}}

: [[error map]]: {{val| -13.803 -17.558 +51.547 }}

* [[CWE]]: ~2 = 1200.000{{c}}, ~6/5 = 231.4822{{c}}

: error map: {{val| 0.000 -7.509 +76.651 }}

[[Badness]] (Sintel): 2.39

== Trisatriyo (5 & 56) ==

[[Comma list]]: {{monzo| 28 -22 3 }} (33554432000/31381059609)

* [[POTE]]: ~2 = 1200.000{{c}}, ~2560/2187 = 235.867{{c}}

{{Mapping|legend=1| 1 -2 -15 | 0 6 29 }}

: mapping generators: ~2, ~243/160

* [[WE]]: ~2 = 1200.3725{{c}}, ~243/160 = 716.9750{{c}}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~243/160 = 716.7671{{c}}

: error map: {{val| 0.000 -1.352 -0.067 }}

{{Optimal ET sequence|legend=1| 5, …, 72, 149, 221, 370, 591b }}

[[Badness]] (Sintel): 16.9

* [[WE]]: ~729/640 = 239.8575{{c}}, ~3/2 = 704.1540{{c}}

* [[CWE]]: ~729/640 = 240.0000{{c}}, ~3/2 = 704.4446{{c}}

[[Badness]] (Sintel): 35.2

{{Mapping|legend=1| 1 -4 -28 | 0 7 38 }}

: mapping generators: ~2, ~177147/102400

* [[WE]]: ~2 = 1199.7741{{c}}, ~177147/102400 = 957.3630{{c}}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~177147/102400 = 957.5367{{c}}

: error map: {{val| 0.000 +0.802 +0.082 }}

{{Optimal ET sequence|legend=1| 5, …, 94, 99, 193, 292, 391, 1074b, 1465bb }}

[[Badness]] (Sintel): 22.8

== Quinla-tritrigu (5 & 118) ==

* [[POTE]]: ~2 = 1200.000{{c}}, ~320/243 = 477.961{{c}}

== Tribilalegu (5 & 137) ==

* [[POTE]]: ~2 = 1200.000{{c}}, ~320/243 = 481.742{{c}}

== 559 & 2513 ==

* [[POTE]]: ~2 = 1200.0000{{c}}, ~3355443200000/2541865828329 = 480.8595{{c}}