Syntonic–diatonic equivalence continuum: Difference between revisions

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|Pythagorean 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 {{nowrap|(81/80)^''n'' ~ 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:

* 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.  

|+ style="font-size: 105%;" | Temperaments with integer ''n''

! rowspan="2" | ''k''

! rowspan="2" | ''n''

| −3

| −4

| Laquadgu (5 & 28)

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

| −2

| −3

| [[Laconic]]

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

| −1

| −2

| {{Monzo| 0 3 -2 }}

| −1

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

| {{Monzo| 8 -5 }}

| {{Monzo| 12 -9 1 }}

| [[Immunity]]

| {{Monzo| 16 -13 2 }}

| {{Monzo| 20 -17 3 }}

| [[10485760000/10460353203|(22 digits)]]

| {{Monzo| 24 -21 4 }}

| [[Quintile]]

| (24 digits)

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

| (28 digits)

| {{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.  

{| class="wikitable center-1"

|+ style="font-size: 105%;" | Temperaments with integer ''m''

! rowspan="2" | ''m''

! rowspan="2" | Temperament

| {{Monzo| 8 -5 }}

|}

 

 

 

[[Subgroup]]: 2.3.5

 

 

: mapping generators: ~2, ~3

[[Optimal tuning]]s:  

* [[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 }}

{{Optimal ET sequence|legend=1| 5, 17, 22, 49, 120b, 169bbc }}

[[Badness]] (Sintel): 3.17

: ''For extensions, see [[Trienstonic clan #Uncle]].''

The 5-limit version of uncle tempers out [[4096/3645]]. It is generated by a fifth that is supposedly sharper than [[5edo|3\5]], so it leads to an [[5L 3s|oneirotonic]] scale, or otherwise a [[5L 2s|diatonic]] scale with negative small steps. The interval class of 5 is found at -6 fifths, as a major 2-step in oneirotonic, or a diminished fifth (C–Gb) in diatonic. It corresponds to {{nowrap| ''n'' {{=}} -1/2 }} or {{nowrap| ''m'' {{=}} 1/3 }}.  

[[Comma list]]: 4096/3645

: mapping generators: ~2, ~3

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

{{Optimal ET sequence|legend=1| 5, 13, 18, 23bc }}

[[Badness]] (Sintel): 6.33

: ''For extensions, see [[Archytas clan #Ultrapyth]].''

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 }}.  

[[Subgroup]]: 2.3.5

[[Comma list]]: 5242880/4782969

{{Mapping|legend=1| 1 0 -20 | 0 1 14 }}

: mapping generators: ~2, ~3

[[Optimal tuning]]s:  

* [[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

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

[[Badness]] (Smith): 1.323443

: ''For extensions, see [[Orwellismic temperaments #Pentaorwell]].''

 

[[Comma list]]: {{monzo| 36 -30 5 }}

{{Mapping|legend=1| 5 0 -36 | 0 1 6 }}

: mapping generators: ~729/640, ~3

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

{{Optimal ET sequence|legend=1| 5, …, 75, 80, 155, 390b, 545bbc }}

[[Badness]] (Sintel): 35.2

: ''For extensions, see [[Hemifamity temperaments #Septiquarter]].''

[[Comma list]]: {{monzo| 44 -38 7 }}

{{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}}

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

[[Badness]] (Sintel): 22.8

: ''For extensions, see [[Wizmic microtemperaments #Tokko]].''

[[Subgroup]]: 2.3.5

[[Comma list]]: {{monzo| -76 67 -13 }}

{{Mapping|legend=1| 1 -1 -11 | 0 13 67 }}

: mapping generators: ~2, ~{{monzo| -35 31 -6 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~{{monzo| -35 31 -6 }} = 238.6015{{c}}

{{Optimal ET sequence|legend=1| 5, …, 166, 171, 860, 1031, 1202, 1373, 1544, 3259, 4803b, 6347b }}

[[Badness]] (Sintel): 20.9

[[Subgroup]]: 2.3.5

[[Comma list]]: {{monzo| -52 46 -9 }}

{{Mapping|legend=1| 1 -2 -16 | 0 9 46 }}

: mapping generators: ~2, ~320/243

[[Optimal tuning]]s:  

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

{{Optimal ET sequence|legend=1| 5, 108c, 113, 118, 1057, 1175, 1293, 1411, 1529, 1647, 1765, 1883, 2001b, 3884b }}

[[Badness]] (Sintel): 14.5

[[Subgroup]]: 2.3.5

[[Comma list]]: {{Monzo| -60 54 -11 }}

: mapping generators: ~2, ~243/160

* [[POTE]]: ~2 = 1200.000{{c}}, ~243/160 = 718.258{{c}}

{{Optimal ET sequence|legend=1| 5, 127c, 132, 137, 553, 690b, 827b, 964b }}

[[Badness]] (Sintel): 84.9

[[Subgroup]]: 2.3.5

[[Comma list]]: {{monzo| -124 109 -21 }}

: mapping generators: ~2, ~{{monzo| -29 26 -5 }}

[[Optimal tuning]]s:  

* [[POTE]]: ~2 = 1200.0000{{c}}, ~{{monzo| -29 26 -5 }} = 719.1405{{c}}

{{Optimal ET sequence|legend=1| 5, …, 277, 559, 1395, 1954, 2513, 40767, 43280, 45793, 48306, 50819, 53332, 55845, 58358, 60871, 63384, 65897, 68410, 70923, 73436, 75949, 78462, 154411b }}