Syntonic–31 equivalence continuum: Difference between revisions

The '''syntonic–31 equivalence continuum''' is a [[equivalence continuum|continuum]] of 5-limit temperaments which equate a number of [[81/80|syntonic commas (81/80)]] with a [[31st-octave temperaments|31-comma ({{monzo| -49 31 }})]]. This continuum is theoretically interesting in that these are all 5-limit temperaments supported by [[31edo]].

All temperaments in the continuum satisfy {{nowrap|(81/80)^''n'' ~ {{monzo| -49 31 }}}}. 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 [[31edo]] due to it being the unique equal temperament that tempers out both commas and thus tempers out all combinations of them. The just value of ''n'' is approximately 7.46781…, and temperaments having ''n'' near this value tend to be the most accurate ones.

{| class="wikitable center-1"

| [[31st-octave temperaments|31-commatic]]

| {{Monzo| -49 31 }}

| 31 & 12c

| {{Monzo| -45 27 1 }}

| [[Miscellaneous 5-limit temperaments #Quasimoha|Quasimoha]]

| {{Monzo| -41 23 2 }}

| [[Miscellaneous 5-limit temperaments #Oncle|Oncle]]

| {{Monzo| -37 19 3 }}

| [[Miscellaneous 5-limit temperaments #Sentinel|Sentinel]]

| {{Monzo| -33 15 4 }}

| [[Miscellaneous 5-limit temperaments #Tritonic|Tritonic]]

| {{Monzo| -29 11 5 }}

| {{Monzo| -25 7 6 }}

| {{Monzo| -21 3 7 }}

| {{Monzo| 17 1 -8 }}

| {{Monzo| 13 5 -9 }}

| {{Monzo| 9 9 -10 }}

| [[Miscellaneous 5-limit temperaments #Nusecond|Nusecond]]

| {{Monzo| 5 13 -11 }}

| [[Miscellaneous 5-limit temperaments #Cypress|Cypress]]

| {{Monzo| 1 17 -12 }}

| [[Miscellaneous 5-limit temperaments #Diesic|Diesic]]

| {{Monzo| -3 21 -13 }}

| 31 & 13c

| {{Monzo| -7 25 -14 }}

| [[Meantone]]

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

== Quadlayo (31 & 12c) ==

In the [[chain-of-fifths notation]], 5/4 is mapped to the quadruple-diminished fifth (C-Gbbbb).

[[Subgroup]]: 2.3.5

[[Comma list]]: {{monzo| -45 27 1 }}

{{Mapping|legend=1| 1 0 45 | 0 1 -27 }}

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1201.6167{{c}}, ~3/2 = 697.8886{{c}}

: [[error map]]: {{val| +1.617 -2.450 -0.204 }}

* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 696.9075{{c}}

{{Optimal ET sequence|legend=1| 12c, 19c, 31, 136bc, 167bc, 198bc, 229bc }}

[[Badness]] (Sintel): 70.2

== Ampersand ==

: ''For extensions, see [[Gamelismic clan #Miracle]].''

Ampersand is the 5-limit version of miracle, tempering out the [[ampersand comma]], which is the difference between a perfect fifth and a stack of six [[16/15|classical diatonic semitones]]. It can be described as the {{nowrap| 31 & 41 }} temperament, corresponding to {{nowrap| ''n'' {{=}} 6 }}.

[[Subgroup]]: 2.3.5

[[Comma list]]: 34171875/33554432

{{Mapping|legend=1| 1 1 3 | 0 6 -7 }}

[[Optimal tuning]]s:  

* [[CWE]]: ~2 = 1200.000{{c}}, ~16/15 = 116.6802{{c}}

: error map: {{val| 0.000 -1.874 -3.075 }}

 

[[Badness]] (Sintel): 3.89

 

== Valentine (5-limit) ==

: ''For extensions, see [[Gamelismic clan #Valentine]].''

The 5-limit version of valentine tempers out the [[valentine comma]], which is the difference between a perfect fifth and a stack of nine [[25/24|classical chromatic semitones]]. It can be described as the {{nowrap| 31 & 46 }} temperament, corresponding to {{nowrap| ''n'' {{=}} 9 }}.

[[Subgroup]]: 2.3.5

[[Comma list]]: 1990656/1953125

{{Mapping|legend=1| 1 1 2 | 0 9 5 }}

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1199.3579{{c}}, ~25/24 = 77.9973{{c}}

: error map: {{val| 0.000 -0.129 +3.590 }}

{{Optimal ET sequence|legend=1| 15, 31, 46, 77, 123 }}

[[Badness]] (Sintel): 2.88

== Quadlaleyo (31 & 70c) ==

[[Comma list]]: {{monzo| -54 18 11 }}

{{Mapping|legend=1| 1 -8 18 | 0 11 -18 }}

: [[error map]]: {{val| +1.042 -0.876 -1.149 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~32768/30375 = 1045.4008{{c}}

: error map: {{val| 0.000 -2.546 -3.529 }}

{{Optimal ET sequence|legend=1| 8c, 31, 101c, 132, 163 }}

[[Badness]] (Sintel): 48.5

[http://x31eq.com/cgi-bin/rt.cgi?ets=31_70c&limit=5 The temperament finder - 5-limit 31 & 70c]

[[Subgroup]]: 2.3.5

[[Comma list]]: 847288609443/781250000000

{{Mapping|legend=1| 1 -7 -13 | 0 14 25 }}

: mapping generators: ~2, ~19683/12500

[[Optimal tuning]]s:  

* [[WE]]: ~2 = 1200.3614{{c}}, ~19683/12500 = 735.7984{{c}}

: [[error map]]: {{val| +0.361 -3.307 +3.498 }}

: error map: {{val| 0.000 -3.625 -3.560 }}

[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~25000/19683 = 464.423{{c}}

{{Optimal ET sequence|legend=1| 13c, 18bc, 31 }}

[[Badness]] (Sintel): 49.1