131edo: Difference between revisions

(11 intermediate revisions by 3 users not shown)

Line 1:

== Theory ==

131edo is the next [[edo]] after [[81edo]] on the [[Golden meantone|Golden Tone System]] (''[[Das Goldene Tonsystem]]'') of Thorvald Kornerup, using the 131b [[val]]. The [[patent val]] has a fifth sharp by 3.389 cents rather than flat like the meantone fifth; rather than tempering out [[81/80]] it tempers out the [[immunity comma]], 1638400/1594323. In the 7-limit it tempers out [[3125/3087]] and [[245/243]], so that it [[support]]s [[~~bophier~~]].

131edo is in[[consistent]] to the [[5-odd-limit]] and the error of [[harmonic]] [[3/1|3]] is quite large. However, it is the next [[edo]] after [[81edo]] on the [[Golden meantone|Golden Tone System]] (''[[Das Goldene Tonsystem]]'') of Thorvald Kornerup, using the 131b [[val]]. The [[patent val]] has a fifth sharp by 3.389 cents rather than flat like the meantone fifth; rather than tempering out [[81/80]] it tempers out the [[immunity comma]], 1638400/1594323. In the 7-limit it tempers out [[3125/3087]] and [[245/243]], so that it [[support]]s [[bohpier]].

131edo is also notable for having a good approximation to [[natave|acoustic ''e'']], at 189\131, which is a [[semiconvergent]]. This number of steps, 189, is particularly well-factorizable, and logarithmic divisors of acoustic ''e'' form a sequence of rapidly converging approximations to small rationals. Among these are [[4/3]] (2\7[[EDN|edn]] = 54\131), [[5/4]] (2\9edn = 42\131), [[15/13]] (1\7edn = 27\131), [[19/17]] (1\9edn = 21\131), [[11/10]] (2\21edn = 18\131), [[14/13]] (2\27edn = 14\131), and [[32/31]] (2\63edn = 6\131), with accuracy increasing the smaller the fraction.

=== Odd harmonics ===

=== Subsets and supersets ===

131edo is the 32nd [[prime]] edo.

131edo is the 32nd [[prime]] edo, following [[127edo]] and before [[137edo]].

== Scales ==

Line 25:

Line 27:

|-

| 19 12 19 19 12 19 19 12

| [[5L_3s|~~Father Tuning~~]] (comparable with [[55edo]])

| [[5L_3s|Oneirotonic tuning]] (comparable with [[55edo]])

|-

| 18 18 18 18 18 18 18 5

| [[7L_1s|Porcupine ~~Tuning~~]] (comparable with [[29edo]] and [[80edo]])

| [[7L_1s|Porcupine tuning]] (comparable with [[29edo]] and [[80edo]])

|-

| 17 17 17 6 17 17 17 17 6

Line 52:

Line 54:

|}

~~[[Category:Armodue]]~~

[[Category:Bohpier]]

[[Category:Immunity]]

[[Category:Meantone]]

[[Category:Golden meantone]]

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|131}}
+{{ED intro}}
 == Theory ==
-edo is the next [[edo]] after [[81edo]] on the [[Golden meantone|Golden Tone System]] (''[[Das Goldene Tonsystem]]'') of Thorvald Kornerup, using the 131b [[val]]. The [[patent val]] has a fifth sharp by 3.389 cents rather than flat like the meantone fifth; rather than tempering out [[81/80]] it tempers out the [[immunity comma]], 1638400/1594323. In the 7-limit it tempers out [[3125/3087]] and [[245/243]], so that it [[support]]s [[bophier]].
+edo is in[[consistent]] to the [[5-odd-limit]] and the error of [[harmonic]] [[3/1|3]] is quite large. However, it is the next [[edo]] after [[81edo]] on the [[Golden meantone|Golden Tone System]] (''[[Das Goldene Tonsystem]]'') of Thorvald Kornerup, using the 131b [[val]]. The [[patent val]] has a fifth sharp by 3.389 cents rather than flat like the meantone fifth; rather than tempering out [[81/80]] it tempers out the [[immunity comma]], 1638400/1594323. In the 7-limit it tempers out [[3125/3087]] and [[245/243]], so that it [[support]]s [[bohpier]].
+edo is also notable for having a good approximation to [[natave|acoustic ''e'']], at 189\131, which is a [[semiconvergent]]. This number of steps, 189, is particularly well-factorizable, and logarithmic divisors of acoustic ''e'' form a sequence of rapidly converging approximations to small rationals. Among these are [[4/3]] (2\7[[EDN|edn]] = 54\131), [[5/4]] (2\9edn = 42\131), [[15/13]] (1\7edn = 27\131), [[19/17]] (1\9edn = 21\131), [[11/10]] (2\21edn = 18\131), [[14/13]] (2\27edn = 14\131), and [[32/31]] (2\63edn = 6\131), with accuracy increasing the smaller the fraction.
 === Odd harmonics ===
-{{Harmonics in equal|131}}
+{{Harmonics in equal|131|columns=15}}
 === Subsets and supersets ===
-edo is the 32nd [[prime]] edo.
+edo is the 32nd [[prime]] edo, following [[127edo]] and before [[137edo]].
 == Scales ==
@@ Line 25: / Line 27: @@
 |-
 | 19 12 19 19 12 19 19 12
-| [[5L_3s|Father Tuning]] (comparable with [[55edo]])
+| [[5L_3s|Oneirotonic tuning]] (comparable with [[55edo]])
 |-
 | 18 18 18 18 18 18 18 5
-| [[7L_1s|Porcupine Tuning]] (comparable with [[29edo]] and [[80edo]])
+| [[7L_1s|Porcupine tuning]] (comparable with [[29edo]] and [[80edo]])
 |-
 | 17 17 17 6 17 17 17 17 6
@@ Line 52: / Line 54: @@
 |}
-[[Category:Armodue]]
 [[Category:Bohpier]]
 [[Category:Immunity]]
 [[Category:Meantone]]
 [[Category:Golden meantone]]