131edo: Difference between revisions

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'''131-EDO''', or '''131-tET''', divides the octave into 131 equal steps of approx. 9.1603 Cents, each one. 131edo is the next [[EDO]], after [[81edo]], on the "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 supports [[Sensamagic_clan#Bohpier|bophier temperament]].

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

== Theory ==

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]].

~~<u>~~'''~~Some MOS Scales in~~ 131-~~EDO:~~'''</u>

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, following [[127edo]] and before [[137edo]].

== Scales ==

=== Mos scales ===

{| class="wikitable"

|-

Line 17:

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

| [[7L_2s|Superdiatonic tuning]] (comparable with [[23edo]])

|-

| 16 16 16 16 16 16 16 16 3

| [[8L 1s|Bohpier tuning]] (comparable with [[41edo]])

|-

| 13 13 9 13 13 13 9 13 13 13 9

| ~~Improper~~ [[Sensi-11 Tuning]]

| [[8L 3s|Sensi-11 Tuning]]

|-

| 11 11 11 11 11 5 11 11 11 11 11 11 5

Line 41:

Line 54:

|}

~~[[Category:131edo| ]] ~~

~~[[Category:Armodue]]~~

[[Category:Bohpier]]

~~[[Category:Golden]]~~

~~[[Category:Golden meantone]]~~

[[Category:Immunity]]

[[Category:Meantone]]

[[Category:~~Prime EDO]]~~

[[Category:Golden meantone]]

~~[[Category:Theory~~]]

@@ Line 1: / Line 1: @@
-'''131-EDO''', or '''131-tET''', divides the octave into 131 equal steps of approx. 9.1603 Cents, each one. 131edo is the next [[EDO]], after [[81edo]], on the "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 supports [[Sensamagic_clan#Bohpier|bophier temperament]].
+{{Infobox ET}}
+{{ED intro}}
-edo is the 32nd [[prime]] EDO.
+== Theory ==
+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]].
-<u>'''Some MOS Scales in 131-EDO:'''</u>
+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|columns=15}}
+=== Subsets and supersets ===
+edo is the 32nd [[prime]] edo, following [[127edo]] and before [[137edo]].
+== Scales ==
+=== Mos scales ===
 {| class="wikitable"
 |-
@@ Line 17: / 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
 | [[7L_2s|Superdiatonic tuning]] (comparable with [[23edo]])
+|-
+| 16 16 16 16 16 16 16 16 3
+| [[8L 1s|Bohpier tuning]] (comparable with [[41edo]])
 |-
 | 13 13 9 13 13 13 9 13 13 13 9
-| Improper [[Sensi-11 Tuning]]
+| [[8L 3s|Sensi-11 Tuning]]
 |-
 | 11 11 11 11 11 5 11 11 11 11 11 11 5
@@ Line 41: / Line 54: @@
 |}
-[[Category:131edo| ]] <!-- main article -->
-[[Category:Armodue]]
 [[Category:Bohpier]]
-[[Category:Golden]]
-[[Category:Golden meantone]]
 [[Category:Immunity]]
 [[Category:Meantone]]
-[[Category:Prime EDO]]
+[[Category:Golden meantone]]
-[[Category:Theory]]