353edo: Difference between revisions

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In the original Hebrew calendar, years number 3, 6, 8, 11, 14, 17, and 19 within a 19-year pattern (makhzor, plural:makhzorim) are leap. When converted to [[19edo]], this results in [[5L 2s]] mode, and simply the diatonic major scale.

Following this logic, a temperament can be constructed for the Rectified Hebrew calendar (see below), containing 130 notes of the 353edo scale. Hebrew[130] scale has 334\353 as its generator, which is a supermajor seventh, or alternately, 19\353, about a third-tone, since inverting the generator has no effect on the scale. Such a temperament gives 19edo a unique stretch: 6\19 corresponds to [[5/4]], 13\19 corresponds to [[13/8]], and 15\19 corresponds to [[7/4]]. When measured relative to the generator, the error is less than 1 in 5000.

Following this logic, a temperament can be constructed for the Rectified Hebrew calendar (see below), containing 130 notes of the 353edo scale. Hebrew[130] scale has 334\353 as its generator, which is a supermajor seventh, or alternately, 19\353, about a third-tone, since inverting the generator has no effect on the scale.

Using such small of a generator helps explore the 353edo's "upside down" side. In addition, every sub-pattern in a 19-note generator is actually a Hebrew makhzor, that is a mini-19edo on its own, until it is truncated to an 11-note pattern. Just as the original calendar reform consists of 18 makhzorim with 1 hendecaeteris, Hebrew[130] scale consists of a stack of naively 18 "major scales" finished with one 11-edo tetratonic.

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=== Specific chords and intervals ===

353bbbbb val offers the following resolution sequence: 7/4 D7 - 13/8 D4/3 - D53 - T53. This has a very pleasant sound, with 13/8 acting as a "doubled resolvant" or "resolution into resolution". In the patent val, 169/168 amounts to 3 steps, which is the L step of the full 93L 37s rectified Hebrew scale.

18L 1s of Rectified Hebrew gives 19edo a unique stretch: 6\19 corresponds to [[5/4]], 13\19 corresponds to [[13/8]], and 15\19 corresponds to [[7/4]]. When measured relative to the generator, the error is less than 1 in 5000. 7\19 corresponds to [[13/10]] when measured using the patent val (1306 - 820 - 353 = 133), however the direct approximation using the number is 134 steps. Since patent val is used to define if a comma is "tempered out", repeatedly stacking 7\19 3 times and reducing arrives at 46\353, an approximation for [[35/32]]. The approach using 134 is inconistent by itself already, so therefore it can't be used.

Just as a large amount of [[12edo]] music can be played consistently in 19edo, it can also be played consistently in the 18L 1s subset of Rectified Hebrew.

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|C

|1/1

|-

|1

|C-C#

|

|-

|2

|C-Db

|[[169/168]]

|-

|3

|C-D

|

|-

|4

|C-D#

|

|-

|19

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|Db

|[[14/13]]

|-

|46

|Db-F

|35/32

|-

|57

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|-

|133

|E#~~/Fb~~

|E#

|[[13/10]] ~~minor~~ (~~best~~ approximation is 134)

|[[13/10]] I (patent val approximation)

|-

|134

|E#-C#

|13/10 II (direct approximation)

|-

|152

@@ Line 16: / Line 16: @@
 In the original Hebrew calendar, years number 3, 6, 8, 11, 14, 17, and 19 within a 19-year pattern (makhzor, plural:makhzorim) are leap. When converted to [[19edo]], this results in [[5L 2s]] mode, and simply the diatonic major scale.
-Following this logic, a temperament can be constructed for the Rectified Hebrew calendar (see below), containing 130 notes of the 353edo scale. Hebrew[130] scale has 334\353 as its generator, which is a supermajor seventh, or alternately, 19\353, about a third-tone, since inverting the generator has no effect on the scale. Such a temperament gives 19edo a unique stretch: 6\19 corresponds to [[5/4]], 13\19 corresponds to [[13/8]], and 15\19 corresponds to [[7/4]]. When measured relative to the generator, the error is less than 1 in 5000.
+Following this logic, a temperament can be constructed for the Rectified Hebrew calendar (see below), containing 130 notes of the 353edo scale. Hebrew[130] scale has 334\353 as its generator, which is a supermajor seventh, or alternately, 19\353, about a third-tone, since inverting the generator has no effect on the scale.
 Using such small of a generator helps explore the 353edo's "upside down" side. In addition, every sub-pattern in a 19-note generator is actually a Hebrew makhzor, that is a mini-19edo on its own, until it is truncated to an 11-note pattern. Just as the original calendar reform consists of 18 makhzorim with 1 hendecaeteris, Hebrew[130] scale consists of a stack of naively 18 "major scales" finished with one 11-edo tetratonic.
@@ Line 26: / Line 26: @@
 === Specific chords and intervals ===
 bbbbb val offers the following resolution sequence: 7/4 D7 - 13/8 D4/3 - D53 - T53. This has a very pleasant sound, with 13/8 acting as a "doubled resolvant" or "resolution into resolution". In the patent val, 169/168 amounts to 3 steps, which is the L step of the full 93L 37s rectified Hebrew scale.
+L 1s of Rectified Hebrew gives 19edo a unique stretch: 6\19 corresponds to [[5/4]], 13\19 corresponds to [[13/8]], and 15\19 corresponds to [[7/4]]. When measured relative to the generator, the error is less than 1 in 5000. 7\19 corresponds to [[13/10]] when measured using the patent val (1306 - 820 - 353 = 133), however the direct approximation using the number is 134 steps. Since patent val is used to define if a comma is "tempered out", repeatedly stacking 7\19 3 times and reducing arrives at 46\353, an approximation for [[35/32]]. The approach using 134 is inconistent by itself already, so therefore it can't be used.
 Just as a large amount of [[12edo]] music can be played consistently in 19edo, it can also be played consistently in the 18L 1s subset of Rectified Hebrew.
@@ Line 41: / Line 43: @@
 |C
 |1/1
+|-
+|1
+|C-C#
+|
+|-
+|2
+|C-Db
+|[[169/168]]
+|-
+|3
+|C-D
+|
+|-
+|4
+|C-D#
+|
 |-
 |19
@@ Line 49: / Line 67: @@
 |Db
 |[[14/13]]
+|-
+|46
+|Db-F
+|35/32
 |-
 |57
@@ Line 67: / Line 89: @@
 |-
 |133
-|E#/Fb
+|E#
-|[[13/10]] minor (best approximation is 134)
+|[[13/10]] I (patent val approximation)
+|-
+|134
+|E#-C#
+|13/10 II (direct approximation)
 |-
 |152