1789edo: Difference between revisions

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1789edo is consistent in the no-threes 13-odd-limit. Since its double, [[3578edo]], is consistent in the 21-odd-limit, it can be thought of as a [[K*N subgroups|2*1789]] 2.9.5.7.11.13.225.289.361.21 subgroup temperament, on which it shares mapping with 3578edo and tempers out the same commas.

On the patent val in the 7-limit, 1789edo supports 99 & 373 temperament called maviloid. In addition, it also tempers out [[2401/2400]]. The 1789bd val in the 13-limit is better tuned than the patent val. It provides a tuning for the [[hemiluna]] temperament.

=== Odd harmonics ===

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1789edo supports the 2.5.11.13.19 subgroup temperament called ''estates general'' defined as 1789 & 3125. This is referencing the fact that Estates General were called by Louis XVI on 5th May 1789, written as 05/05, and 3125 is 5 to the 5th power and also provides an optimal patent val for tempering out the jacobin comma, contuing the lore.

~~Other temperaments it supports in this commatic realm are sextilimeans and double Bastille.~~

=== Other ===

1789edo can be used for the finite "French decimal" temperament - that is, where all the interval targets in just intonation are expressed as terminating decimals. For example, [[5/4]], [[25/16]], [[128/125]], [[32/25]], 625/512, etc. This rings particularly true for the French attempts to decimalize a lot more things than we are used to today. Using the maximal evenness method of finding rank-2 temperaments, we get a 1525 & 1789 temperament with comma basis 28824005/28792192, 200126927/200000000, 6106906624/6103515625 in the 2.5.7.11.13 subgroup.

1789edo ~~has an essentially perfect [[9/8]]~~, a ~~very common interval. The associated comma is {{monzo| 5671 -1789 }} in 2.9. This is~~ a ~~direct consequence~~ of ~~1789edo being a~~ [[~~dual-fifth system~~]]~~. 1046th and 1047th steps are not associated with JI intervals by themselves, but satisfy 3+ × 3~~-~~ = 9,~~ which ~~represents 9/1~~, and ~~therefore when octave reduced this leads~~ to 9/~~8. One such example~~, ~~1789edo supports the 2.9.5.11.13 subgroup temperament called ''commatose'' which uses the Pythagorean comma as a generator, which is excess of six 9~~/~~8s over the octave in this case. It is defined as a 460 & 1789 temperament~~.

Since the 5/4 of 1789edo is on the 576th step, a highly divisible number, 1789edo can replicate a lot of [[Ed5/4]] temperaments - more exactly those which are divisors of 576, and that includes all from [[2ed5/4]] to [[9ed5/4]], skipping [[7ed5/4]].

~~Since the 5/4 of~~ 1789edo ~~is on the 576th step, a highly divisible number, 1789edo can replicate a lot of~~ [[~~Ed5~~/4]] ~~temperaments - more exactly those~~ which ~~are divisors~~ of ~~576, and that includes all from [[2ed5~~/~~4]] to [[9ed5/4]], skipping [[7ed5/4]]~~.

1789edo has an essentially perfect [[9/8]], a very common interval. 1789edo supports the 2.9.5.11.13 subgroup temperament called ''commatose'' which uses the Pythagorean comma as a generator, which is excess of six 9/8s over the octave in this case. It is defined as a 460 & 1789 temperament.

Since 1789edo has a very precise 31/29, it supports tricesimoprimal miracloid - a version of secor with 31/29 as the generator and a flat, meantone-esque fifth of about 692.23 cents. Using the maximal evenness method, we find a 52 & 1789 temperament. Best subgroup for it is 2.5.7.11.19.29.31, since both 52edo and 1789edo support it well, and the comma basis is 10241/10240, 5858783/5856400, 4093705/4090624, 15109493/15089800, 102942875/102834688.

On the patent val in the 7-limit, 1789edo supports 99 & 373 temperament called maviloid. In addition, it also tempers out [[2401/2400]]. The 1789bd val in the 13-limit is better tuned than the patent val. It provides a tuning for the [[hemiluna]] temperament.

== Table of selected intervals ==

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|

| 65/62

|-

|125

|Sextilimeans generator

|16807/16000

|-

| 172

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| Jacobin naiadic

| [[13/10]]

|-

|750

|Sextilimeans fourth

|

|-

| 777

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Line 119:

| Secor fifth, Tricesimoprimal Miracle fifth

| (31/29)6

|-

|1039

|Sextilimeans fifth

|

|-

| 1046

@@ Line 7: / Line 7: @@
 edo is consistent in the no-threes 13-odd-limit. Since its double, [[3578edo]], is consistent in the 21-odd-limit, it can be thought of as a [[K*N subgroups|2*1789]] 2.9.5.7.11.13.225.289.361.21 subgroup temperament, on which it shares mapping with 3578edo and tempers out the same commas.
-On the patent val in the 7-limit, 1789edo supports 99 & 373 temperament called maviloid. In addition, it also tempers out [[2401/2400]]. The 1789bd val in the 13-limit is better tuned than the patent val. It provides a tuning for the [[hemiluna]] temperament.
 === Odd harmonics ===
@@ Line 19: / Line 17: @@
 edo supports the 2.5.11.13.19 subgroup temperament called ''estates general'' defined as 1789 & 3125. This is referencing the fact that Estates General were called by Louis XVI on 5th May 1789, written as 05/05, and 3125 is 5 to the 5th power and also provides an optimal patent val for tempering out the jacobin comma, contuing the lore.
-Other temperaments it supports in this commatic realm are sextilimeans and double Bastille.
 === Other ===
 edo can be used for the finite "French decimal" temperament - that is, where all the interval targets in just intonation are expressed as terminating decimals. For example, [[5/4]], [[25/16]], [[128/125]], [[32/25]], 625/512, etc. This rings particularly true for the French attempts to decimalize a lot more things than we are used to today. Using the maximal evenness method of finding rank-2 temperaments, we get a 1525 & 1789 temperament with comma basis 28824005/28792192, 200126927/200000000, 6106906624/6103515625 in the 2.5.7.11.13 subgroup.
-edo has an essentially perfect [[9/8]], a very common interval. The associated comma is {{monzo| 5671 -1789 }} in 2.9. This is a direct consequence of 1789edo being a [[dual-fifth system]]. 1046th and 1047th steps are not associated with JI intervals by themselves, but satisfy 3<sup>+</sup> × 3<sup>-</sup> = 9, which represents 9/1, and therefore when octave reduced this leads to 9/8. One such example, 1789edo supports the 2.9.5.11.13 subgroup temperament called ''commatose'' which uses the Pythagorean comma as a generator, which is excess of six 9/8s over the octave in this case. It is defined as a 460 & 1789 temperament.
+Since the 5/4 of 1789edo is on the 576th step, a highly divisible number, 1789edo can replicate a lot of [[Ed5/4]] temperaments - more exactly those which are divisors of 576, and that includes all from [[2ed5/4]] to [[9ed5/4]], skipping [[7ed5/4]].
-Since the 5/4 of 1789edo is on the 576th step, a highly divisible number, 1789edo can replicate a lot of [[Ed5/4]] temperaments - more exactly those which are divisors of 576, and that includes all from [[2ed5/4]] to [[9ed5/4]], skipping [[7ed5/4]].
+edo has an essentially perfect [[9/8]], a very common interval. 1789edo supports the 2.9.5.11.13 subgroup temperament called ''commatose'' which uses the Pythagorean comma as a generator, which is excess of six 9/8s over the octave in this case. It is defined as a 460 & 1789 temperament.
 Since 1789edo has a very precise 31/29, it supports tricesimoprimal miracloid - a version of secor with 31/29 as the generator and a flat, meantone-esque fifth of about 692.23 cents. Using the maximal evenness method, we find a 52 & 1789 temperament. Best subgroup for it is 2.5.7.11.19.29.31, since both 52edo and 1789edo support it well, and the comma basis is 10241/10240, 5858783/5856400, 4093705/4090624, 15109493/15089800, 102942875/102834688.
+On the patent val in the 7-limit, 1789edo supports 99 & 373 temperament called maviloid. In addition, it also tempers out [[2401/2400]]. The 1789bd val in the 13-limit is better tuned than the patent val. It provides a tuning for the [[hemiluna]] temperament.
 == Table of selected intervals ==
@@ Line 65: / Line 63: @@
 |
 | 65/62
+|-
+|125
+|Sextilimeans generator
+|16807/16000
 |-
 | 172
@@ Line 101: / Line 103: @@
 | Jacobin naiadic
 | [[13/10]]
+|-
+|750
+|Sextilimeans fourth
+|
 |-
 | 777
@@ Line 113: / Line 119: @@
 | Secor fifth, Tricesimoprimal Miracle fifth
 | (31/29)<sup>6</sup>
+|-
+|1039
+|Sextilimeans fifth
+|
 |-
 | 1046