388edo: Difference between revisions

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== Theory ==

388edo is the first edo that is ~~distinctly~~ [[consistent]] through to the [[27-odd-limit]]; it is also consistent through the 37-odd-limit.

388edo is the first edo that is [[consistency|distinctly consistent]] through to the [[27-odd-limit]]; it is also consistent through the [[37-odd-limit]].

~~388et~~ tempers out the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[tricot comma]], {{monzo| 39 -29 3 }}, the [[minortone comma]], {{monzo| -16 35 -17 }}, and the [[Very high accuracy temperaments #Raider|raider comma]], {{monzo| 71 -99 31 }}, in the 5-limit, ~~and provides~~ a tuning ~~with less error than any previous equal temperaments~~. It tempers out [[4375/4374]] and 235298/234375 in the 7-limit, and [[3025/3024]], [[5632/5625]] and [[9801/9800]] in the 11-limit and [[847/845]], [[1001/1000]] and [[4096/4095]] in the 13-limit. It is the [[optimal patent val]] for the rank-5 cuthbert temperament, which tempers out ~~cuthbert, the~~ [[847/845]] comma, and for a number of other temperaments tempering it out, e.g. [[neusec]], the 190 & 198 temperament. By tempering out cuthbert it [[support]]s [[cuthbert chords]], in addition to [[sinbadmic chords]].

The equal temperament [[tempering out|tempers out]] the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[tricot comma]], {{monzo| 39 -29 3 }}, the [[minortone comma]], {{monzo| -16 35 -17 }}, and the [[Very high accuracy temperaments #Raider|raider comma]], {{monzo| 71 -99 31 }}, in the 5-limit, giving a strong tuning. It tempers out [[4375/4374]] and 235298/234375 in the 7-limit, and [[3025/3024]], [[5632/5625]] and [[9801/9800]] in the 11-limit and [[847/845]], [[1001/1000]] and [[4096/4095]] in the 13-limit.

It provides the [[optimal patent val]] for the rank-5 cuthbert temperament, which tempers out [[847/845]], the cuthbert comma, and for a number of other temperaments tempering it out, e.g. [[neusec]], the 190 & 198 temperament. By tempering out cuthbert it [[support]]s [[cuthbert chords]], in addition to [[sinbadmic chords]].

=== Prime harmonics ===

=== Subsets and supersets ===

Since 388 factors into {{factorization|388}}, 388edo has subset edos {{EDOs| 2, 4, 97, and 194 }}.

== Regular temperament properties ==

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| 2.3

| {{monzo| 615 -388 }}

| [{{~~val~~| 388 615 }}]

| {{mapping| 388 615 }}

| +0.0337

| 0.0337

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| 2.3.5

| {{monzo| 23 6 -14 }}, {{monzo| 39 -29 3 }}

| [{{~~val~~| 388 615 901 }}]

| {{mapping| 388 615 901 }}

| -0.0633

| 0.0501

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| 2.3.5.7

| 4375/4374, 235298/234375, 2100875/2097152

| [{{~~val~~| 388 615 901 1089 }}]

| {{mapping| 388 615 901 1089 }}

| +0.0224

| 0.1546

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| 2.3.5.7.11

| 3025/3024, 4375/4374, 5632/5625, 235298/234375

| [{{~~val~~| 388 615 901 1089 1342 }}]

| {{mapping| 388 615 901 1089 1342 }}

| +0.0643

| 0.1617

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| 2.3.5.7.11.13

| 847/845, 1001/1000, 3025/3024, 4096/4095, 4375/4374

| [{{~~val~~| 388 615 901 1089 1342 1436 }}]

| {{mapping| 388 615 901 1089 1342 1436 }}

| +0.0216

| 0.1758

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| 2.3.5.7.11.13.17

| 833/832, 847/845, 1001/1000, 1089/1088, 1225/1224, 1701/1700

| [{{~~val~~| 388 615 901 1089 1342 1436 1586 }}]

| {{mapping| 388 615 901 1089 1342 1436 1586 }}

| +0.0116

| 0.1646

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| 2.3.5.7.11.13.17.19

| 833/832, 847/845, 1001/1000, 1089/1088, 1216/1215, 1225/1224, 1331/1330

| [{{~~val~~| 388 615 901 1089 1342 1436 1586 1648 }}]

| {{mapping| 388 615 901 1089 1342 1436 1586 1648 }}

| +0.0280

| 0.1600

| 5.17

|}

* 388et has a lower absolute error in the 5-limit than any previous equal temperaments, past [[323edo|323]] and followed by [[441edo|441]].

=== Rank-2 temperaments ===

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|+Table of rank-2 temperaments by generator

! Periods per 8ve

! Generator~~ (Reduced)~~

! Generator*

! Cents~~ (Reduced)~~

! Cents*

! Associated Ratio

! Associated Ratio*

! Temperaments

|-

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| [[Berkelium]]

|}

<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct

[[Category:Cuthbert]]

@@ Line 3: / Line 3: @@
 == Theory ==
-edo is the first edo that is distinctly [[consistent]] through to the [[27-odd-limit]]; it is also consistent through the 37-odd-limit.
+edo is the first edo that is [[consistency|distinctly consistent]] through to the [[27-odd-limit]]; it is also consistent through the [[37-odd-limit]].
-et tempers out the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[tricot comma]], {{monzo| 39 -29 3 }}, the [[minortone comma]], {{monzo| -16 35 -17 }}, and the [[Very high accuracy temperaments #Raider|raider comma]], {{monzo| 71 -99 31 }}, in the 5-limit, and provides a tuning with less error than any previous equal temperaments. It tempers out [[4375/4374]] and 235298/234375 in the 7-limit, and [[3025/3024]], [[5632/5625]] and [[9801/9800]] in the 11-limit and [[847/845]], [[1001/1000]] and [[4096/4095]] in the 13-limit. It is the [[optimal patent val]] for the rank-5 cuthbert temperament, which tempers out cuthbert, the [[847/845]] comma, and for a number of other temperaments tempering it out, e.g. [[neusec]], the 190 &amp; 198 temperament. By tempering out cuthbert it [[support]]s [[cuthbert chords]], in addition to [[sinbadmic chords]].
+The equal temperament [[tempering out|tempers out]] the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[tricot comma]], {{monzo| 39 -29 3 }}, the [[minortone comma]], {{monzo| -16 35 -17 }}, and the [[Very high accuracy temperaments #Raider|raider comma]], {{monzo| 71 -99 31 }}, in the 5-limit, giving a strong tuning. It tempers out [[4375/4374]] and 235298/234375 in the 7-limit, and [[3025/3024]], [[5632/5625]] and [[9801/9800]] in the 11-limit and [[847/845]], [[1001/1000]] and [[4096/4095]] in the 13-limit.
+It provides the [[optimal patent val]] for the rank-5 cuthbert temperament, which tempers out [[847/845]], the cuthbert comma, and for a number of other temperaments tempering it out, e.g. [[neusec]], the 190 &amp; 198 temperament. By tempering out cuthbert it [[support]]s [[cuthbert chords]], in addition to [[sinbadmic chords]].
 === Prime harmonics ===
 {{Harmonics in equal|388|columns=12}}
+=== Subsets and supersets ===
+Since 388 factors into {{factorization|388}}, 388edo has subset edos {{EDOs| 2, 4, 97, and 194 }}.
 == Regular temperament properties ==
@@ Line 23: / Line 28: @@
 | 2.3
 | {{monzo| 615 -388 }}
-| [{{val| 388 615 }}]
+| {{mapping| 388 615 }}
 | +0.0337
 | 0.0337
@@ Line 30: / Line 35: @@
 | 2.3.5
 | {{monzo| 23 6 -14 }}, {{monzo| 39 -29 3 }}
-| [{{val| 388 615 901 }}]
+| {{mapping| 388 615 901 }}
 | -0.0633
 | 0.0501
@@ Line 37: / Line 42: @@
 | 2.3.5.7
 | 4375/4374, 235298/234375, 2100875/2097152
-| [{{val| 388 615 901 1089 }}]
+| {{mapping| 388 615 901 1089 }}
 | +0.0224
 | 0.1546
@@ Line 44: / Line 49: @@
 | 2.3.5.7.11
 | 3025/3024, 4375/4374, 5632/5625, 235298/234375
-| [{{val| 388 615 901 1089 1342 }}]
+| {{mapping| 388 615 901 1089 1342 }}
 | +0.0643
 | 0.1617
@@ Line 51: / Line 56: @@
 | 2.3.5.7.11.13
 | 847/845, 1001/1000, 3025/3024, 4096/4095, 4375/4374
-| [{{val| 388 615 901 1089 1342 1436 }}]
+| {{mapping| 388 615 901 1089 1342 1436 }}
 | +0.0216
 | 0.1758
@@ Line 58: / Line 63: @@
 | 2.3.5.7.11.13.17
 | 833/832, 847/845, 1001/1000, 1089/1088, 1225/1224, 1701/1700
-| [{{val| 388 615 901 1089 1342 1436 1586 }}]
+| {{mapping| 388 615 901 1089 1342 1436 1586 }}
 | +0.0116
 | 0.1646
@@ Line 65: / Line 70: @@
 | 2.3.5.7.11.13.17.19
 | 833/832, 847/845, 1001/1000, 1089/1088, 1216/1215, 1225/1224, 1331/1330
-| [{{val| 388 615 901 1089 1342 1436 1586 1648 }}]
+| {{mapping| 388 615 901 1089 1342 1436 1586 1648 }}
 | +0.0280
 | 0.1600
 | 5.17
 |}
+* 388et has a lower absolute error in the 5-limit than any previous equal temperaments, past [[323edo|323]] and followed by [[441edo|441]].
 === Rank-2 temperaments ===
@@ Line 75: / Line 81: @@
 |+Table of rank-2 temperaments by generator
 ! Periods<br>per 8ve
-! Generator<br>(Reduced)
+! Generator*
-! Cents<br>(Reduced)
+! Cents*
-! Associated<br>Ratio
+! Associated<br>Ratio*
 ! Temperaments
 |-
@@ Line 128: / Line 134: @@
 | [[Berkelium]]
 |}
+<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
 [[Category:Cuthbert]]