388edo: Difference between revisions

(15 intermediate revisions by 4 users not shown)

Line 1:

The '''388 equal divisions of the octave''' ('''388edo'''), or the '''388(-tone) equal temperament''' ('''388tet''', '''388et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 388 [[equal]] parts of about 3.09 [[cent]]s each.

== Theory ==

388edo is the first edo that is ~~uniquely~~ [[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 ~~5632~~/~~5625~~, [[~~3025~~/~~3024~~]] 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 cuthbert temperament, which tempers out ~~cuthbert, the~~ 847/845 comma, and for a number of other temperaments tempering out ~~cuthbert~~, e.g. [[neusec]], the 190&~~amp;~~198 temperament. By tempering out cuthbert it [[support]]s ~~the~~ [[cuthbert ~~triad~~]], in addition to [[sinbadmic chords]].

The equal temperament [[tempering out|tempers out]] the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[alphatricot 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 {{nowrap| 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 primes as {{nowrap| 22 × 97 }}, 388edo has subset edos {{EDOs| 2, 4, 97, and 194 }}.

== Approximation to JI ==

This edo has a high consistency limit, although due to [[311edo]] having a higher consistency limit, among other things, it is mostly unexplored. However, it still makes for an interesting comparison.

388edo has a consistency limit of 37 compared to 311edo's consistency limit of 41, so most people wishing to approximate higher-limit JI choose the latter. 311edo also approximates harmonics up to 42 with under 25% error, while 388edo fails as early as harmonic [[7/1|7]]. One thing to note is that 388edo's distinct consistency limit is 27 compared to 311edo's 23, so those who want distinct consistency in the 27-odd-limit may choose 388.

311edo also deals better with composite harmonics than 388edo. 311edo is consistent to the 41-limit 77-odd-limit, while 388edo has inconsistencies involving composite harmonics as low as 39, and harmonic 49 itself is inconsistent. The 7th and 11th harmonics both being flat by just over 25% of a step is less than ideal. However, it approximates some higher primes better than 311 does. The only inconsistencies in the 41-odd-limit in 388edo are 39/28, 39/22 ,39/37, 41/28, 41/22, 41/37 and their octave complements. This is due to the fact that harmonics 39 and 41 are quite sharp, both just over 1/4 of a step. 311edo misses most primes after 41, though it hits 73, 89, (101,) 109, and 113. 388, on the other hand, hits primes 47, 61, 71, 79, 97, 109, and 113. Still, 311 does much better at composite harmonics due to having much lower error in the 13-limit, which is also important to note by itself, though if one wants to approximate the 13-limit specifically they may prefer [[270edo]] or [[494edo]]. Note that 311edo has generally higher absolute errors than 388edo due to its smaller size, but having a smaller size also means the system is easier to handle.

Another system notable in high limits around this size is [[422edo]], using the sharp-tending 422[[wart|l]] [[val]] for prime 37.

=== 37-odd-limit intervals ===

== Regular temperament properties ==

{| class="wikitable center-4 center-5 center-6"

! rowspan="2" | Subgroup

|-

! rowspan="2" | [[Subgroup]]

! rowspan="2" | [[Comma list]]

! rowspan="2" | [[Mapping]]

Line 22:

Line 42:

|-

| 2.3

| {{~~monzo~~| 615 -388 }}

| {{Monzo| 615 -388 }}

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

| {{Mapping| 388 615 }}

| +0.0337

| 0.0337

Line 29:

Line 49:

|-

| 2.3.5

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

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

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

| {{Mapping| 388 615 901 }}

| -0.0633

| −0.0633

| 0.0501

| 1.62

Line 37:

Line 57:

| 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 64:

| 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 71:

| 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 78:

| 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 85:

| 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 ===

{| class="wikitable center-all left-5"

|+Table of rank-2 temperaments by generator

|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator

! Periods per ~~octave~~

|-

! Generator~~ (reduced)~~

! Periods per 8ve

! Cents~~ (reduced)~~

! Generator*

! Associated ratio

! Cents*

! Associated ratio*

! Temperaments

|-

Line 102:

Line 124:

| 565.97

| 75/52

| [[~~Trillium~~]] / [[pseudotrillium]]

| [[Alphatrillium]] / [[pseudotrillium]]

|-

| 2

Line 121:

Line 143:

| 81/65 (22/21)

| [[Quasithird]]

|-

| 97

| 161\388 (1\388)

| 497.938 (3.09)

| 4/3 (?)

| [[Berkelium]]

|}

<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct

~~[[Category:Equal divisions of the octave|###]] ~~

[[Category:Cuthbert]]

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-The '''388 equal divisions of the octave''' ('''388edo'''), or the '''388(-tone) equal temperament''' ('''388tet''', '''388et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 388 [[equal]] parts of about 3.09 [[cent]]s each.
+{{ED intro}}
 == Theory ==
-edo is the first edo that is uniquely [[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 5632/5625, [[3025/3024]] 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 cuthbert temperament, which tempers out cuthbert, the 847/845 comma, and for a number of other temperaments tempering out cuthbert, e.g. [[neusec]], the 190&amp;198 temperament. By tempering out cuthbert it [[support]]s the [[cuthbert triad]], in addition to [[sinbadmic chords]].
+The equal temperament [[tempering out|tempers out]] the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[alphatricot 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 {{nowrap| 190 & 198 }} temperament. By tempering out cuthbert it [[support]]s [[cuthbert chords]], in addition to [[sinbadmic chords]].
 === Prime harmonics ===
-{{Harmonics in equal|388|columns=11}}
+{{Harmonics in equal|388|columns=12}}
+{{Harmonics in equal|388|start=13|columns=12|collapsed=1}}
+=== Subsets and supersets ===
+Since 388 factors into primes as {{nowrap| 2<sup>2</sup> × 97 }}, 388edo has subset edos {{EDOs| 2, 4, 97, and 194 }}.
+== Approximation to JI ==
+This edo has a high consistency limit, although due to [[311edo]] having a higher consistency limit, among other things, it is mostly unexplored. However, it still makes for an interesting comparison.
+edo has a consistency limit of 37 compared to 311edo's consistency limit of 41, so most people wishing to approximate higher-limit JI choose the latter. 311edo also approximates harmonics up to 42 with under 25% error, while 388edo fails as early as harmonic [[7/1|7]]. One thing to note is that 388edo's distinct consistency limit is 27 compared to 311edo's 23, so those who want distinct consistency in the 27-odd-limit may choose 388.
+edo also deals better with composite harmonics than 388edo. 311edo is consistent to the 41-limit 77-odd-limit, while 388edo has inconsistencies involving composite harmonics as low as 39, and harmonic 49 itself is inconsistent. The 7th and 11th harmonics both being flat by just over 25% of a step is less than ideal. However, it approximates some higher primes better than 311 does. The only inconsistencies in the 41-odd-limit in 388edo are 39/28, 39/22 ,39/37, 41/28, 41/22, 41/37 and their octave complements. This is due to the fact that harmonics 39 and 41 are quite sharp, both just over 1/4 of a step. 311edo misses most primes after 41, though it hits 73, 89, (101,) 109, and 113. 388, on the other hand, hits primes 47, 61, 71, 79, 97, 109, and 113. Still, 311 does much better at composite harmonics due to having much lower error in the 13-limit, which is also important to note by itself, though if one wants to approximate the 13-limit specifically they may prefer [[270edo]] or [[494edo]]. Note that 311edo has generally higher absolute errors than 388edo due to its smaller size, but having a smaller size also means the system is easier to handle.
+Another system notable in high limits around this size is [[422edo]], using the sharp-tending 422[[wart|l]] [[val]] for prime 37.
+=== 37-odd-limit intervals ===
+{{Q-odd-limit intervals|388|limit=37}}
 == Regular temperament properties ==
 {| class="wikitable center-4 center-5 center-6"
-! rowspan="2" | Subgroup
+|-
+! rowspan="2" | [[Subgroup]]
 ! rowspan="2" | [[Comma list]]
 ! rowspan="2" | [[Mapping]]
@@ Line 22: / Line 42: @@
 |-
 | 2.3
-| {{monzo| 615 -388 }}
+| {{Monzo| 615 -388 }}
-| [{{val| 388 615 }}]
+| {{Mapping| 388 615 }}
 | +0.0337
 | 0.0337
@@ Line 29: / Line 49: @@
 |-
 | 2.3.5
-| {{monzo| 23 6 -14 }}, {{monzo| 39 -29 3 }}
+| {{Monzo| 23 6 -14 }}, {{monzo| 39 -29 3 }}
-| [{{val| 388 615 901 }}]
+| {{Mapping| 388 615 901 }}
-| -0.0633
+| −0.0633
 | 0.0501
 | 1.62
@@ Line 37: / Line 57: @@
 | 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 64: @@
 | 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 71: @@
 | 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 78: @@
 | 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 85: @@
 | 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 ===
 {| class="wikitable center-all left-5"
-|+Table of rank-2 temperaments by generator
+|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
-! Periods<br>per octave
+|-
-! Generator<br>(reduced)
+! Periods<br>per 8ve
-! Cents<br>(reduced)
+! Generator*
-! Associated<br>ratio
+! Cents*
+! Associated<br>ratio*
 ! Temperaments
 |-
@@ Line 102: / Line 124: @@
 | 565.97
 | 75/52
-| [[Trillium]] / [[pseudotrillium]]
+| [[Alphatrillium]] / [[pseudotrillium]]
 |-
 | 2
@@ Line 121: / Line 143: @@
 | 81/65<br>(22/21)
 | [[Quasithird]]
+|-
+| 97
+| 161\388<br>(1\388)
+| 497.938<br>(3.09)
+| 4/3<br>(?)
+| [[Berkelium]]
 |}
+<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct
-[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
 [[Category:Cuthbert]]