190edo: Difference between revisions

(12 intermediate revisions by 6 users not shown)

Line 1:

{{Infobox ET

~~| Prime factorization = 2 × 5 × 19~~

~~| Step size = 6.31579¢~~

~~| Fifth = 111\190 (701.05¢)~~

~~| Semitones = 17:15 (107.37¢ : 94.74¢)~~

~~| Consistency = 15~~

}}

The '''190 equal divisions of the octave''' ('''190edo''') or '''190(-tone) equal temperament''' ('''190tet''', '''190et''') when view from a [[regular temperament]] perspective, divides the [[octave]] into 190 equal parts of about 6.32 [[cent]]s each.

== Theory ==

190edo is interesting because of the utility of its approximations; it tempers out [[1029/1024]], [[4375/4374]], [[385/384]], [[441/440]], [[3025/3024]] and [[9801/9800]]. It provides the [[optimal patent val]] for both the 7- and 11-limit versions of [[unidec]], the 72 & 118 temperament, which tempers out 1029/1024, 4375/4374, and in the 11-limit, 385/384 and 441/440. It also provides the optimal patent val for the rank-3 11-limit temperament [[portent]], which tempers out 385/384 and 441/440, and [[gamelan]], the rank-3 7-limit temperament which tempers out 1029/1024, as well as [[slendric]], the 2.3.7 subgroup temperament featured in the [[#Music]] section. In the 13-limit, 190et tempers out [[~~847~~/~~845~~]], [[~~625~~/~~624~~]], [[~~729~~/~~728~~]], [[~~1575~~/~~1573~~]] and [[~~1001~~/~~1000~~]], and provides the optimal patent val for the [[ekadash]] temperament and the rank-3 [[portentous]] temperament.

190edo is [[consistency|distinctly consistent]] in the [[15-odd-limit]] with a flat tendency, as [[harmonic]]s 3 through 13 are all tuned flat.

190edo is interesting because of the utility of its approximations; it [[tempering out|tempers out]] [[1029/1024]], [[4375/4374]], [[385/384]], [[441/440]], [[3025/3024]], and [[9801/9800]]. It provides the [[optimal patent val]] for both the 7- and 11-limit versions of [[unidec]], the {{nowrap|72 & 118}} temperament, which tempers out 1029/1024, 4375/4374, and in the 11-limit, 385/384 and 441/440. It also provides the optimal patent val for the rank-3 11-limit temperament [[portent]], which tempers out 385/384 and 441/440, and [[gamelan]], the rank-3 7-limit temperament which tempers out 1029/1024, as well as [[slendric]], the 2.3.7 subgroup temperament featured in the [[#Music]] section. In the 13-limit, 190et tempers out [[625/624]], [[729/728]], [[847/845]], [[1001/1000]] and [[1575/1573]], and provides the optimal patent val for the [[ekadash]] temperament and the rank-3 [[portentous]] temperament.

The 190g [[val]] shows us a smooth path to the even higher limits. This extension tempers out [[289/288]], [[561/560]], [[595/594]] in the 17-limit; [[343/342]], [[476/475]], [[495/494]] in the 19-limit; and [[391/390]], [[529/528]] in the 23-limit.

=== Prime harmonics ===

=== Subsets and supersets ===

Since 190 factors into {{factorization|190}}, 190edo has subset edos {{EDOs| 2, 5, 10, 19, 38, and 95 }}.

== Regular temperament properties ==

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

|-

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

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

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

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

! rowspan="2" | Optimal 8ve stretch (¢)

Line 26:

Line 28:

|-

| 2.3

| {{~~monzo~~| -301 190 }}

| {{Monzo| -301 190 }}

| [{{~~val~~| 190 301 }}]

| {{Mapping| 190 301 }}

| +0.285

| 0.285

Line 34:

Line 36:

| 2.3.5

| 2109375/2097152, {{monzo| -7 22 -12 }}

| [{{~~val~~| 190 301 441 }}]

| {{Mapping| 190 301 441 }}

| +0.341

| 0.246

Line 41:

Line 43:

| 2.3.5.7

| 1029/1024, 4375/4374, 235298/234375

| [{{~~val~~| 190 301 441 533 }}]

| {{Mapping| 190 301 441 533 }}

| +0.479

| 0.321

Line 48:

Line 50:

| 2.3.5.7.11

| 385/384, 441/440, 4375/4374, 234375/234256

| [{{~~val~~| 190 301 441 533 657 }}]

| {{Mapping| 190 301 441 533 657 }}

| +0.490

| 0.288

Line 55:

Line 57:

| 2.3.5.7.11.13

| 385/384, 441/440, 625/624, 729/728, 847/845

| [{{~~val~~| 190 301 441 533 657 703 }}]

| {{Mapping| 190 301 441 533 657 703 }}

| +0.432

| 0.293

| 4.63

|-

| 2.3.5.7.11.13.17

| 289/288, 385/384, 441/440, 561/560, 625/624, 847/845

| {{Mapping| 190 301 441 533 657 703 776 }} (190g)

| +0.507

| 0.327

| 5.18

|-

| 2.3.5.7.11.13.17.19

| 289/288, 343/342, 385/384, 441/440, 476/475, 495/494, 847/845

| {{Mapping| 190 301 441 533 657 703 776 807 }} (190g)

| +0.463

| 0.327

| 5.17

|-

| 2.3.5.7.11.13.17.19.23

| 289/288, 343/342, 385/384, 391/390, 441/440, 476/475, 495/494, 529/528

| {{Mapping| 190 301 441 533 657 703 776 807 859 }} (190g)

| +0.486

| 0.315

| 4.98

|}

* 190et (190g val) has a lower relative error in the 23-limit than any previous equal temperaments, being the first to beat [[94edo|94]]. However, [[193edo|193]], only slightly larger, beats it.

* It is also prominent in the 13- and 19-limit, with lower absolute errors than any previous equal temperaments. It beats [[183edo|183]] in either subgroup and is bettered by [[198edo|198]] in the 13-limit, and by 193 in the 19-limit.

=== 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*

! ~~Temperaments~~

! Associated ratio*

! Temperament

|-

| 1

Line 80:

Line 106:

| 271.58

| 75/64

| [[~~Orson]] / [[sabric~~]]

| [[Sabric]]

|-

| 1

Line 128:

Line 154:

| 498.95 (18.95)

| 4/3 (81/80)

| [[~~Pental~~]]

| [[Quintile]]

|-

| 10

Line 140:

Line 166:

| 498.95 (18.95)

| 4/3 (81/80)

| [[~~Decal~~]]

| [[Decile]]

|-

| 19

Line 160:

Line 186:

| [[Semihemienneadecal]]

|}

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

== Scales ==

Line 168:

Line 195:

== Music ==

* [http://micro.soonlabel.com/tuning-survey/daily20111026-the-11th-slendric-fanfare.mp3 The 11th Slendric Fanfare]

; [[Chris Vaisvil]]

* [http://micro.soonlabel.com/tuning-survey/daily20111026-16-slendric-virgins.mp3 16 Slendric Virgins~~] by [[Chris Vaisvil]~~]

* [http://micro.soonlabel.com/tuning-survey/daily20111026-the-11th-slendric-fanfare.mp3 ''The 11th Slendric Fanfare'']

* [http://micro.soonlabel.com/tuning-survey/daily20111026-16-slendric-virgins.mp3 ''16 Slendric Virgins'']

~~[[Category:190edo| ]] ~~

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

~~[[Category:Unidec]]~~

[[Category:Ekadash]]

[[Category:~~Gamelan~~]]

[[Category:Gamelismic]]

[[Category:Listen]]

[[Category:Portent]]

[[Category:Portentous]]

[[Category:Unidec]]

@@ Line 1: / Line 1: @@
-{{Infobox ET
+{{Infobox ET}}
-| Prime factorization = 2 × 5 × 19
+{{ED intro}}
-| Step size = 6.31579¢
-| Fifth = 111\190 (701.05¢)
-| Semitones = 17:15 (107.37¢ : 94.74¢)
-| Consistency = 15
-}}
-The '''190 equal divisions of the octave''' ('''190edo''') or '''190(-tone) equal temperament''' ('''190tet''', '''190et''') when view from a [[regular temperament]] perspective, divides the [[octave]] into 190 equal parts of about 6.32 [[cent]]s each.
 == Theory ==
-edo is interesting because of the utility of its approximations; it tempers out [[1029/1024]], [[4375/4374]], [[385/384]], [[441/440]], [[3025/3024]] and [[9801/9800]]. It provides the [[optimal patent val]] for both the 7- and 11-limit versions of [[unidec]], the 72 & 118 temperament, which tempers out 1029/1024, 4375/4374, and in the 11-limit, 385/384 and 441/440. It also provides the optimal patent val for the rank-3 11-limit temperament [[portent]], which tempers out 385/384 and 441/440, and [[gamelan]], the rank-3 7-limit temperament which tempers out 1029/1024, as well as [[slendric]], the 2.3.7 subgroup temperament featured in the [[#Music]] section. In the 13-limit, 190et tempers out [[847/845]], [[625/624]], [[729/728]], [[1575/1573]] and [[1001/1000]], and provides the optimal patent val for the [[ekadash]] temperament and the rank-3 [[portentous]] temperament.
+edo is [[consistency|distinctly consistent]] in the [[15-odd-limit]] with a flat tendency, as [[harmonic]]s 3 through 13 are all tuned flat.
+edo is interesting because of the utility of its approximations; it [[tempering out|tempers out]] [[1029/1024]], [[4375/4374]], [[385/384]], [[441/440]], [[3025/3024]], and [[9801/9800]]. It provides the [[optimal patent val]] for both the 7- and 11-limit versions of [[unidec]], the {{nowrap|72 &amp; 118}} temperament, which tempers out 1029/1024, 4375/4374, and in the 11-limit, 385/384 and 441/440. It also provides the optimal patent val for the rank-3 11-limit temperament [[portent]], which tempers out 385/384 and 441/440, and [[gamelan]], the rank-3 7-limit temperament which tempers out 1029/1024, as well as [[slendric]], the 2.3.7 subgroup temperament featured in the [[#Music]] section. In the 13-limit, 190et tempers out [[625/624]], [[729/728]], [[847/845]], [[1001/1000]] and [[1575/1573]], and provides the optimal patent val for the [[ekadash]] temperament and the rank-3 [[portentous]] temperament.
+The 190g [[val]] shows us a smooth path to the even higher limits. This extension tempers out [[289/288]], [[561/560]], [[595/594]] in the 17-limit; [[343/342]], [[476/475]], [[495/494]] in the 19-limit; and [[391/390]], [[529/528]] in the 23-limit.
 === Prime harmonics ===
-{{Harmonics in equal|190|columns=11}}
+{{Harmonics in equal|190|intervals=prime}}
+=== Subsets and supersets ===
+Since 190 factors into {{factorization|190}}, 190edo has subset edos {{EDOs| 2, 5, 10, 19, 38, and 95 }}.
 == Regular temperament properties ==
 {| class="wikitable center-4 center-5 center-6"
+|-
 ! rowspan="2" | [[Subgroup]]
-! rowspan="2" | [[Comma list|Comma List]]
+! rowspan="2" | [[Comma list]]
 ! rowspan="2" | [[Mapping]]
 ! rowspan="2" | Optimal<br>8ve stretch (¢)
@@ Line 26: / Line 28: @@
 |-
 | 2.3
-| {{monzo| -301 190 }}
+| {{Monzo| -301 190 }}
-| [{{val| 190 301 }}]
+| {{Mapping| 190 301 }}
 | +0.285
 | 0.285
@@ Line 34: / Line 36: @@
 | 2.3.5
 | 2109375/2097152, {{monzo| -7 22 -12 }}
-| [{{val| 190 301 441 }}]
+| {{Mapping| 190 301 441 }}
 | +0.341
 | 0.246
@@ Line 41: / Line 43: @@
 | 2.3.5.7
 | 1029/1024, 4375/4374, 235298/234375
-| [{{val| 190 301 441 533 }}]
+| {{Mapping| 190 301 441 533 }}
 | +0.479
 | 0.321
@@ Line 48: / Line 50: @@
 | 2.3.5.7.11
 | 385/384, 441/440, 4375/4374, 234375/234256
-| [{{val| 190 301 441 533 657 }}]
+| {{Mapping| 190 301 441 533 657 }}
 | +0.490
 | 0.288
@@ Line 55: / Line 57: @@
 | 2.3.5.7.11.13
 | 385/384, 441/440, 625/624, 729/728, 847/845
-| [{{val| 190 301 441 533 657 703 }}]
+| {{Mapping| 190 301 441 533 657 703 }}
 | +0.432
 | 0.293
 | 4.63
+|-
+| 2.3.5.7.11.13.17
+| 289/288, 385/384, 441/440, 561/560, 625/624, 847/845
+| {{Mapping| 190 301 441 533 657 703 776 }} (190g)
+| +0.507
+| 0.327
+| 5.18
+|-
+| 2.3.5.7.11.13.17.19
+| 289/288, 343/342, 385/384, 441/440, 476/475, 495/494, 847/845
+| {{Mapping| 190 301 441 533 657 703 776 807 }} (190g)
+| +0.463
+| 0.327
+| 5.17
+|-
+| 2.3.5.7.11.13.17.19.23
+| 289/288, 343/342, 385/384, 391/390, 441/440, 476/475, 495/494, 529/528
+| {{Mapping| 190 301 441 533 657 703 776 807 859 }} (190g)
+| +0.486
+| 0.315
+| 4.98
 |}
+* 190et (190g val) has a lower relative error in the 23-limit than any previous equal temperaments, being the first to beat [[94edo|94]]. However, [[193edo|193]], only slightly larger, beats it.
+* It is also prominent in the 13- and 19-limit, with lower absolute errors than any previous equal temperaments. It beats [[183edo|183]] in either subgroup and is bettered by [[198edo|198]] in the 13-limit, and by 193 in the 19-limit.
 === 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*
-! Temperaments
+! Associated<br>ratio*
+! Temperament
 |-
 | 1
@@ Line 80: / Line 106: @@
 | 271.58
 | 75/64
-| [[Orson]] / [[sabric]]
+| [[Sabric]]
 |-
 | 1
@@ Line 128: / Line 154: @@
 | 498.95<br>(18.95)
 | 4/3<br>(81/80)
-| [[Pental]]
+| [[Quintile]]
 |-
 | 10
@@ Line 140: / Line 166: @@
 | 498.95<br>(18.95)
 | 4/3<br>(81/80)
-| [[Decal]]
+| [[Decile]]
 |-
 | 19
@@ Line 160: / Line 186: @@
 | [[Semihemienneadecal]]
 |}
+<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
 == Scales ==
@@ Line 168: / Line 195: @@
 == Music ==
-* [http://micro.soonlabel.com/tuning-survey/daily20111026-the-11th-slendric-fanfare.mp3 The 11th Slendric Fanfare]
+; [[Chris Vaisvil]]
-* [http://micro.soonlabel.com/tuning-survey/daily20111026-16-slendric-virgins.mp3 16 Slendric Virgins] by [[Chris Vaisvil]]
+* [http://micro.soonlabel.com/tuning-survey/daily20111026-the-11th-slendric-fanfare.mp3 ''The 11th Slendric Fanfare'']
+* [http://micro.soonlabel.com/tuning-survey/daily20111026-16-slendric-virgins.mp3 ''16 Slendric Virgins'']
-[[Category:190edo| ]] <!-- main article -->
-[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
-[[Category:Unidec]]
 [[Category:Ekadash]]
-[[Category:Gamelan]]
+[[Category:Gamelismic]]
+[[Category:Listen]]
 [[Category:Portent]]
 [[Category:Portentous]]
+[[Category:Unidec]]