190edo: Difference between revisions

(3 intermediate revisions by 2 users not shown)

Line 1:

== Theory ==

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

~~The equal temperament~~ 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 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.

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.

Line 16:

== Regular temperament properties ==

{~~{comma basis begin}}~~

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

|-

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

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

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

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

! colspan="2" | Tuning error

|-

! [[TE error|Absolute]] (¢)

! [[TE simple badness|Relative]] (%)

|-

| 2.3

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

| {{Monzo| -301 190 }}

| {{~~mapping~~| 190 301 }}

| {{Mapping| 190 301 }}

| +0.285

| 0.285

Line 27:

Line 36:

| 2.3.5

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

| {{~~mapping~~| 190 301 441 }}

| {{Mapping| 190 301 441 }}

| +0.341

| 0.246

Line 34:

Line 43:

| 2.3.5.7

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

| {{~~mapping~~| 190 301 441 533 }}

| {{Mapping| 190 301 441 533 }}

| +0.479

| 0.321

Line 41:

Line 50:

| 2.3.5.7.11

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

| {{~~mapping~~| 190 301 441 533 657 }}

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

| +0.490

| 0.288

Line 48:

Line 57:

| 2.3.5.7.11.13

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

| {{~~mapping~~| 190 301 441 533 657 703 }}

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

| +0.432

| 0.293

Line 55:

Line 64:

| 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)

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

| +0.507

| 0.327

Line 62:

Line 71:

| 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)

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

| +0.463

| 0.327

Line 69:

Line 78:

| 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)

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

| +0.486

| 0.315

| 4.98

~~{{comma basis end}~~}

|}

* 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 ===

{{rank-2 ~~begin}}~~

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

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

|-

! Periods per 8ve

! Generator*

! Cents*

! Associated ratio*

! Temperament

|-

| 1

Line 123:

Line 139:

|-

| 2

| 59\190 (36\190)

| 59\190 (36\190)

| 372.63 (227.37)

| 372.63 (227.37)

| 26/21 (297/260)

| 26/21 (297/260)

| [[Essence]]

|-

| 2

| 71\190 (24\190)

| 71\190 (24\190)

| 448.42 (151.58)

| 448.42 (151.58)

| 35/27 (12/11)

| 35/27 (12/11)

| [[Neusec]]

|-

| 5

| 79\190 (3\190)

| 79\190 (3\190)

| 498.95 (18.95)

| 498.95 (18.95)

| 4/3 (81/80)

| 4/3 (81/80)

| [[~~Pental (temperament)|Pental~~]]

| [[Quintile]]

|-

| 10

| 50\190 (7\190)

| 50\190 (7\190)

| 315.79 (45.79)

| 315.79 (45.79)

| 6/5 (40/39)

| 6/5 (40/39)

| [[Deca]]

|-

| 10

| 79\190 (3\190)

| 79\190 (3\190)

| 498.95 (18.95)

| 498.95 (18.95)

| 4/3 (81/80)

| 4/3 (81/80)

| [[~~Decal~~]]

| [[Decile]]

|-

| 19

| 79\190 (1\190)

| 79\190 (1\190)

| 498.95 (6.32)

| 498.95 (6.32)

| 4/3 (225/224)

| 4/3 (225/224)

| [[Enneadecal]]

|-

| 38

| 79\190 (1\190)

| 79\190 (1\190)

| 265.26 (6.32)

| 265.26 (6.32)

| 4/3 (225/224)

| 4/3 (225/224)

| [[Hemienneadecal]]

|-

| 38

| 42\190 (2\190)

| 42\190 (2\190)

| 265.26 (12.63)

| 265.26 (12.63)

| 500/429 (144/143)

| 500/429 (144/143)

| [[Semihemienneadecal]]

~~{{rank-2 end}~~}

|}

~~{{orf}}~~

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

== Scales ==

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|190}}
+{{ED intro}}
 == Theory ==
 edo is [[consistency|distinctly consistent]] in the [[15-odd-limit]] with a flat tendency, as [[harmonic]]s 3 through 13 are all tuned flat.
-The equal temperament 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 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.
+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.
@@ Line 16: / Line 16: @@
 == Regular temperament properties ==
-{{comma basis begin}}
+{| class="wikitable center-4 center-5 center-6"
+|-
+! rowspan="2" | [[Subgroup]]
+! rowspan="2" | [[Comma list]]
+! rowspan="2" | [[Mapping]]
+! rowspan="2" | Optimal<br>8ve stretch (¢)
+! colspan="2" | Tuning error
+|-
+! [[TE error|Absolute]] (¢)
+! [[TE simple badness|Relative]] (%)
 |-
 | 2.3
-| {{monzo| -301 190 }}
+| {{Monzo| -301 190 }}
-| {{mapping| 190 301 }}
+| {{Mapping| 190 301 }}
 | +0.285
 | 0.285
@@ Line 27: / Line 36: @@
 | 2.3.5
 | 2109375/2097152, {{monzo| -7 22 -12 }}
-| {{mapping| 190 301 441 }}
+| {{Mapping| 190 301 441 }}
 | +0.341
 | 0.246
@@ Line 34: / Line 43: @@
 | 2.3.5.7
 | 1029/1024, 4375/4374, 235298/234375
-| {{mapping| 190 301 441 533 }}
+| {{Mapping| 190 301 441 533 }}
 | +0.479
 | 0.321
@@ Line 41: / Line 50: @@
 | 2.3.5.7.11
 | 385/384, 441/440, 4375/4374, 234375/234256
-| {{mapping| 190 301 441 533 657 }}
+| {{Mapping| 190 301 441 533 657 }}
 | +0.490
 | 0.288
@@ Line 48: / Line 57: @@
 | 2.3.5.7.11.13
 | 385/384, 441/440, 625/624, 729/728, 847/845
-| {{mapping| 190 301 441 533 657 703 }}
+| {{Mapping| 190 301 441 533 657 703 }}
 | +0.432
 | 0.293
@@ Line 55: / Line 64: @@
 | 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)
+| {{Mapping| 190 301 441 533 657 703 776 }} (190g)
 | +0.507
 | 0.327
@@ Line 62: / Line 71: @@
 | 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)
+| {{Mapping| 190 301 441 533 657 703 776 807 }} (190g)
 | +0.463
 | 0.327
@@ Line 69: / Line 78: @@
 | 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)
+| {{Mapping| 190 301 441 533 657 703 776 807 859 }} (190g)
 | +0.486
 | 0.315
 | 4.98
-{{comma basis end}}
+|}
 * 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 ===
-{{rank-2 begin}}
+{| class="wikitable center-all left-5"
+|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
+|-
+! Periods<br>per 8ve
+! Generator*
+! Cents*
+! Associated<br>ratio*
+! Temperament
 |-
 | 1
@@ Line 123: / Line 139: @@
 |-
 | 2
-| 59\190<br />(36\190)
+| 59\190<br>(36\190)
-| 372.63<br />(227.37)
+| 372.63<br>(227.37)
-| 26/21<br />(297/260)
+| 26/21<br>(297/260)
 | [[Essence]]
 |-
 | 2
-| 71\190<br />(24\190)
+| 71\190<br>(24\190)
-| 448.42<br />(151.58)
+| 448.42<br>(151.58)
-| 35/27<br />(12/11)
+| 35/27<br>(12/11)
 | [[Neusec]]
 |-
 | 5
-| 79\190<br />(3\190)
+| 79\190<br>(3\190)
-| 498.95<br />(18.95)
+| 498.95<br>(18.95)
-| 4/3<br />(81/80)
+| 4/3<br>(81/80)
-| [[Pental (temperament)|Pental]]
+| [[Quintile]]
 |-
 | 10
-| 50\190<br />(7\190)
+| 50\190<br>(7\190)
-| 315.79<br />(45.79)
+| 315.79<br>(45.79)
-| 6/5<br />(40/39)
+| 6/5<br>(40/39)
 | [[Deca]]
 |-
 | 10
-| 79\190<br />(3\190)
+| 79\190<br>(3\190)
-| 498.95<br />(18.95)
+| 498.95<br>(18.95)
-| 4/3<br />(81/80)
+| 4/3<br>(81/80)
-| [[Decal]]
+| [[Decile]]
 |-
 | 19
-| 79\190<br />(1\190)
+| 79\190<br>(1\190)
-| 498.95<br />(6.32)
+| 498.95<br>(6.32)
-| 4/3<br />(225/224)
+| 4/3<br>(225/224)
 | [[Enneadecal]]
 |-
 | 38
-| 79\190<br />(1\190)
+| 79\190<br>(1\190)
-| 265.26<br />(6.32)
+| 265.26<br>(6.32)
-| 4/3<br />(225/224)
+| 4/3<br>(225/224)
 | [[Hemienneadecal]]
 |-
 | 38
-| 42\190<br />(2\190)
+| 42\190<br>(2\190)
-| 265.26<br />(12.63)
+| 265.26<br>(12.63)
-| 500/429<br />(144/143)
+| 500/429<br>(144/143)
 | [[Semihemienneadecal]]
-{{rank-2 end}}
+|}
-{{orf}}
+<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
 == Scales ==