197edo: Difference between revisions

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~~==Theory==~~

197edo gives excellent results for tuning both marvel, the planar temperament tempering out 225/224, and [[Kleismic_family|catakleismic]], the temperament [[tempering_out|tempering out]] both 225/224 and 4375/4374, which has wedgie <<6 5 22 -6 18 37||. Among [[Patent_val|patent vals]], in fact, it gives the best results for both. In fact, the [[11-limit]] patent val <197 312 457 553 682| has a [[comma basis]] [225/224, 4375/4374, 441/440, 65536/65219], so taking 225/224 and 441/440 together ([[Marvel family|prodigy temperament]]) also works well with 197edo, and taking 225/224, 441/440, and 4375/4374 (an alternative 11-limit catakleismic) is once again excellently tuned by 197edo.

If we use 197e, the ~~<~~197 312 457 553 681| val, we can also use 197edo as an excellent tuning for the 11-limit version of marvel temperament, tempering out 385/384 as well as 225/224. If we add 4375/4374 to the comma list for 11-limit ~~[[Marvel family|~~marvel]], we get 11-limit catakleismic, and 197edo with the above val is also an excellent tuning for that.

== Theory ==

===Odd harmonics===

197edo gives excellent results for tuning both [[marvel]], the planar temperament [[tempering out]] [[225/224]], and [[catakleismic]], the temperament tempering out both 225/224 and [[4375/4374]]. Among [[patent val]]s, it gives the best results for both. In fact, the [[11-limit]] patent val {{val| 197 312 457 553 682 }} has a [[comma basis]] {225/224, 441/440, 4375/4374, 65536/65219}, so taking 225/224 and [[441/440]] together ([[prodigy]] temperament) also works well with 197edo, and taking 225/224, 441/440, and 4375/4374 (an alternative 11-limit catakleismic) is once again excellently tuned by 197edo.

If we use 197e, the {{val| 197 312 457 553 681 }} val, we can also use 197edo as an excellent tuning for the 11-limit version of marvel temperament, tempering out [[385/384]] as well as 225/224. If we add 4375/4374 to the comma list for 11-limit marvel, we get 11-limit catakleismic, and 197edo with the above val is also an excellent tuning for that. The 197ef val, {{val| 197 312 457 553 681 728}}, is an excellent tuning for the 13-limit version of catakleismic.

=== Odd harmonics ===

~~===Subsets and supersets===~~

~~197edo is the 45th [[prime_numbers|prime]] edo.~~

==Regular temperament properties==

=== Subsets and supersets ===

197edo is the 45th [[prime edo]].

== Regular temperament properties ==

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

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

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

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

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

~~! colspan="2" |Tuning Error~~

|-

~~![[TE error|Absolute]] (¢)~~

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

|-

|2.3

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

|{{monzo|-312 197}}

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

|{{~~val~~|197 312}}

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

| 0.4566

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

! colspan="2" | Tuning error

|-

! [[TE error|Absolute]] (¢)

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

|-

| 2.3

| {{monzo| -312 197 }}

| {{mapping| 197 312 }}

| +0.4566

| 0.4568

| 7.50

|-

|2.3.5

| 2.3.5

|15625/15552, {{monzo|-53 32 1}}

| 15625/15552, {{monzo| -53 32 1 }}

|{{~~val~~|197 312 457}}

| {{mapping| 197 312 457 }}

| 0.6717

| +0.6717

| 0.4813

| 7.90

|-

|2.3.5.7

| 2.3.5.7

|225/224, 4375/4374, ~~15625/15552~~

| 225/224, 4375/4374, {{monzo| -25 6 -3 8 }}

|{{~~val~~|197 312 457 553}}

| {{mapping| 197 312 457 553 }}

| 0.5302

| +0.5302

| 0.4834

| 7.94

|}

=== 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 8ve

|-

! Generator~~ (reduced)~~

! Periods per 8ve

! Cents~~ (reduced)~~

! Generator*

! Associated ratio

! Cents*

! Associated ratio*

! Temperaments

|-

|1

| 1

|52\197

| 52\197

|316.75

| 316.75

|6/5

| 6/5

|[[~~Hanson]] / [[catakleismic~~]]

| [[Catakleismic]]

|-

|1

| 1

|53\197

| 53\197

|322.84

| 322.84

|3087/2560

| 3087/2560

|[[Seniority]]

| [[Seniority]]

|}

<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

==Scales==

== Scales ==

* [[Prismarv]]

* [[Marveldene]]

* [[~~Pump12_1~~]~~] [http://micro.soonlabel.com/pump_tunings/pump1/daily20111029-a-pump1-woodwinds-no-verb.mp3 Pump1 by Chris Vaisvil~~]

* [[Pump12 1]]

* [[~~Pump12_2~~]]

* [[Pump12 2]]

* [[Pump13]]

* [[Pump14]]

Line 76:

Line 83:

* [[Pump18]]

[[~~Category~~:~~197edo~~| ~~]] <!~~-- ~~main article~~ -->

== Music ==

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

; [[Francium]]

* "Have You Ever Eaten A Square Cutlet?" from ''Questions'' (2024) – [https://open.spotify.com/track/7cA0T79hEkzEoMGvufGZ5R Spotify] | [https://francium223.bandcamp.com/track/have-you-ever-eaten-a-square-cutlet Bandcamp] | [https://www.youtube.com/watch?v=_LkriwwqbIY YouTube]

; [[Chris Vaisvil]]

* [http://micro.soonlabel.com/pump_tunings/pump1/daily20111029-a-pump1-woodwinds-no-verb.mp3 ''Pump1''] – in pump12 1

[[Category:Catakleismic]]

[[Category:Marvel]]

~~[[Category:Prime EDO]]~~

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|197}}
+{{ED intro}}
-==Theory==
-edo gives excellent results for tuning both marvel, the planar temperament tempering out 225/224, and [[Kleismic_family|catakleismic]], the temperament [[tempering_out|tempering out]] both 225/224 and 4375/4374, which has wedgie &lt;&lt;6 5 22 -6 18 37||. Among [[Patent_val|patent vals]], in fact, it gives the best results for both. In fact, the [[11-limit]] patent val &lt;197 312 457 553 682| has a [[comma basis]] [225/224, 4375/4374, 441/440, 65536/65219], so taking 225/224 and 441/440 together ([[Marvel family|prodigy temperament]]) also works well with 197edo, and taking 225/224, 441/440, and 4375/4374 (an alternative 11-limit catakleismic) is once again excellently tuned by 197edo.
-If we use 197e, the &lt;197 312 457 553 681| val, we can also use 197edo as an excellent tuning for the 11-limit version of marvel temperament, tempering out 385/384 as well as 225/224. If we add 4375/4374 to the comma list for 11-limit [[Marvel family|marvel]], we get 11-limit catakleismic, and 197edo with the above val is also an excellent tuning for that.
+== Theory ==
-===Odd harmonics===
+edo gives excellent results for tuning both [[marvel]], the planar temperament [[tempering out]] [[225/224]], and [[catakleismic]], the temperament tempering out both 225/224 and [[4375/4374]]. Among [[patent val]]s, it gives the best results for both. In fact, the [[11-limit]] patent val {{val| 197 312 457 553 682 }} has a [[comma basis]] {225/224, 441/440, 4375/4374, 65536/65219}, so taking 225/224 and [[441/440]] together ([[prodigy]] temperament) also works well with 197edo, and taking 225/224, 441/440, and 4375/4374 (an alternative 11-limit catakleismic) is once again excellently tuned by 197edo.
+If we use 197e, the {{val| 197 312 457 553 681 }} val, we can also use 197edo as an excellent tuning for the 11-limit version of marvel temperament, tempering out [[385/384]] as well as 225/224. If we add 4375/4374 to the comma list for 11-limit marvel, we get 11-limit catakleismic, and 197edo with the above val is also an excellent tuning for that. The 197ef val, {{val| 197 312 457 553 681 728}}, is an excellent tuning for the 13-limit version of catakleismic.
+=== Odd harmonics ===
 {{Harmonics in equal|197}}
-===Subsets and supersets===
-edo is the 45th [[prime_numbers|prime]] edo.
-==Regular temperament properties==
+=== Subsets and supersets ===
+edo is the 45th [[prime edo]].
+== Regular temperament properties ==
 {| class="wikitable center-4 center-5 center-6"
-! rowspan="2" |[[Subgroup]]
-! rowspan="2" |[[Comma list|Comma List]]
-! rowspan="2" |[[Mapping]]
-! rowspan="2" |Optimal<br>8ve Stretch (¢)
-! colspan="2" |Tuning Error
-|-
-![[TE error|Absolute]] (¢)
-![[TE simple badness|Relative]] (%)
 |-
-|2.3
+! rowspan="2" | [[Subgroup]]
-|{{monzo|-312 197}}
+! rowspan="2" | [[Comma list]]
-|{{val|197 312}}
+! rowspan="2" | [[Mapping]]
-| 0.4566
+! rowspan="2" | Optimal<br />8ve stretch (¢)
+! colspan="2" | Tuning error
+|-
+! [[TE error|Absolute]] (¢)
+! [[TE simple badness|Relative]] (%)
+|-
+| 2.3
+| {{monzo| -312 197 }}
+| {{mapping| 197 312 }}
+| +0.4566
 | 0.4568
 | 7.50
 |-
-|2.3.5
+| 2.3.5
-|15625/15552, {{monzo|-53 32 1}}
+| 15625/15552, {{monzo| -53 32 1 }}
-|{{val|197 312 457}}
+| {{mapping| 197 312 457 }}
-| 0.6717
+| +0.6717
 | 0.4813
 | 7.90
 |-
-|2.3.5.7
+| 2.3.5.7
-|225/224, 4375/4374, 15625/15552
+| 225/224, 4375/4374, {{monzo| -25 6 -3 8 }}
-|{{val|197 312 457 553}}
+| {{mapping| 197 312 457 553 }}
-| 0.5302
+| +0.5302
 | 0.4834
 | 7.94
 |}
 === 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 8ve
+|-
-! Generator<br>(reduced)
+! Periods<br />per 8ve
-! Cents<br>(reduced)
+! Generator*
-! Associated<br>ratio
+! Cents*
+! Associated<br />ratio*
 ! Temperaments
 |-
-|1
+| 1
-|52\197
+| 52\197
-|316.75
+| 316.75
-|6/5
+| 6/5
-|[[Hanson]] / [[catakleismic]]
+| [[Catakleismic]]
 |-
-|1
+| 1
-|53\197
+| 53\197
-|322.84
+| 322.84
-|3087/2560
+| 3087/2560
-|[[Seniority]]
+| [[Seniority]]
 |}
+<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
-==Scales==
+== Scales ==
 * [[Prismarv]]
 * [[Marveldene]]
-* [[Pump12_1]]<br>[http://micro.soonlabel.com/pump_tunings/pump1/daily20111029-a-pump1-woodwinds-no-verb.mp3 Pump1 by Chris Vaisvil]
+* [[Pump12 1]]
-* [[Pump12_2]]
+* [[Pump12 2]]
 * [[Pump13]]
 * [[Pump14]]
@@ Line 76: / Line 83: @@
 * [[Pump18]]
-[[Category:197edo| ]] <!-- main article -->
+== Music ==
-[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
+; [[Francium]]
+* "Have You Ever Eaten A Square Cutlet?" from ''Questions'' (2024) – [https://open.spotify.com/track/7cA0T79hEkzEoMGvufGZ5R Spotify] | [https://francium223.bandcamp.com/track/have-you-ever-eaten-a-square-cutlet Bandcamp] | [https://www.youtube.com/watch?v=_LkriwwqbIY YouTube]
+; [[Chris Vaisvil]]
+* [http://micro.soonlabel.com/pump_tunings/pump1/daily20111029-a-pump1-woodwinds-no-verb.mp3 ''Pump1''] – in pump12 1
 [[Category:Catakleismic]]
 [[Category:Marvel]]
-[[Category:Prime EDO]]