197edo: Difference between revisions

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m Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct"
 
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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|197}}
{{ED intro}}


== Theory ==
== Theory ==
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.
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.
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 ===
=== Odd harmonics ===
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== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve Stretch (¢)
! rowspan="2" | Optimal<br />8ve stretch (¢)
! colspan="2" | Tuning Error
! colspan="2" | Tuning error
|-
|-
! [[TE error|Absolute]] (¢)
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
! [[TE simple badness|Relative]] (%)
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| {{monzo| -312 197 }}
| {{monzo| -312 197 }}
| {{mapping| 197 312 }}
| {{mapping| 197 312 }}
| 0.4566
| +0.4566
| 0.4568
| 0.4568
| 7.50
| 7.50
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| 15625/15552, {{monzo| -53 32 1 }}
| 15625/15552, {{monzo| -53 32 1 }}
| {{mapping| 197 312 457 }}
| {{mapping| 197 312 457 }}
| 0.6717
| +0.6717
| 0.4813
| 0.4813
| 7.90
| 7.90
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| 225/224, 4375/4374, {{monzo| -25 6 -3 8 }}
| 225/224, 4375/4374, {{monzo| -25 6 -3 8 }}
| {{mapping| 197 312 457 553 }}
| {{mapping| 197 312 457 553 }}
| 0.5302
| +0.5302
| 0.4834
| 0.4834
| 7.94
| 7.94
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=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{| 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
|-
! Periods<br />per 8ve
! Generator*
! Generator*
! Cents*
! Cents*
! Associated<br>Ratio
! Associated<br />ratio*
! Temperaments
! Temperaments
|-
|-
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| [[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 ==
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== Music ==
== Music ==
; [[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]]
; [[Chris Vaisvil]]
* [http://micro.soonlabel.com/pump_tunings/pump1/daily20111029-a-pump1-woodwinds-no-verb.mp3 ''Pump1''] – in pump12 1
* [http://micro.soonlabel.com/pump_tunings/pump1/daily20111029-a-pump1-woodwinds-no-verb.mp3 ''Pump1''] – in pump12 1

Latest revision as of 13:32, 13 March 2026

← 196edo 197edo 198edo →
Prime factorization 197 (prime)
Step size 6.09137 ¢ 
Fifth 115\197 (700.508 ¢)
Semitones (A1:m2) 17:16 (103.6 ¢ : 97.46 ¢)
Consistency limit 9
Distinct consistency limit 9

197 equal divisions of the octave (abbreviated 197edo or 197ed2), also called 197-tone equal temperament (197tet) or 197 equal temperament (197et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 197 equal parts of about 6.09 ¢ each. Each step represents a frequency ratio of 21/197, or the 197th root of 2.

Theory

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 vals, 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, 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 ⟨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, ⟨197 312 457 553 681 728], is an excellent tuning for the 13-limit version of catakleismic.

Odd harmonics

Approximation of odd harmonics in 197edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) -1.45 -2.56 -0.30 -2.89 +3.00 +0.08 +2.09 -1.40 +0.96 -1.75 -0.86
Relative (%) -23.8 -42.0 -4.9 -47.5 +49.2 +1.3 +34.3 -23.0 +15.8 -28.7 -14.2
Steps
(reduced) 		312
(115) 		457
(63) 		553
(159) 		624
(33) 		682
(91) 		729
(138) 		770
(179) 		805
(17) 		837
(49) 		865
(77) 		891
(103)

Subsets and supersets

197edo is the 45th prime edo.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [-312 197⟩ [⟨197 312]] +0.4566 0.4568 7.50
2.3.5 15625/15552, [-53 32 1⟩ [⟨197 312 457]] +0.6717 0.4813 7.90
2.3.5.7 225/224, 4375/4374, [-25 6 -3 8⟩ [⟨197 312 457 553]] +0.5302 0.4834 7.94

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
1 52\197 316.75 6/5 Catakleismic
1 53\197 322.84 3087/2560 Seniority

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct

Scales

Music

Francium
Chris Vaisvil
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