197edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
'''197edo''' is the [[EDO|equal division of the octave]] into 197 parts of 6.0914 cents each. It 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.
{{EDO intro|197}}
==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.
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.
===Odd harmonics===
{{Harmonics in equal|197}}
===Subsets and supersets===
197edo is the 45th [[prime_numbers|prime]] edo.


197edo is the 45th [[prime_numbers|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
|{{monzo|-312 197}}
|{{val|197 312}}
| 0.4566
| 0.4568
| 7.50
|-
|2.3.5
|15625/15552, {{monzo|-53 32 1}}
|{{val|197 312 457}}
| 0.6717
| 0.4813
| 7.90
|-
|2.3.5.7
|225/224, 4375/4374, 15625/15552
|{{val|197 312 457 553}}
| 0.5302
| 0.4834
| 7.94
|}
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods<br>per 8ve
! Generator<br>(reduced)
! Cents<br>(reduced)
! Associated<br>ratio
! Temperaments
|-
|1
|52\197
|316.75
|6/5
|[[Hanson]] / [[catakleismic]]
|-
|1
|53\197
|322.84
|3087/2560
|[[Seniority]]
|}


==Scales==
==Scales==

Revision as of 20:25, 3 November 2023

← 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

Template:EDO intro

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, which has wedgie <<6 5 22 -6 18 37||. Among 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 (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.

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, 15625/15552 ⟨197 312 457 553] 0.5302 0.4834 7.94

Rank-2 temperaments

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

Scales

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