397edo: Difference between revisions

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== Theory ==
== Theory ==
397et is only consistent to the [[5-odd-limit]], with three mappings possible for the 7-limit:
397edo is only [[consistent]] to the [[5-odd-limit]], with three mappings possible for the 7-limit:
* {{val|397 629 922 1115}} (patent val)
* {{val| 397 629 922 1115 }} ([[patent val]])
* {{val|397 629 922 '''1114'''}} (397d val)
* {{val| 397 629 922 '''1114''' }} (397d val)
* {{val|397 629 '''921''' '''1114'''}} (397cd val)
* {{val| 397 629 '''921''' '''1114''' }} (397cd val)


Using the patent val, it tempers out [[129140163/128000000]] and {{monzo|55 -1 -23}} in the 5-limit; [[6144/6125]], 16875/16807 and 129140163/128000000 in the 7-limit; [[support]]ing [[grendel]], [[porwell]] and [[mirkwai]].
Using the patent val, it tempers out [[129140163/128000000]] and {{monzo| 55 -1 -23 }} in the 5-limit; [[6144/6125]], [[16875/16807]] and 129140163/128000000 in the 7-limit; [[support]]ing [[grendel]].


Using the 397d val, it tempers out 129140163/128000000 and {{monzo|55 -1 -23}} in the 5-limit; [[3136/3125]], [[420175/419904]] and 33756345/33554432 in the 7-limit; supporting [[sengagen]] and [[parahemwuer]].
Using the 397d val, it tempers out 129140163/128000000 and {{monzo| 55 -1 -23 }} in the 5-limit; [[3136/3125]], [[420175/419904]] and 33756345/33554432 in the 7-limit; supporting [[sengagen]].


Using the 397cd val, it tempers out 390625000/387420489 and {{monzo|-55 23 8}} in the 5-limit; [[2401/2400]], 390625/387072 and [[14348907/14336000]] in the 7-limit; supporting [[cotritone]] and [[breed]].
Using the 397cd val, it tempers out 390625000/387420489 and {{monzo| -55 23 8 }} in the 5-limit; [[2401/2400]], 390625/387072 and [[14348907/14336000]] in the 7-limit; supporting [[cotritone]].


=== Prime harmonics ===
=== Prime harmonics ===
Line 18: Line 18:


=== Subsets and supersets ===
=== Subsets and supersets ===
397edo is the 78th [[prime EDO]]. [[1588edo]], which quadruples it, gives a good correction to the harmonic 7.
397edo is the 78th [[prime edo]]. [[1588edo]], which quadruples it, gives a good correction to the harmonic 7.


== 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|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]] (%)
|-
|-
|2.3
| 2.3
|{{monzo|-629 397}}
| {{monzo| -629 397 }}
|{{mapping|397 629}}
| {{mapping| 397 629 }}
| 0.2194
| 0.2194
| 0.2195
| 0.2195
| 7.26
| 7.26
|-
|-
|2.3.5
| 2.3.5
|129140163/128000000, {{monzo|55 -1 -23}}
| 129140163/128000000, {{monzo| 55 -1 -23 }}
|{{mapping|397 629 922}}
| {{mapping| 397 629 922 }}
| 0.0618
| 0.0618
| 0.2859
| 0.2859
Line 55: Line 55:
! Temperaments
! Temperaments
|-
|-
|1
| 1
|8\397
| 8\397
|24.18
| 24.18
|686/675
| 686/675
|[[Sengagen]] (397d)
| [[Sengagen]] (397d)
|-
|-
|1
| 1
|37\397
| 37\397
|111.84
| 111.84
|16/15
| 16/15
|[[Vavoom]]
| [[Vavoom]]
|-
|-
|1
| 1
|128\397
| 128\397
|386.90
| 386.90
|5/4
| 5/4
|[[Grendel]] (397)
| [[Grendel]] (397)
|-
|-
|1
| 1
|171\397
| 171\397
|516.88
| 516.88
|27/20
| 27/20
|[[Gravity]]
| [[Gravity]]
|-
|-
|1
| 1
|193\397
| 193\397
|583.38
| 583.38
|7/5
| 7/5
|[[Cotritone]] (397cd)
| [[Cotritone]] (397cd)
|}
|}
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct

Revision as of 14:46, 16 January 2024

← 396edo 397edo 398edo →
Prime factorization 397 (prime)
Step size 3.02267 ¢ 
Fifth 232\397 (701.259 ¢)
Semitones (A1:m2) 36:31 (108.8 ¢ : 93.7 ¢)
Consistency limit 5
Distinct consistency limit 5

Template:EDO intro

Theory

397edo is only consistent to the 5-odd-limit, with three mappings possible for the 7-limit:

  • ⟨397 629 922 1115] (patent val)
  • ⟨397 629 922 1114] (397d val)
  • ⟨397 629 921 1114] (397cd val)

Using the patent val, it tempers out 129140163/128000000 and [55 -1 -23⟩ in the 5-limit; 6144/6125, 16875/16807 and 129140163/128000000 in the 7-limit; supporting grendel.

Using the 397d val, it tempers out 129140163/128000000 and [55 -1 -23⟩ in the 5-limit; 3136/3125, 420175/419904 and 33756345/33554432 in the 7-limit; supporting sengagen.

Using the 397cd val, it tempers out 390625000/387420489 and [-55 23 8⟩ in the 5-limit; 2401/2400, 390625/387072 and 14348907/14336000 in the 7-limit; supporting cotritone.

Prime harmonics

Approximation of prime harmonics in 397edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 -0.70 +0.59 +1.45 -1.19 -0.23 +0.84 -1.29 +0.44 +1.15 +0.56
Relative (%) +0.0 -23.0 +19.5 +48.0 -39.4 -7.5 +27.7 -42.7 +14.6 +38.2 +18.4
Steps
(reduced) 		397
(0) 		629
(232) 		922
(128) 		1115
(321) 		1373
(182) 		1469
(278) 		1623
(35) 		1686
(98) 		1796
(208) 		1929
(341) 		1967
(379)

Subsets and supersets

397edo is the 78th prime edo. 1588edo, which quadruples it, gives a good correction to the harmonic 7.

Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢) 		Tuning Error
Absolute (¢) Relative (%)
2.3 [-629 397⟩ [⟨397 629]] 0.2194 0.2195 7.26
2.3.5 129140163/128000000, [55 -1 -23⟩ [⟨397 629 922]] 0.0618 0.2859 9.46

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
Ratio* 		Temperaments
1 8\397 24.18 686/675 Sengagen (397d)
1 37\397 111.84 16/15 Vavoom
1 128\397 386.90 5/4 Grendel (397)
1 171\397 516.88 27/20 Gravity
1 193\397 583.38 7/5 Cotritone (397cd)

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

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