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397edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|397}}
{{ED intro}}


== 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 ===
{{Harmonics in equal|397}}
{{Harmonics in equal|397}}


=== 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" |[[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|-629 397}}
! rowspan="2" | [[Comma list]]
|{{mapping|397 629}}
! rowspan="2" | [[Mapping]]
| 0.2194
! rowspan="2" | Optimal<br />8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -629 397 }}
| {{mapping| 397 629 }}
| +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
| 9.46
| 9.46
Line 48: Line 49:
=== 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
|-
|-
|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 />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
== Music ==
; [[Francium]]
* "Mysteries Within Us" from ''End Of Sartorius Membranes'' (2024) – [https://open.spotify.com/track/4fbOQF5p1YITudOw866GvH Spotify] | [https://francium223.bandcamp.com/track/mysteries-within-us Bandcamp] | [https://www.youtube.com/watch?v=kDdh9XZ4UNE YouTube]


<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
[[Category:Listen]]
Retrieved from "https://en.xen.wiki/w/397edo"