497edo: Difference between revisions

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Created page with "{{Infobox ET}} {{EDO intro|497}} == Theory == 497et only is consistent to the 5-limit. Using the patent val, it tempers out 67108864/66976875, 48828125/48771072, 2100875/20..."
 
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|497}}
{{ED intro}}


== Theory ==
== Theory ==
497et only is consistent to the 5-limit. Using the patent val, it tempers out 67108864/66976875, 48828125/48771072, [[2100875/2097152]] and 200120949/200000000 in the 7-limit; [[117440512/117406179]], 26796875/26763264, 151263/151250, [[131072/130977]], 42875/42768, 4302592/4296875, [[5632/5625]], 537109375/536870912, 9453125/9437184, 160083/160000, 1362944/1361367, 391314/390625, 43923/43904 and 644204/643125 in the 11-limit.
497et only is [[consistent]] to the [[5-odd-limit]]. Using the [[patent val]], the equal temperament [[tempering out|tempers out]] [[2100875/2097152]], 48828125/48771072, 67108864/66976875, and 200120949/200000000 in the 7-limit; [[5632/5625]], 42875/42768, 43923/43904, [[131072/130977]], 151263/151250, 160083/160000, and 391314/390625 in the 11-limit.


=== Odd harmonics ===
=== Odd harmonics ===
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=== Subsets and supersets ===
=== Subsets and supersets ===
497 factors into 7 × 41, with [[7edo]] and [[41edo]] as its subset edos. [[1491edo]], which triples it, gives a good correction to the harmonic 7.
497 factors into {{nowrap|7 × 71}}, with [[7edo]] and [[71edo]] as its subset edos. [[1491edo]], which triples it, gives a good correction to harmonics 3 and 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|788 -497}}
! rowspan="2" | [[Comma list]]
|{{mapping|497 788}}
! rowspan="2" | [[Mapping]]
| -0.2084
! rowspan="2" | Optimal<br />8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 788 -497 }}
| {{mapping| 497 788 }}
| −0.2084
| 0.2084
| 0.2084
| 8.63
| 8.63
|-
|-
|2.3.5
| 2.3.5
|{{monzo|38 -2 -15}}, {{val|12 -31 16}}
| {{monzo| 38 -2 -15 }}, {{monzo| 12 -31 16 }}
|{{mapping|497 788 1154}}
| {{mapping| 497 788 1154 }}
| -0.1396
| −0.1396
| 0.1961
| 0.1961
| 8.12
| 8.12
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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
|-
! Generator<br>(reduced)*
! Periods<br />per 8ve
! Cents<br>(reduced)*
! Generator*
! Associated<br>Ratio*
! Cents*
! Associated<br />ratio*
! Temperaments
! Temperaments
|-
|-
|1
| 1
|80\497
| 80\497
|193.16
| 193.16
|262144/234375
| 262144/234375
|[[Luna]]
| [[Luna]]
|}
|}
 
<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
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct

Latest revision as of 13:32, 13 March 2026

← 496edo 497edo 498edo →
Prime factorization 7 × 71
Step size 2.41449 ¢ 
Fifth 291\497 (702.616 ¢)
Semitones (A1:m2) 49:36 (118.3 ¢ : 86.92 ¢)
Consistency limit 5
Distinct consistency limit 5

497 equal divisions of the octave (abbreviated 497edo or 497ed2), also called 497-tone equal temperament (497tet) or 497 equal temperament (497et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 497 equal parts of about 2.41 ¢ each. Each step represents a frequency ratio of 21/497, or the 497th root of 2.

Theory

497et only is consistent to the 5-odd-limit. Using the patent val, the equal temperament tempers out 2100875/2097152, 48828125/48771072, 67108864/66976875, and 200120949/200000000 in the 7-limit; 5632/5625, 42875/42768, 43923/43904, 131072/130977, 151263/151250, 160083/160000, and 391314/390625 in the 11-limit.

Odd harmonics

Approximation of odd harmonics in 497edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) +0.66 +0.00 -0.62 -1.09 -0.81 -0.29 +0.66 -1.13 -0.53 +0.04 -0.51
Relative (%) +27.4 +0.2 -25.5 -45.3 -33.8 -11.9 +27.5 -46.9 -22.0 +1.8 -21.0
Steps
(reduced) 		788
(291) 		1154
(160) 		1395
(401) 		1575
(84) 		1719
(228) 		1839
(348) 		1942
(451) 		2031
(43) 		2111
(123) 		2183
(195) 		2248
(260)

Subsets and supersets

497 factors into 7 × 71, with 7edo and 71edo as its subset edos. 1491edo, which triples it, gives a good correction to harmonics 3 and 7.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [788 -497⟩ [⟨497 788]] −0.2084 0.2084 8.63
2.3.5 [38 -2 -15⟩, [12 -31 16⟩ [⟨497 788 1154]] −0.1396 0.1961 8.12

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
1 80\497 193.16 262144/234375 Luna

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

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