700edo: Difference between revisions

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{{EDO intro|700}}
{{EDO intro|700}}


==== Odd harmonics ====
== Theory ==
700edo is [[consistent]] to the [[7-odd-limit]] and its [[harmonic]] [[3/1|3]] is about halfway its steps. It is strong in the 2.9.15.7.31 [[subgroup]], [[tempering out]] [[3969/3968]], [[420175/419904]], 10255177611/10240000000 and 2202927104/2197265625. The equal temperament can also be used in the 2.9.5.7.13.17.23 subgroup, tempering out [[1225/1224]], 46000/45927, 8281/8280, 426496/426465, 31250/31213 and 262395/262144.
 
=== Odd harmonics ===
{{Harmonics in equal|700}}
{{Harmonics in equal|700}}


=== Subsets and supersets ===
Since 700 factors into 2<sup>2</sup> × 5<sup>2</sup> × 7, 700edo has subset edos {{EDOs| 2, 4, 5, 7, 10, 14, 20, 25, 28, 35, 50, 70, 100, 140, 175, and 350 }}. [[1400edo]], which doubles it, gives a good correction to the harmonic 3.


{{todo|inline=1|explain its xenharmonic value}}
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br />8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.9
| {{monzo|317 -100}}
| {{mapping|700 2219}}
| -0.0142
| 0.0142
| 0.83
|-
| 2.9.5
| {{monzo|-65 0 28}}, {{monzo|63 -25 7}}
| {{mapping|700 2219 1625}}
| +0.0766
| 0.1289
| 7.52
|-
| 2.9.5.7
| 2100875/2097152, 184528125/184473632, 3500000000/3486784401
| {{mapping|700 2219 1625 1965}}
| +0.0801
| 0.1118
| 6.52
|}

Revision as of 11:22, 14 January 2025

← 699edo 700edo 701edo →
Prime factorization 22 × 52 × 7
Step size 1.71429 ¢ 
Fifth 409\700 (701.143 ¢)
Semitones (A1:m2) 63:55 (108 ¢ : 94.29 ¢)
Dual sharp fifth 410\700 (702.857 ¢) (→ 41\70)
Dual flat fifth 409\700 (701.143 ¢)
Dual major 2nd 119\700 (204 ¢) (→ 17\100)
Consistency limit 7
Distinct consistency limit 7

Template:EDO intro

Theory

700edo is consistent to the 7-odd-limit and its harmonic 3 is about halfway its steps. It is strong in the 2.9.15.7.31 subgroup, tempering out 3969/3968, 420175/419904, 10255177611/10240000000 and 2202927104/2197265625. The equal temperament can also be used in the 2.9.5.7.13.17.23 subgroup, tempering out 1225/1224, 46000/45927, 8281/8280, 426496/426465, 31250/31213 and 262395/262144.

Odd harmonics

Approximation of odd harmonics in 700edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) -0.812 -0.599 -0.254 +0.090 +0.682 -0.528 +0.303 -0.384 +0.773 +0.648 -0.846
Relative (%) -47.4 -35.0 -14.8 +5.2 +39.8 -30.8 +17.7 -22.4 +45.1 +37.8 -49.3
Steps
(reduced) 		1109
(409) 		1625
(225) 		1965
(565) 		2219
(119) 		2422
(322) 		2590
(490) 		2735
(635) 		2861
(61) 		2974
(174) 		3075
(275) 		3166
(366)

Subsets and supersets

Since 700 factors into 22 × 52 × 7, 700edo has subset edos 2, 4, 5, 7, 10, 14, 20, 25, 28, 35, 50, 70, 100, 140, 175, and 350. 1400edo, which doubles it, gives a good correction to the harmonic 3.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.9 [317 -100⟩ [⟨700 2219]] -0.0142 0.0142 0.83
2.9.5 [-65 0 28⟩, [63 -25 7⟩ [⟨700 2219 1625]] +0.0766 0.1289 7.52
2.9.5.7 2100875/2097152, 184528125/184473632, 3500000000/3486784401 [⟨700 2219 1625 1965]] +0.0801 0.1118 6.52
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