353edo: Difference between revisions

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== Theory ==
== Theory ==
From the prime number standpoint, 353edo is suitable for use with 2.7.11.17.23.29.31.37 subgroup. This makes 353edo an "upside-down" EDO – poor approximation of the low harmonics, but an improvement over the high ones. Nonetheless, it provides the [[optimal patent val]] for [[didacus]], the 2.5.7 subgroup temperament tempering out [[3136/3125]].
=== Odd harmonics ===
{{Harmonics in equal|353}}
{{Harmonics in equal|353}}
From the prime number standpoint, 353edo is suitable for use with 2.7.11.17.23.29.31.37 subgroup. This makes 353edo an "upside-down" EDO – poor approximation of the low harmonics, but an improvement over the high ones. Nonetheless, it provides the [[optimal patent val]] for [[didacus]], the 2.5.7 subgroup temperament tempering out [[3136/3125]].


353edo is the 71st [[prime EDO]].
=== Subsets and supersets ===
353edo is the 71st [[prime edo]].


=== Relation to a calendar reform ===
=== Relation to a calendar reform ===
Line 149: Line 152:
Assuming 353edo is treated as the 2.5.7.11.13.17 subgroup temperament.
Assuming 353edo is treated as the 2.5.7.11.13.17 subgroup temperament.
{| 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
! rowspan="2" | Optimal<br>8ve Stretch (¢)
8ve stretch (¢)
! colspan="2" | Tuning Error
! colspan="2" |Tuning error
|-
|-
![[TE error|Absolute]] (¢)
! [[TE error|Absolute]] (¢)
![[TE simple badness|Relative]] (%)
! [[TE simple badness|Relative]] (%)
|-
|-
|2.5
| 2.5
|{{monzo|820 -353}}
| {{monzo| 820 -353 }}
|[{{val|353 820}}]
| {{mapping| 353 820 }}
| -0.263
| -0.263
|0.263
| 0.263
|7.74
| 7.74
|-
|-
|2.5.7
| 2.5.7
|3136/3125, {{monzo|209 -9 -67}}
| 3136/3125, {{monzo| 209 -9 -67 }}
|[{{val|353 820 991}}]
| {{mapping| 353 820 991 }}
| -0.177
| -0.177
|0.247
| 0.247
|7.26
| 7.26
|-
|-
|2.5.7.11
| 2.5.7.11
|3136/3125, 5767168/5764801, {{monzo|-20 -6  1 9}}
| 3136/3125, 5767168/5764801, {{monzo| -20 -6  1 9 }}
|[{{val|353 820 991 1221}}]
| {{mapping| 353 820 991 1221 }}
| -0.089
| -0.089
|0.263
| 0.263
|7.73
| 7.73
|-
|-
|2.5.7.11.13
| 2.5.7.11.13
|3136/3125, 4394/4375, 6656/6655, 5767168/5764801
| 3136/3125, 4394/4375, 6656/6655, 5767168/5764801
|[{{val|353 820 991 1221 1306}}]
| {{mapping| 353 820 991 1221 1306 }}
| -0.024
| -0.024
|0.268
| 0.268
|7.89
| 7.89
|-
|-
|2.5.7.11.13.17
| 2.5.7.11.13.17
|3136/3125, 4394/4375, 7744/7735, 60112/60025, 64141/64000
| 3136/3125, 4394/4375, 7744/7735, 60112/60025, 64141/64000
|[{{val|353 820 991 1221 1306 1443}}]
| {{mapping| 353 820 991 1221 1306 1443 }}
| -0.037
| -0.037
|0.247
| 0.247
|7.26
| 7.26
|}
|}
=== Rank-2 temperaments  ===
=== Rank-2 temperaments  ===
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
!Periods
! Periods<br>per 8ve
per octave
! Generator*
!Generator
! Cents*
(reduced)
! Associated<br>Ratio*
!Cents
! Temperaments
(reduced)
!Associated
ratio
!Temperaments
|-
|-
|1
| 1
|19\353
| 19\353
|64.59
| 64.59
|26/25
| 26/25
|[[Hemimean clan#Rectified Hebrew|Rectified Hebrew]]
| [[Rectified hebrew]]
|-
|-
|1
| 1
|34\353
| 34\353
|115.58
| 115.58
|77/72
| 77/72
|[[Subgroup temperaments#Apparatus|Apparatus]]
| [[Subgroup temperaments#Apparatus|Apparatus]]
|-
|-
|1
| 1
|152\353
| 152\353
|516.71
| 516.71
|27/20
| 27/20
|[[Marvo]] (353c) / [[zarvo]] (353cd)
| [[Marvo]] (353c) / [[zarvo]] (353cd)
|}
|}
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct


== Scales ==
== Scales ==
Line 235: Line 235:


== Music ==
== Music ==
; [[Mercury Amalgam]]
* [https://www.youtube.com/watch?v=z-SxvrnkTzU ''Bottom Text''] (2022)


* [https://www.youtube.com/watch?v=z-SxvrnkTzU Bottom Text] by Mercury Amalgam
== External links ==
 
== Links ==
* [http://individual.utoronto.ca/kalendis/hebrew/rect.htm Rectified Hebrew Calendar]
* [http://individual.utoronto.ca/kalendis/hebrew/rect.htm Rectified Hebrew Calendar]


Revision as of 09:53, 12 November 2023

← 352edo 353edo 354edo →
Prime factorization 353 (prime)
Step size 3.39943 ¢ 
Fifth 206\353 (700.283 ¢)
Semitones (A1:m2) 30:29 (102 ¢ : 98.58 ¢)
Dual sharp fifth 207\353 (703.683 ¢)
Dual flat fifth 206\353 (700.283 ¢)
Dual major 2nd 60\353 (203.966 ¢)
Consistency limit 3
Distinct consistency limit 3

Template:EDO intro

Theory

From the prime number standpoint, 353edo is suitable for use with 2.7.11.17.23.29.31.37 subgroup. This makes 353edo an "upside-down" EDO – poor approximation of the low harmonics, but an improvement over the high ones. Nonetheless, it provides the optimal patent val for didacus, the 2.5.7 subgroup temperament tempering out 3136/3125.

Odd harmonics

Approximation of odd harmonics in 353edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) -1.67 +1.22 +0.01 +0.06 -0.61 -0.87 -0.45 +0.43 +1.64 -1.66 +0.62
Relative (%) -49.2 +35.9 +0.4 +1.6 -17.9 -25.5 -13.2 +12.6 +48.2 -48.8 +18.3
Steps
(reduced) 		559
(206) 		820
(114) 		991
(285) 		1119
(60) 		1221
(162) 		1306
(247) 		1379
(320) 		1443
(31) 		1500
(88) 		1550
(138) 		1597
(185)

Subsets and supersets

353edo is the 71st prime edo.

Relation to a calendar reform

Main article: Rectified Hebrew

In the original Hebrew calendar, years number 3, 6, 8, 11, 14, 17, and 19 within a 19-year pattern (makhzor (מחזור), plural:makhzorim) are leap. When converted to 19edo, this results in 5L 2s mode, and simply the diatonic major scale. Following this logic, a temperament can be constructed for the Rectified Hebrew calendar.

The 11-step perfect fifth in this scale becomes 209\353, and it corresponds to 98/65, which is sharp of 3/2 by 196/195.

In addition, every sub-pattern in a 19-note generator is actually a Hebrew makhzor, that is a mini-19edo on its own, until it is truncated to an 11-note pattern. Just as the original calendar reform consists of 18 makhzorim with 1 hendecaeteris, Hebrew[130] scale consists of a stack of naively 18 "major scales" finished with one 11-edo tetratonic.

The number 353 in this version of the Hebrew calendar must not be confused with the number of days in shanah chaserah (שנה חסרה), the deficient year.

Other

It's possible to use superpyth fifth of Rectified Hebrew fifth, 209\353, as a generator. In this case, 76 & 353 temperament is obtained. In the 2.5.7.13 subgroup, this results in the fifth being equal to 98/65 and the comma basis of 10985/10976, [-103 0 -38 51 0 13⟩.

353edo also supports apparatus temperaments, and marvo and zarvo.

Table of intervals

Step Note name

(diatonic Hebrew[19] version)

Associated ratio

(2.5.7.13 subgroup)

0 C 1/1
1 C-C#
2 C-Db
3 C-D 196/195
4 C-D#
19 C# 26/25
38 Db 14/13
41 Db-D 13/12
46 Db-F 35/32
57 D
76 D#
95 Eb
114 E 5/4
133 E# 13/10 I (patent val approximation)
134 E#-C# 13/10 II (direct approximation)
152 F
171 F# 7/5
190 Gb
206 Gb-Bb 3/2
209 G 98/65
228 G#
247 Ab 13/8
266 A
285 A# 7/4
304 Bb
323 B
342 B#/Cb
353 C 2/1

Regular temperament properties

Assuming 353edo is treated as the 2.5.7.11.13.17 subgroup temperament.

Subgroup Comma List Mapping Optimal
8ve Stretch (¢) 		Tuning Error
Absolute (¢) Relative (%)
2.5 [820 -353⟩ [⟨353 820]] -0.263 0.263 7.74
2.5.7 3136/3125, [209 -9 -67⟩ [⟨353 820 991]] -0.177 0.247 7.26
2.5.7.11 3136/3125, 5767168/5764801, [-20 -6  1 9⟩ [⟨353 820 991 1221]] -0.089 0.263 7.73
2.5.7.11.13 3136/3125, 4394/4375, 6656/6655, 5767168/5764801 [⟨353 820 991 1221 1306]] -0.024 0.268 7.89
2.5.7.11.13.17 3136/3125, 4394/4375, 7744/7735, 60112/60025, 64141/64000 [⟨353 820 991 1221 1306 1443]] -0.037 0.247 7.26

Rank-2 temperaments

Periods
per 8ve 		Generator* Cents* Associated
Ratio* 		Temperaments
1 19\353 64.59 26/25 Rectified hebrew
1 34\353 115.58 77/72 Apparatus
1 152\353 516.71 27/20 Marvo (353c) / zarvo (353cd)

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

Scales

  • RectifiedHebrew[19] - 18L 1s
  • RectifiedHebrew[130] - 93L 37s
  • Austro-Hungarian Minor[9] - 57 38 38 38 38 38 38 38 30

See also

Music

Mercury Amalgam

External links

Retrieved from "https://en.xen.wiki/index.php?title=353edo&oldid=127266"