353edo: Difference between revisions

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== Theory ==

353edo is in[[consistent]] in the [[5-odd-limit]] and [[harmonic]] [[3/1|3]] is about halfway between its steps. It is suitable for use with the 2.9.15.7.11.13.17.23.29.31.37 [[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 in[[consistent]] in the [[5-odd-limit]] and [[harmonic]] [[3/1|3]] is about halfway between its steps. It is suitable for use with the 2.9.15.7.11.13.17.23.29.31.37 [[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]], and serves as a very close approximation of its just-[[7/4]] tuning.

Using the [[patent val]] nonetheless, 353edo supports [[apparatus]], [[marvo]] and [[zarvo]].

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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.

It is possible to use a superpyth-ish 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, {{Monzo|-103 0 -38 51 0 13}}.

It is possible to use a superpyth-ish fifth of Rectified Hebrew fifth, 209\353, as a generator. In this case, {{nowrap|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, {{Monzo|-103 0 -38 51 0 13}}.

== Table of intervals ==

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|-

! Step

! Note name~~ (diatonic Hebrew[19] version)~~

! Note name*

! Associated ratio~~ (2.5.7.13 subgroup)~~

! Associated ratio**

|-

| 0

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| 2/1

|}

<nowiki />* Diatonic Hebrew[19] version

<nowiki />** 2.5.7.13 subgroup

== Regular temperament properties ==

Assuming 353edo is treated as the 2.5.7.11.13.17 subgroup temperament.

{~~{comma basis begin}}~~

{| class="wikitable center-4 center-5 center-6"

|-

! rowspan="2" | [[Subgroup]]

! rowspan="2" | [[Comma list]]

! rowspan="2" | [[Mapping]]

! rowspan="2" | Optimal 8ve stretch (¢)

! colspan="2" | Tuning error

|-

! [[TE error|Absolute]] (¢)

! [[TE simple badness|Relative]] (%)

|-

| 2.5

| {{monzo| 820 -353 }}

| {{mapping| 353 820 }}

| ~~−0~~.263

| −0.263

| 0.263

| 7.74

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| 3136/3125, {{monzo| 209 -9 -67 }}

| {{mapping| 353 820 991 }}

| ~~−0~~.177

| −0.177

| 0.247

| 7.26

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| 3136/3125, 5767168/5764801, {{monzo| -20 -6 1 9 }}

| {{mapping| 353 820 991 1221 }}

| ~~−0~~.089

| −0.089

| 0.263

| 7.73

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| 3136/3125, 4394/4375, 6656/6655, 5767168/5764801

| {{mapping| 353 820 991 1221 1306 }}

| ~~−0~~.024

| −0.024

| 0.268

| 7.89

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| 3136/3125, 4394/4375, 7744/7735, 60112/60025, 64141/64000

| {{mapping| 353 820 991 1221 1306 1443 }}

| ~~−0~~.037

| −0.037

| 0.247

| 7.26

~~{{comma basis end}~~}

|}

=== Rank-2 temperaments ===

{{rank-2 ~~begin}}~~

{| class="wikitable center-all left-5"

|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator

|-

! Periods per 8ve

! Generator*

! Cents*

! Associated ratio*

! Temperament

|-

| 1

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| 27/20

| [[Marvo]] (353c) / [[zarvo]] (353cd)

~~{{rank-2 end}~~}

|}

~~{{orf}}~~

<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Scales ==

* RectifiedHebrew[19] ~~–~~ 18L 1s

* RectifiedHebrew[19] – 18L 1s

* RectifiedHebrew[130] ~~–~~ 93L 37s

* RectifiedHebrew[130] – 93L 37s

* Austro-Hungarian Minor[9] ~~–~~ 57 38 38 38 38 38 38 38 30

* Austro-Hungarian Minor[9] – 57 38 38 38 38 38 38 38 30

== See also ==

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== Music ==

; [[Eliora]]

* [https://www.youtube.com/watch?v=JrSEGE6_oys ''Snow On My City''] (2022) ~~–~~ cover of [[wikipedia:Naomi Shemer|Naomi Shemer]] in Rectified Hebrew and apparatus

* [https://www.youtube.com/watch?v=JrSEGE6_oys ''Snow On My City''] (2022) – cover of [[wikipedia:Naomi Shemer|Naomi Shemer]] in Rectified Hebrew and apparatus

; [[Mercury Amalgam]]

* [https://www.youtube.com/watch?v=z-SxvrnkTzU ''Bottom Text''] (2022) in Rectified Hebrew

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|353}}
+{{ED intro}}
 == Theory ==
-edo is in[[consistent]] in the [[5-odd-limit]] and [[harmonic]] [[3/1|3]] is about halfway between its steps. It is suitable for use with the 2.9.15.7.11.13.17.23.29.31.37 [[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]].
+edo is in[[consistent]] in the [[5-odd-limit]] and [[harmonic]] [[3/1|3]] is about halfway between its steps. It is suitable for use with the 2.9.15.7.11.13.17.23.29.31.37 [[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]], and serves as a very close approximation of its just-[[7/4]] tuning.
 Using the [[patent val]] nonetheless, 353edo supports [[apparatus]], [[marvo]] and [[zarvo]].
@@ Line 20: / Line 20: @@
 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.
-It is possible to use a superpyth-ish 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, {{Monzo|-103 0 -38 51 0 13}}.
+It is possible to use a superpyth-ish fifth of Rectified Hebrew fifth, 209\353, as a generator. In this case, {{nowrap|76 &amp; 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, {{Monzo|-103 0 -38 51 0 13}}.
 == Table of intervals ==
@@ Line 26: / Line 26: @@
 |-
 ! Step
-! Note name<br /><span style="font-size: 0.75em;">(diatonic Hebrew[19] version)</span>
+! Note name*
-! Associated ratio<br /><span style="font-size: 0.75em;">(2.5.7.13 subgroup)</span>
+! Associated ratio**
 |-
 | 0
@@ Line 141: / Line 141: @@
 | 2/1
 |}
+<nowiki />* Diatonic Hebrew[19] version
+<nowiki />** 2.5.7.13 subgroup
 == Regular temperament properties ==
 Assuming 353edo is treated as the 2.5.7.11.13.17 subgroup temperament.
-{{comma basis begin}}
+{| 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]] (%)
 |-
 | 2.5
 | {{monzo| 820 -353 }}
 | {{mapping| 353 820 }}
-| &minus;0.263
+| −0.263
 | 0.263
 | 7.74
@@ Line 156: / Line 168: @@
 | 3136/3125, {{monzo| 209 -9 -67 }}
 | {{mapping| 353 820 991 }}
-| &minus;0.177
+| −0.177
 | 0.247
 | 7.26
@@ Line 163: / Line 175: @@
 | 3136/3125, 5767168/5764801, {{monzo| -20 -6  1 9 }}
 | {{mapping| 353 820 991 1221 }}
-| &minus;0.089
+| −0.089
 | 0.263
 | 7.73
@@ Line 170: / Line 182: @@
 | 3136/3125, 4394/4375, 6656/6655, 5767168/5764801
 | {{mapping| 353 820 991 1221 1306 }}
-| &minus;0.024
+| −0.024
 | 0.268
 | 7.89
@@ Line 177: / Line 189: @@
 | 3136/3125, 4394/4375, 7744/7735, 60112/60025, 64141/64000
 | {{mapping| 353 820 991 1221 1306 1443 }}
-| &minus;0.037
+| −0.037
 | 0.247
 | 7.26
-{{comma basis end}}
+|}
-=== Rank-2 temperaments  ===
+=== Rank-2 temperaments ===
-{{rank-2 begin}}
+{| class="wikitable center-all left-5"
+|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
+|-
+! Periods<br />per 8ve
+! Generator*
+! Cents*
+! Associated<br />ratio*
+! Temperament
 |-
 | 1
@@ Line 202: / Line 221: @@
 | 27/20
 | [[Marvo]] (353c) / [[zarvo]] (353cd)
-{{rank-2 end}}
+|}
-{{orf}}
+<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
 == Scales ==
-* RectifiedHebrew[19] &ndash; 18L 1s
+* RectifiedHebrew[19] – 18L 1s
-* RectifiedHebrew[130] &ndash; 93L 37s
+* RectifiedHebrew[130] – 93L 37s
-* Austro-Hungarian Minor[9] &ndash; 57 38 38 38 38 38 38 38 30
+* Austro-Hungarian Minor[9] – 57 38 38 38 38 38 38 38 30
 == See also ==
@@ Line 216: / Line 235: @@
 == Music ==
 ; [[Eliora]]
-* [https://www.youtube.com/watch?v=JrSEGE6_oys ''Snow On My City''] (2022) &ndash; cover of [[wikipedia:Naomi Shemer|Naomi Shemer]] in Rectified Hebrew and apparatus
+* [https://www.youtube.com/watch?v=JrSEGE6_oys ''Snow On My City''] (2022) – cover of [[wikipedia:Naomi Shemer|Naomi Shemer]] in Rectified Hebrew and apparatus
 ; [[Mercury Amalgam]]
 * [https://www.youtube.com/watch?v=z-SxvrnkTzU ''Bottom Text''] (2022) in Rectified Hebrew