Würschmidt: Difference between revisions

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== Extensions ==

Another useful interpretation of the würschmidt comma is that it makes the interval of [[25/24]] equal to two-thirds the size of [[16/15]]. This can be exploited, as 16/15 factorizes into near-2:1 parts as ([[24/23]])×([[46/45]]), and therefore it is illogical ''not'' to set 25/24 equal to 24/23 (and [[128/125]] equal to 46/45) as well and set the remainder, 46/45, equal to a third of 16/15, by tempering S24 = [[576/575]] and S462 × S47 = [[12167/12150]] in the 2.3.5.23 [[subgroup]]. 14 generators turn out to stack to [[23/1]], and notably, 6/1 stacked 7 times and 23/1 stacked four times (at 56 generators) differ only by the 0.59-cent comma [[279936/279841]].

Another useful interpretation of the würschmidt comma is that it makes the interval of [[25/24]] equal to two-thirds the size of [[16/15]]. This can be exploited, as 16/15 factorizes into near-2:1 parts as {{nowrap|([[24/23]]) × ([[46/45]])}}, and therefore it is illogical ''not'' to set 25/24 equal to 24/23 (and [[128/125]] equal to 46/45) as well and set the remainder, 46/45, equal to a third of 16/15, by tempering {{nowrap|S24 {{=}} [[576/575]]}} and {{nowrap|S462 × S47 {{=}} [[12167/12150]]}} in the 2.3.5.23 [[subgroup]]. 14 generators turn out to stack to [[23/1]], and notably, 6/1 stacked 7 times and 23/1 stacked four times (at 56 generators) differ only by the 0.59-cent comma [[279936/279841]].

Strong extensions to the [[7-limit]] include [[würschmidt family#septimal würschmidt|septimal würschmidt]] (tempering out [[225/224]], finding 7 at +18 generator steps), [[worschmidt]] (tempering out [[126/125]], finding 7 at -13 generator steps), and [[whirrschmidt]] (tempering out [[4375/4374]], finding 7 at +52 generator steps), but these are either considerably higher-damage or much higher-complexity than 5-limit würschmidt. In fact, the best septimal extension may be the weak extension [[hemiwürschmidt]], which splits the ~5/4 generator into two ~[[28/25]]'s by tempering out [[3136/3125]] alongside [[6144/6125]] (notably, in the 2.3.5.7.23 subgroup, this is the extension that tempers out the tiny comma S161 = [[25921/25920]]).

Strong extensions to the [[7-limit]] include [[würschmidt family#septimal würschmidt|septimal würschmidt]] (tempering out [[225/224]], finding 7 at +18 generator steps), [[worschmidt]] (tempering out [[126/125]], finding 7 at -13 generator steps), and [[whirrschmidt]] (tempering out [[4375/4374]], finding 7 at +52 generator steps), but these are either considerably higher-damage or much higher-complexity than 5-limit würschmidt. In fact, the best septimal extension may be the weak extension [[hemiwürschmidt]], which splits the ~5/4 generator into two ~[[28/25]]'s by tempering out [[3136/3125]] alongside [[6144/6125]] (notably, in the 2.3.5.7.23 subgroup, this is the extension that tempers out the tiny comma {{nowrap|S161 {{=}} [[25921/25920]]}}).

Therefore, it may be advisable to consider würschmidt a no-sevens system, specifically in the 2.3.5.11 subgroup, where an extension that equates 128/125 with [[45/44]] and therefore 625/512 with [[11/9]] (by tempering out [[243/242]] and [[5632/5625]]), finding the 11th harmonic at 20 generators up, is highly natural in addition to the aforementioned extension to prime 23.

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{| class="wikitable center-1 right-2"

|-

! rowspan="2" | # !! rowspan="2" | Cents* !! colspan="2" | Approximate ratios

|-

! 2.3.5.23 subgroup !! Add-11 extension

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| 31 || 21.06 || 81/80 || 121/120

|}

<nowiki>*~~</nowiki>~~ In 5-limit CWE tuning

<nowiki />* In 5-limit CWE tuning

== Tunings ==

=== Optimized tunings ===

{| class="wikitable mw-collapsible mw-collapsed"

|+ style="font-size: 105%; white-space: nowrap;" | ~~Constrained prime~~-optimized tunings

|+ style="font-size: 105%; white-space: nowrap;" | Prime-optimized tunings

|-

! Weight-skew\Order !! Euclidean

! rowspan="2" | Weight-skew\Order !! colspan="2" | Euclidean

|-

~~| Tenney || (2.3.5) CTE: ~5/4 = 387.734¢~~

! Constrained !! Destretched

|-

| ~~Weil~~ || (2.3.5) ~~CWE~~: ~5/4 = 387.~~776¢~~

! Tenney

| (2.3.5) CTE: ~5/4 = 387.734¢ || (2.3.5) POTE: ~5/4 = 387.7993¢

|-

~~| Tenney |~~| (2.3.5~~.23~~) ~~CTE~~: ~5/4 = 387.~~734¢~~

! Weil

| (2.3.5) CWE: ~5/4 = 387.776¢ ||

|-

~~| Weil |~~| (2.3.5.23) ~~CWE~~: ~5/4 = 387.~~781¢~~

! Tenney

|}

| (2.3.5.23) CTE: ~5/4 = 387.734¢ || (2.3.5.23) POTE: ~5/4 = 387.8051¢

~~{| class="wikitable mw-collapsible mw-collapsed"~~

~~|+ style="font-size: 105%; white-space: nowrap;" | Destretched prime-optimized tunings~~

|-

~~! Weight-skew\Order !! Euclidean~~

|-

~~| Tenney~~ || (2.3.5) POTE: ~5/4 = 387.~~7993¢~~

|-

~~| Tenney |~~| (2.3.5.23) ~~POTE~~: ~5/4 = 387.~~8051¢~~

! Weil

| (2.3.5.23) CWE: ~5/4 = 387.781¢ ||

|}

{| class="wikitable mw-collapsible mw-collapsed"

|+style="font-size: 105%; white-space: nowrap;" | [[Delta-rational chord|DR]] and equal-beating tunings

|+ style="font-size: 105%; white-space: nowrap;" | [[Delta-rational chord|DR]] and equal-beating tunings

|-

! Optimized chord !! Generator value !! Polynomial !! Further notes

|-

| 3:4:5 (+1 +1) || ~5/4 = 387.4975 || ''g''8 + 8''g'' - 16 = 0 || 1-3-5 equal-beating tuning, close to 3/29-comma

| 3:4:5 (+1 +1) || ~5/4 = 387.4975 || ''g''8 + 8''g'' − 16 = 0 || {{dash|1, 3, 5|med}} equal-beating tuning, close to 3/29-comma

|-

| 4:5:6 (+1 +1) || ~5/4 = 388.1207 || ''g''8 - 8''g'' + 8 = 0 || 1-3-5 equal-beating tuning, close to 3/19-comma

| 4:5:6 (+1 +1) || ~5/4 = 388.1207 || ''g''8 − 8''g'' + 8 = 0 || {{dash|1, 3, 5|med}} equal-beating tuning, close to 3/19-comma

|-

| 10:12:15 (+2 +3) || ~5/4 = 388.2216 || ''g''8 - 2''g''7 + 4 = 0 || Close to 1/6-comma

| 10:12:15 (+2 +3) || ~5/4 = 388.2216 || ''g''8 − 2''g''7 + 4 = 0 || Close to 1/6-comma

|-

| 15:18:23 (+3 +5) || ~5/4 = 387.9215 || 4''g''7 - 3''g''5 - 10 = 0 ||

| 15:18:23 (+3 +5) || ~5/4 = 387.9215 || 4''g''7 − 3''g''5 − 10 = 0 ||

|}

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{| class="wikitable center-all left-4"

! Edo generator

! Edo generator

! [[Eigenmonzo|Eigenmonzo (unchanged-interval)]]*

! [[Eigenmonzo|Eigenmonzo (unchanged-interval)]]*

! Generator (¢)

! Comments

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| '''Upper bound of 2.3.5.23-subgroup 25-odd-limit diamond monotone'''

|}

<nowiki>*~~</nowiki> besides~~ the octave

<nowiki />* Besides the octave

=== Other tunings ===

@@ Line 4: / Line 4: @@
 == Extensions ==
-Another useful interpretation of the würschmidt comma is that it makes the interval of [[25/24]] equal to two-thirds the size of [[16/15]]. This can be exploited, as 16/15 factorizes into near-2:1 parts as ([[24/23]])×([[46/45]]), and therefore it is illogical ''not'' to set 25/24 equal to 24/23 (and [[128/125]] equal to 46/45) as well and set the remainder, 46/45, equal to a third of 16/15, by tempering S24 = [[576/575]] and S46<sup>2</sup> × S47 = [[12167/12150]] in the 2.3.5.23 [[subgroup]]. 14 generators turn out to stack to [[23/1]], and notably, 6/1 stacked 7 times and 23/1 stacked four times (at 56 generators) differ only by the 0.59-cent comma [[279936/279841]].
+Another useful interpretation of the würschmidt comma is that it makes the interval of [[25/24]] equal to two-thirds the size of [[16/15]]. This can be exploited, as 16/15 factorizes into near-2:1 parts as {{nowrap|([[24/23]]) &times; ([[46/45]])}}, and therefore it is illogical ''not'' to set 25/24 equal to 24/23 (and [[128/125]] equal to 46/45) as well and set the remainder, 46/45, equal to a third of 16/15, by tempering {{nowrap|S24 {{=}} [[576/575]]}} and {{nowrap|S46<sup>2</sup> &times; S47 {{=}} [[12167/12150]]}} in the 2.3.5.23 [[subgroup]]. 14 generators turn out to stack to [[23/1]], and notably, 6/1 stacked 7 times and 23/1 stacked four times (at 56 generators) differ only by the 0.59-cent comma [[279936/279841]].
-Strong extensions to the [[7-limit]] include [[würschmidt family#septimal würschmidt|septimal würschmidt]] (tempering out [[225/224]], finding 7 at +18 generator steps), [[worschmidt]] (tempering out [[126/125]], finding 7 at -13 generator steps), and [[whirrschmidt]] (tempering out [[4375/4374]], finding 7 at +52 generator steps), but these are either considerably higher-damage or much higher-complexity than 5-limit würschmidt. In fact, the best septimal extension may be the weak extension [[hemiwürschmidt]], which splits the ~5/4 generator into two ~[[28/25]]'s by tempering out [[3136/3125]] alongside [[6144/6125]] (notably, in the 2.3.5.7.23 subgroup, this is the extension that tempers out the tiny comma S161 = [[25921/25920]]).
+Strong extensions to the [[7-limit]] include [[würschmidt family#septimal würschmidt|septimal würschmidt]] (tempering out [[225/224]], finding 7 at +18 generator steps), [[worschmidt]] (tempering out [[126/125]], finding 7 at -13 generator steps), and [[whirrschmidt]] (tempering out [[4375/4374]], finding 7 at +52 generator steps), but these are either considerably higher-damage or much higher-complexity than 5-limit würschmidt. In fact, the best septimal extension may be the weak extension [[hemiwürschmidt]], which splits the ~5/4 generator into two ~[[28/25]]'s by tempering out [[3136/3125]] alongside [[6144/6125]] (notably, in the 2.3.5.7.23 subgroup, this is the extension that tempers out the tiny comma {{nowrap|S161 {{=}} [[25921/25920]]}}).
 Therefore, it may be advisable to consider würschmidt a no-sevens system, specifically in the 2.3.5.11 subgroup, where an extension that equates 128/125 with [[45/44]] and therefore 625/512 with [[11/9]] (by tempering out [[243/242]] and [[5632/5625]]), finding the 11th harmonic at 20 generators up, is highly natural in addition to the aforementioned extension to prime 23.
@@ Line 15: / Line 15: @@
 {| class="wikitable center-1 right-2"
 |-
-! rowspan="2" | # !! rowspan="2" | Cents* !! colspan="2" | Approximate ratios
+! rowspan="2" | &#35; !! rowspan="2" | Cents* !! colspan="2" | Approximate ratios
 |-
 ! 2.3.5.23 subgroup !! Add-11 extension
@@ Line 83: / Line 83: @@
 | 31 || 21.06 || 81/80 || 121/120
 |}
-<nowiki>*</nowiki> In 5-limit CWE tuning
+<nowiki />* In 5-limit CWE tuning
 == Tunings ==
 === Optimized tunings ===
 {| class="wikitable mw-collapsible mw-collapsed"
-|+ style="font-size: 105%; white-space: nowrap;" | Constrained prime-optimized tunings
+|+ style="font-size: 105%; white-space: nowrap;" | Prime-optimized tunings
 |-
-! Weight-skew\Order !! Euclidean
+! rowspan="2" | Weight-skew\Order !! colspan="2" | Euclidean
 |-
-| Tenney || (2.3.5) CTE: ~5/4 = 387.734¢
+! Constrained !! Destretched
 |-
-| Weil || (2.3.5) CWE: ~5/4 = 387.776¢
+! Tenney
+| (2.3.5) CTE: ~5/4 = 387.734¢ || (2.3.5) POTE: ~5/4 = 387.7993¢
 |-
-| Tenney || (2.3.5.23) CTE: ~5/4 = 387.734¢
+! Weil
+| (2.3.5) CWE: ~5/4 = 387.776¢ ||
 |-
-| Weil || (2.3.5.23) CWE: ~5/4 = 387.781¢
+! Tenney
-|}
+| (2.3.5.23) CTE: ~5/4 = 387.734¢ || (2.3.5.23) POTE: ~5/4 = 387.8051¢
-{| class="wikitable mw-collapsible mw-collapsed"
-|+ style="font-size: 105%; white-space: nowrap;" | Destretched prime-optimized tunings
-|-
-! Weight-skew\Order !! Euclidean
-|-
-| Tenney || (2.3.5) POTE: ~5/4 = 387.7993¢
 |-
-| Tenney || (2.3.5.23) POTE: ~5/4 = 387.8051¢
+! Weil
+| (2.3.5.23) CWE: ~5/4 = 387.781¢ ||
 |}
 {| class="wikitable mw-collapsible mw-collapsed"
-|+style="font-size: 105%; white-space: nowrap;" | [[Delta-rational chord|DR]] and equal-beating tunings
+|+ style="font-size: 105%; white-space: nowrap;" | [[Delta-rational chord|DR]] and equal-beating tunings
 |-
 ! Optimized chord !! Generator value !! Polynomial !! Further notes
 |-
-| 3:4:5 (+1 +1) || ~5/4 = 387.4975 || ''g''<sup>8</sup> + 8''g'' - 16 = 0 || 1-3-5 equal-beating tuning, close to 3/29-comma
+| 3:4:5 (+1 +1) || ~5/4 = 387.4975 || ''g''<sup>8</sup> + 8''g'' &minus; 16 = 0 || {{dash|1, 3, 5|med}} equal-beating tuning, close to 3/29-comma
 |-
-| 4:5:6 (+1 +1) || ~5/4 = 388.1207 || ''g''<sup>8</sup> - 8''g'' + 8 = 0 || 1-3-5 equal-beating tuning, close to 3/19-comma
+| 4:5:6 (+1 +1) || ~5/4 = 388.1207 || ''g''<sup>8</sup> &minus; 8''g'' + 8 = 0 || {{dash|1, 3, 5|med}} equal-beating tuning, close to 3/19-comma
 |-
-| 10:12:15 (+2 +3) || ~5/4 = 388.2216 || ''g''<sup>8</sup> - 2''g''<sup>7</sup> + 4 = 0 || Close to 1/6-comma
+| 10:12:15 (+2 +3) || ~5/4 = 388.2216 || ''g''<sup>8</sup> &minus; 2''g''<sup>7</sup> + 4 = 0 || Close to 1/6-comma
 |-
-| 15:18:23 (+3 +5) || ~5/4 = 387.9215 || 4''g''<sup>7</sup> - 3''g''<sup>5</sup> - 10 = 0 ||
+| 15:18:23 (+3 +5) || ~5/4 = 387.9215 || 4''g''<sup>7</sup> &minus; 3''g''<sup>5</sup> &minus; 10 = 0 ||
 |}
@@ Line 129: / Line 125: @@
 {| class="wikitable center-all left-4"
-! Edo<br>generator
+! Edo<br />generator
-! [[Eigenmonzo|Eigenmonzo<br>(unchanged-interval)]]*
+! [[Eigenmonzo|Eigenmonzo<br />(unchanged-interval)]]*
 ! Generator (¢)
 ! Comments
@@ Line 339: / Line 335: @@
 | '''Upper bound of 2.3.5.23-subgroup 25-odd-limit diamond monotone'''
 |}
-<nowiki>*</nowiki> besides the octave
+<nowiki />* Besides the octave
 === Other tunings ===