Würschmidt: Difference between revisions

Line 3:

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]].

Strong extensions to the [[7-limit]] include [[würschmidt family#septimal würschmidt|septimal würschmidt]], [[worschmidt]], and [[whirrschmidt]], but these are either considerably higher-damage or much higher-complexity than 5-limit würschmidt. In fact, the best septimal extension may be [[hemiwürschmidt]], which splits the ~5/4 generator into two ~[[28/25]]'s. 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.

Strong extensions to the [[7-limit]] include [[würschmidt family#septimal würschmidt|septimal würschmidt]], [[worschmidt]], and [[whirrschmidt]], 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. 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.

== Interval chains ==

In the below, octave-reduced harmonics below 125 are indicated in '''bold'''.

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

|+ style="font-size: 105%;" | Würschmidt

|-

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

|-

! 2.3.5.23 subgroup !! Add-11 extension

|-

| -8 || 497.59 || '''4/3''' || 162/121

|-

| -7 || 885.39 || 5/3 || 92/55

|-

| -6 || 73.19 || 23/12, 48/25 || 23/22, 288/275

|-

| -5 || 460.99 || 30/23, 125/96 || 72/55, 176/135

|-

| -4 || 848.79 || 75/46, 368/225, 625/384 || 18/11, 44/27

|-

| -3 || 36.60 || 46/45, 128/125 || 45/44, 55/54

|-

| -2 || 424.40 || 23/18, 32/25 || 88/69, 225/176

|-

| -1 || 812.20 || 8/5 || 110/69

|-

| 0 || 0.0 || '''1/1''' ||

|-

| 1 || 387.80 || '''5/4''' || 69/55

|-

| 2 || 775.60 || '''25/16''', 36/23 || 69/44, 352/225

|-

| 3 || 1163.40 || 45/23, '''125/64''' || 88/45, 108/55

|-

| 4 || 351.21 || 92/75, 225/184, 625/512 || 11/9, 27/22

|-

| 5 || 739.01 || 23/15, 192/125 || 55/36, 135/88

|-

| 6 || 1126.81 || 23/12, 48/25 || 44/23, 275/144

|-

| 7 || 314.61 || 6/5 || 55/46

|-

| 8 || 702.41 || '''3/2''' || 121/81

|-

| 9 || 1090.21 || '''15/8''' || 207/110, 253/135

|-

| 10 || 278.01 || 27/23, '''75/64''' || 88/75, 207/176

|-

| 11 || 665.82 || 135/92, 184/125, 375/256 || 22/15, 81/55

|-

| 12 || 1053.62 || 46/25, 675/368 || 11/6, 81/44

|-

| 13 || 241.42 || 23/20, 144/125 || 55/48, 132/115

|-

| 14 || 629.22 || '''23/16''', 36/25 || 33/23, 275/192

|-

| 15 || 1017.02 || 9/5 || 165/92, 242/135

|-

| 16 || 204.82 || '''9/8''' || 121/108

|-

| 17 || 592.62 || '''45/32''' || 253/180

|-

| 18 || 980.43 || 81/46, 225/128 || 44/25

|-

| 19 || 168.23 || 138/125, 405/368 || 11/10, 243/220

|-

| 20 || 556.03 || 69/50, 864/625 || '''11/8''', 243/176

{{table notes|cols=4

| In 2.3.5-targeted [[DKW theory|DKW]] tuning

}}

|}

</div>

=== Other tunings ===

@@ Line 3: / Line 3: @@
 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]].
-Strong extensions to the [[7-limit]] include [[würschmidt family#septimal würschmidt|septimal würschmidt]], [[worschmidt]], and [[whirrschmidt]], but these are either considerably higher-damage or much higher-complexity than 5-limit würschmidt. In fact, the best septimal extension may be [[hemiwürschmidt]], which splits the ~5/4 generator into two ~[[28/25]]'s. 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.
+Strong extensions to the [[7-limit]] include [[würschmidt family#septimal würschmidt|septimal würschmidt]], [[worschmidt]], and [[whirrschmidt]], 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. 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.
 {{tdlink|Würschmidt family #Würschmidt}}
+== Interval chains ==
+In the below, octave-reduced harmonics below 125 are indicated in '''bold'''.
+<div><div style="display: inline-grid; margin-right: 25px;">
+{| class="wikitable center-1 right-2"
+|+ style="font-size: 105%;" | Würschmidt
+|-
+! rowspan="2" | &#35; !! rowspan="2" | Cents&#42; !! colspan="2" | Approximate Ratios
+|-
+! 2.3.5.23 subgroup !! Add-11 extension
+|-
+| -8 || 497.59 || '''4/3''' || 162/121
+|-
+| -7 || 885.39 || 5/3 || 92/55
+|-
+| -6 || 73.19 || 23/12, 48/25 || 23/22, 288/275
+|-
+| -5 || 460.99 || 30/23, 125/96 || 72/55, 176/135
+|-
+| -4 || 848.79 || 75/46, 368/225, 625/384 || 18/11, 44/27
+|-
+| -3 || 36.60 || 46/45, 128/125 || 45/44, 55/54
+|-
+| -2 || 424.40 || 23/18, 32/25 || 88/69, 225/176
+|-
+| -1 || 812.20 || 8/5 || 110/69
+|-
+| 0 || 0.0 || '''1/1''' ||
+|-
+| 1 || 387.80 || '''5/4''' || 69/55
+|-
+| 2 || 775.60 || '''25/16''', 36/23 || 69/44, 352/225
+|-
+| 3 || 1163.40 || 45/23, '''125/64''' || 88/45, 108/55
+|-
+| 4 || 351.21 || 92/75, 225/184, 625/512 || 11/9, 27/22
+|-
+| 5 || 739.01 || 23/15, 192/125 || 55/36, 135/88
+|-
+| 6 || 1126.81 || 23/12, 48/25 || 44/23, 275/144
+|-
+| 7 || 314.61 || 6/5 || 55/46
+|-
+| 8 || 702.41 || '''3/2''' || 121/81
+|-
+| 9 || 1090.21 || '''15/8''' || 207/110, 253/135
+|-
+| 10 || 278.01 || 27/23, '''75/64''' || 88/75, 207/176
+|-
+| 11 || 665.82 || 135/92, 184/125, 375/256 || 22/15, 81/55
+|-
+| 12 || 1053.62 || 46/25, 675/368 || 11/6, 81/44
+|-
+| 13 || 241.42 || 23/20, 144/125 || 55/48, 132/115
+|-
+| 14 || 629.22 || '''23/16''', 36/25 || 33/23, 275/192
+|-
+| 15 || 1017.02 || 9/5 || 165/92, 242/135
+|-
+| 16 || 204.82 || '''9/8''' || 121/108
+|-
+| 17 || 592.62 || '''45/32''' || 253/180
+|-
+| 18 || 980.43 || 81/46, 225/128 || 44/25
+|-
+| 19 || 168.23 || 138/125, 405/368 || 11/10, 243/220
+|-
+| 20 || 556.03 || 69/50, 864/625 || '''11/8''', 243/176
+{{table notes|cols=4
+| In 2.3.5-targeted [[DKW theory|DKW]] tuning
+}}
+|}
+</div>
 === Other tunings ===