Würschmidt: Difference between revisions

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'''Würschmidt''' is a [[rank-2 temperament|rank-2]] [[regular temperament|temperament]] and parent of the [[würschmidt family]], characterized by tempering out the würschmidt comma, ([[393216/390625]]). It can be treated as analogous to [[schismatic]] with the roles of the primes 3 and 5 reversed, since würschmidt is [[generator|generated]] by a [[5/4|classical major third (5/4)]], very slightly sharpened so that eight of them make the sixth harmonic ([[6/1]]), giving [[3/2]] the same complexity [[5/4]] does in schismatic, but with comparable accuracy on the part of the generator. Four generators, therefore, reach the interval [[625/512]], which is equated to [[768/625]] and functions as a neutral third.

'''Würschmidt''' is a [[rank-2 temperament|rank-2]] [[regular temperament|temperament]] and parent of the [[würschmidt family]], characterized by tempering out the würschmidt comma, [[393216/390625]]. It can be treated as analogous to [[schismatic]] with the roles of the primes 3 and 5 reversed, since würschmidt is [[generator|generated]] by a [[5/4|classical major third (5/4)]], very slightly sharpened so that eight of them make the sixth harmonic ([[6/1]]), giving [[3/2]] the same complexity [[5/4]] does in schismatic, but with comparable accuracy on the part of the generator. Four generators, therefore, reach the interval [[625/512]], which is equated to [[768/625]] and functions as a neutral third.

== Extensions ==

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== Interval chain ==

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

In the below, octave-reduced harmonics 1–23 are indicated in '''bold'''.

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

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! 2.3.5.23 subgroup !! Add-11 extension

|-

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

|-

~~| -7 || 885.39 || 5/3, 192/115 || 92/55~~

|-

~~| -6 || 73.19 || 24/23, 25/24 || 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, 115/72 || 110/69~~

~~|- style="background-color: #C7BFDF;"~~

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

|-

| 1 || 387.80 || '''5/4''', 144/115 || 69/55

|-

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

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

|-

| 3 || 1163.40 || 45/23, ~~'''~~125/64~~'''~~, 736/375 || 88/45, 108/55

| 3 || 1163.40 || 45/23, 125/64, 736/375 || 88/45, 108/55

|-

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

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| 9 || 1090.21 || '''15/8''', 216/115 || 207/110, 253/135

|-

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

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

|-

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

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

|-

| 15 || 1017.02 || 9/5, ~~'''~~115/64~~'''~~ || 165/92, 242/135

| 15 || 1017.02 || 9/5, 115/64 || 165/92, 242/135

|-

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

|-

| 17 || 592.62 || ~~'''~~45/32~~'''~~, 162/115 || 253/180

| 17 || 592.62 || 45/32, 162/115 || 253/180

|-

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

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|-

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

|-

| 21 || 943.83 || 69/40 || 55/32

|-

| 22 || 131.63 || 69/64 || 165/128

|}

<nowiki>*</nowiki> In 2.3.5-targeted [[DKW theory|DKW]] tuning

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=== Optimized tunings ===

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

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

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

|-

! Weight-skew\Order !! Euclidean

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{| 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 1: / Line 1: @@
-'''Würschmidt''' is a [[rank-2 temperament|rank-2]] [[regular temperament|temperament]] and parent of the [[würschmidt family]], characterized by tempering out the würschmidt comma, ([[393216/390625]]). It can be treated as analogous to [[schismatic]] with the roles of the primes 3 and 5 reversed, since würschmidt is [[generator|generated]] by a [[5/4|classical major third (5/4)]], very slightly sharpened so that eight of them make the sixth harmonic ([[6/1]]), giving [[3/2]] the same complexity [[5/4]] does in schismatic, but with comparable accuracy on the part of the generator. Four generators, therefore, reach the interval [[625/512]], which is equated to [[768/625]] and functions as a neutral third.
+'''Würschmidt''' is a [[rank-2 temperament|rank-2]] [[regular temperament|temperament]] and parent of the [[würschmidt family]], characterized by tempering out the würschmidt comma, [[393216/390625]]. It can be treated as analogous to [[schismatic]] with the roles of the primes 3 and 5 reversed, since würschmidt is [[generator|generated]] by a [[5/4|classical major third (5/4)]], very slightly sharpened so that eight of them make the sixth harmonic ([[6/1]]), giving [[3/2]] the same complexity [[5/4]] does in schismatic, but with comparable accuracy on the part of the generator. Four generators, therefore, reach the interval [[625/512]], which is equated to [[768/625]] and functions as a neutral third.
-{{tdlink|Würschmidt family #Würschmidt}}
+{{Tdlink|Würschmidt family #Würschmidt}}
 == Extensions ==
@@ Line 11: / Line 11: @@
 == Interval chain ==
-In the below, octave-reduced harmonics below 125 are indicated in '''bold'''.
+In the below, octave-reduced harmonics 1–23 are indicated in '''bold'''.
 {| class="wikitable center-1 right-2"
@@ Line 19: / Line 19: @@
 ! 2.3.5.23 subgroup !! Add-11 extension
 |-
-| -8 || 497.59 || 4/3 || 162/121
-|-
-| -7 || 885.39 || 5/3, 192/115 || 92/55
-|-
-| -6 || 73.19 || 24/23, 25/24 || 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, 115/72 || 110/69
-|- style="background-color: #C7BFDF;"
 | 0 || 0.0 || '''1/1''' ||
 |-
 | 1 || 387.80 || '''5/4''', 144/115 || 69/55
 |-
-| 2 || 775.60 || '''25/16''', 36/23 || 69/44, 352/225
+| 2 || 775.60 || 25/16, 36/23 || 69/44, 352/225
 |-
-| 3 || 1163.40 || 45/23, '''125/64''', 736/375 || 88/45, 108/55
+| 3 || 1163.40 || 45/23, 125/64, 736/375 || 88/45, 108/55
 |-
 | 4 || 351.21 || 92/75, 225/184, 625/512 || 11/9, 27/22
@@ Line 55: / Line 39: @@
 | 9 || 1090.21 || '''15/8''', 216/115 || 207/110, 253/135
 |-
-| 10 || 278.01 || 27/23, '''75/64''' || 88/75, 207/176
+| 10 || 278.01 || 27/23, 75/64 || 88/75, 207/176
 |-
 | 11 || 665.82 || 184/125, 135/92, 375/256 || 22/15, 81/55
@@ Line 65: / Line 49: @@
 | 14 || 629.22 || '''23/16''', 36/25 || 33/23, 275/192
 |-
-| 15 || 1017.02 || 9/5, '''115/64''' || 165/92, 242/135
+| 15 || 1017.02 || 9/5, 115/64 || 165/92, 242/135
 |-
 | 16 || 204.82 || '''9/8''' || 121/108
 |-
-| 17 || 592.62 || '''45/32''', 162/115 || 253/180
+| 17 || 592.62 || 45/32, 162/115 || 253/180
 |-
 | 18 || 980.43 || 81/46, 225/128 || 44/25
@@ Line 76: / Line 60: @@
 |-
 | 20 || 556.03 || 69/50, 864/625 || '''11/8''', 243/176
+|-
+| 21 || 943.83 || 69/40 || 55/32
+|-
+| 22 || 131.63 || 69/64 || 165/128
 |}
 <nowiki>*</nowiki> In 2.3.5-targeted [[DKW theory|DKW]] tuning
@@ Line 82: / Line 70: @@
 === Optimized tunings ===
 {| class="wikitable mw-collapsible mw-collapsed"
-|+ style="font-size: 105%; white-space: nowrap;" | Prime-optimized tunings
+|+ style="font-size: 105%; white-space: nowrap;" | Prime-Optimized Tunings
 |-
 ! Weight-skew\Order !! Euclidean
@@ Line 113: / Line 101: @@
 {| class="wikitable center-all left-4"
-! EDO<br>generator
+! Edo<br>generator
-! [[Eigenmonzo|Eigenmonzo<br>(unchanged interval)]]*
+! [[Eigenmonzo|Eigenmonzo<br>(unchanged-interval)]]*
 ! Generator (¢)
 ! Comments