25edo: Difference between revisions

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'''25edo''' divides the [[octave]] in 25 [[equal]] steps of exact size 48 [[cent]]s each.

== Theory ==

~~{{Odd harmonics in edo|edo=25}}~~

~~25EDO divides the [[octave]] in 25 equal steps of exact size 48 [[cent]]s each. It~~ is a good way to tune the [[~~Archytas_clan#7-limit|~~blackwood ~~temperament~~]], which takes the very sharp fifths of [[5edo]] as a given, tempers out 28/27 and 49/48, and attempts to optimize the tunings for 5 ([[5/4]]) and 7 ([[7/4]]). It also tunes [[sixix]] temperament with a sharp fifth. It supplies the optimal patent val for the 11-limit 6&25 temperament tempering out 49/48, 77/75 and 605/576, and the 13-limit extension also tempering out 66/65.

25edo is a good way to tune the [[blackwood]] temperament, which takes the very sharp fifths of [[5edo]] as a given, tempers out 28/27 and 49/48, and attempts to optimize the tunings for 5 ([[5/4]]) and 7 ([[7/4]]). It also tunes [[sixix]] temperament with a sharp fifth. It supplies the optimal patent val for the 11-limit 6&25 temperament tempering out 49/48, 77/75 and 605/576, and the 13-limit extension also tempering out 66/65.

~~25EDO~~ has fifths 18 cents sharp, but its major thirds are excellent and its 7/4 is acceptable. Moreover, in full 7-limit including the 3, it is not [[consistent]]. It therefore makes sense to use it as a 2.5.7 [[~~Just intonation subgroups|~~subgroup]] tuning. Looking just at 2, 5, and 7, it equates five [[8/7]]s with the octave, and so tempers out (8/7)^5 / 2 = 16807/16384. It also equates a [[128/125]] [[diesis]] and two [[septimal tritones]] of [[7/5]] with the octave, and hence tempers out 3136/3125. If we want to temper out both of these and also have decent fifths, the obvious solution is [[50edo]]. An alternative fifth, 14\25, which is 672 cents, provides an alternative very flat fifth which can be used for [[mavila]] temperament.

25edo has fifths 18 cents sharp, but its major thirds of 5/4 are excellent and its 7/4 is acceptable. Moreover, in full 7-limit including the 3, it is not [[consistent]]. It therefore makes sense to use it as a 2.5.7 [[subgroup]] tuning. Looking just at 2, 5, and 7, it equates five [[8/7]]'s with the octave, and so tempers out (8/7)<sup>5</sup> / 2 = 16807/16384. It also equates a [[128/125]] [[diesis]] and two [[septimal tritones]] of [[7/5]] with the octave, and hence tempers out [[3136/3125]]. If we want to temper out both of these and also have decent fifths, the obvious solution is [[50edo]]. An alternative fifth, 14\25, which is 672 cents, provides an alternative very flat fifth which can be used for [[mavila]] temperament.

If 5/4 and 7/4 ~~aren't~~ good enough, it also does 17/16 and 19/16, just like ~~12EDO~~. In fact, on the [[k*~~N_subgroups~~|2*25 subgroup]] 2.9.5.7.33.39.17.19 it provides the same tuning and tempers out the same commas as 50et, which makes for a wide range of harmony.

If 5/4 and 7/4 are not good enough, it also does 17/16 and 19/16, just like 12edo. In fact, on the [[k*N subgroups|2*25 subgroup]] 2.9.5.7.33.39.17.19 it provides the same tuning and tempers out the same commas as 50et, which makes for a wide range of harmony.

=== Odd harmonics ===

== Intervals ==

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== Relationship to Armodue ==

Like [[16edo~~|16-EDO~~]] and [[23edo~~|23-EDO~~]], ~~25-EDO~~ contains the 9-note ~~"Superdiatonic"~~ scale of [[~~7L_2s|7L2s~~]] (LLLsLLLLs) that is generated by a circle of heavily-flattened 3/2s (ranging in size from 5\9~~-EDO~~ or 666.67 cents, to 4\7~~-EDO~~ or 685.71 cents). The ~~25-EDO~~ generator for this scale is the 672-cent interval. This allows ~~25-EDO~~ to be used with the [[~~Armodue_theory~~|Armodue]] notation system in much the same way that [[19edo~~|19-EDO~~]] is used with the standard diatonic notation; see the above interval chart for the Armodue names. Because the ~~25-EDO~~ Armodue 6th is flatter than that of ~~16-EDO~~ (the middle of the Armodue spectrum), sharps are lower in pitch than enharmonic flats.

Like [[16edo]] and [[23edo]], 25edo contains the 9-note superdiatonic scale of [[7L 2s]] (LLLsLLLLs) that is generated by a circle of heavily-flattened 3/2s (ranging in size from 5\9 or 666.67 cents, to 4\7 or 685.71 cents). The 25edo generator for this scale is the 672-cent interval. This allows 25edo to be used with the [[Armodue theory|Armodue]] notation system in much the same way that [[19edo]] is used with the standard diatonic notation; see the above interval chart for the Armodue names. Because the 25edo Armodue 6th is flatter than that of 16edo (the middle of the Armodue spectrum), sharps are lower in pitch than enharmonic flats.

== Commas ==

~~25 EDO~~ [[tempers out]] the following [[comma]]s. (Note: This assumes the [[val]] {{val| 25 40 58 70 86 93 }}.)

25edo [[tempers out]] the following [[comma]]s. (Note: This assumes the [[val]] {{val| 25 40 58 70 86 93 }}.)

{| class="wikitable center-all left-3 right-4 left-6"

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== ~~A 25edo keyboard~~ ==

== Keyboard layout ==

[[File:mm25.PNG|alt=mm25.PNG|mm25.PNG]]

@@ Line 7: / Line 7: @@
 | Consistency = 5
 }}
+'''25edo''' divides the [[octave]] in 25 [[equal]] steps of exact size 48 [[cent]]s each.
 == Theory ==
-{{Odd harmonics in edo|edo=25}}
-EDO divides the [[octave]] in 25 equal steps of exact size 48 [[cent]]s each. It is a good way to tune the [[Archytas_clan#7-limit|blackwood temperament]], which takes the very sharp fifths of [[5edo]] as a given, tempers out 28/27 and 49/48, and attempts to optimize the tunings for 5 ([[5/4]]) and 7 ([[7/4]]). It also tunes [[sixix]] temperament with a sharp fifth. It supplies the optimal patent val for the 11-limit 6&amp;25 temperament tempering out 49/48, 77/75 and 605/576, and the 13-limit extension also tempering out 66/65.
+edo is a good way to tune the [[blackwood]] temperament, which takes the very sharp fifths of [[5edo]] as a given, tempers out 28/27 and 49/48, and attempts to optimize the tunings for 5 ([[5/4]]) and 7 ([[7/4]]). It also tunes [[sixix]] temperament with a sharp fifth. It supplies the optimal patent val for the 11-limit 6&amp;25 temperament tempering out 49/48, 77/75 and 605/576, and the 13-limit extension also tempering out 66/65.
-EDO has fifths 18 cents sharp, but its major thirds are excellent and its 7/4 is acceptable. Moreover, in full 7-limit including the 3, it is not [[consistent]]. It therefore makes sense to use it as a 2.5.7 [[Just intonation subgroups|subgroup]] tuning. Looking just at 2, 5, and 7, it equates five [[8/7]]s with the octave, and so tempers out (8/7)^5 / 2 = 16807/16384. It also equates a [[128/125]] [[diesis]] and two [[septimal tritones]] of [[7/5]] with the octave, and hence tempers out 3136/3125. If we want to temper out both of these and also have decent fifths, the obvious solution is [[50edo]]. An alternative fifth, 14\25, which is 672 cents, provides an alternative very flat fifth which can be used for [[mavila]] temperament.
+edo has fifths 18 cents sharp, but its major thirds of 5/4 are excellent and its 7/4 is acceptable. Moreover, in full 7-limit including the 3, it is not [[consistent]]. It therefore makes sense to use it as a 2.5.7 [[subgroup]] tuning. Looking just at 2, 5, and 7, it equates five [[8/7]]'s with the octave, and so tempers out (8/7)<sup>5</sup> / 2 = 16807/16384. It also equates a [[128/125]] [[diesis]] and two [[septimal tritones]] of [[7/5]] with the octave, and hence tempers out [[3136/3125]]. If we want to temper out both of these and also have decent fifths, the obvious solution is [[50edo]]. An alternative fifth, 14\25, which is 672 cents, provides an alternative very flat fifth which can be used for [[mavila]] temperament.
-If 5/4 and 7/4 aren't good enough, it also does 17/16 and 19/16, just like 12EDO. In fact, on the [[k*N_subgroups|2*25 subgroup]] 2.9.5.7.33.39.17.19 it provides the same tuning and tempers out the same commas as 50et, which makes for a wide range of harmony.
+If 5/4 and 7/4 are not good enough, it also does 17/16 and 19/16, just like 12edo. In fact, on the [[k*N subgroups|2*25 subgroup]] 2.9.5.7.33.39.17.19 it provides the same tuning and tempers out the same commas as 50et, which makes for a wide range of harmony.
+=== Odd harmonics ===
+{{Harmonics in equal|25}}
 == Intervals ==
@@ Line 331: / Line 334: @@
 == Relationship to Armodue ==
-Like [[16edo|16-EDO]] and [[23edo|23-EDO]], 25-EDO contains the 9-note "Superdiatonic" scale of [[7L_2s|7L2s]] (LLLsLLLLs) that is generated by a circle of heavily-flattened 3/2s (ranging in size from 5\9-EDO or 666.67 cents, to 4\7-EDO or 685.71 cents). The 25-EDO generator for this scale is the 672-cent interval. This allows 25-EDO to be used with the [[Armodue_theory|Armodue]] notation system in much the same way that [[19edo|19-EDO]] is used with the standard diatonic notation; see the above interval chart for the Armodue names. Because the 25-EDO Armodue 6th is flatter than that of 16-EDO (the middle of the Armodue spectrum), sharps are lower in pitch than enharmonic flats.
+Like [[16edo]] and [[23edo]], 25edo contains the 9-note superdiatonic scale of [[7L 2s]] (LLLsLLLLs) that is generated by a circle of heavily-flattened 3/2s (ranging in size from 5\9 or 666.67 cents, to 4\7 or 685.71 cents). The 25edo generator for this scale is the 672-cent interval. This allows 25edo to be used with the [[Armodue theory|Armodue]] notation system in much the same way that [[19edo]] is used with the standard diatonic notation; see the above interval chart for the Armodue names. Because the 25edo Armodue 6th is flatter than that of 16edo (the middle of the Armodue spectrum), sharps are lower in pitch than enharmonic flats.
 == Commas ==
-EDO [[tempers out]] the following [[comma]]s. (Note: This assumes the [[val]] {{val| 25 40 58 70 86 93 }}.)
+edo [[tempers out]] the following [[comma]]s. (Note: This assumes the [[val]] {{val| 25 40 58 70 86 93 }}.)
 {| class="wikitable center-all left-3 right-4 left-6"
@@ Line 452: / Line 455: @@
 <references/>
-== A 25edo keyboard ==
+== Keyboard layout ==
 [[File:mm25.PNG|alt=mm25.PNG|mm25.PNG]]