Diaschismic family: Difference between revisions

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The 5-limit parent comma for the '''diaschismic family''' is 2048/2025, the [[diaschisma]]. Its monzo is {{monzo| 11 -4 -2 }}, and flipping that yields {{multival| 2 -4 -11 }} for the wedgie for 5-limit '''diaschismic''', or '''srutal''', temperament. This tells us the period is half an octave, the [[Wikipedia: Greatest common divisor|GCD]] of 2 and -4, and that the generator is a fifth. Three periods gives 1800 cents, and decreasing this by two fifths gives the major third. [[~~34edo|34EDO~~]] is a good tuning choice, with [[~~46edo|46EDO~~]], [[~~56edo|56EDO~~]], [[~~58edo|58EDO~~]] or [[~~80edo|80EDO~~]] being other possibilities. Both [[~~12edo|12EDO~~]] and [[~~22edo|22EDO~~]] support it, and retuning them to a MOS of diaschismic gives two scale possibilities.

The 5-limit parent comma for the '''diaschismic family''' is 2048/2025, the [[diaschisma]]. Its monzo is {{monzo| 11 -4 -2 }}, and flipping that yields {{multival| 2 -4 -11 }} for the wedgie for 5-limit '''diaschismic''', or '''srutal''', temperament. This tells us the period is half an octave, the [[Wikipedia: Greatest common divisor|GCD]] of 2 and -4, and that the generator is a fifth. Three periods gives 1800 cents, and decreasing this by two fifths gives the major third. [[34 EDO]] is a good tuning choice, with [[46 EDO]], [[56 EDO]], [[58 EDO]] or [[80 EDO]] being other possibilities. Both [[12 EDO]] and [[22 EDO]] support it, and retuning them to a MOS of diaschismic gives two scale possibilities.

== Srutal (12&34, aka diaschismic) ==

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Pajara is closely associated with ~~22EDO~~ (not to mention [[Paul Erlich]]) but other tunings are possible. The 1/2 octave period serves as both a [[10/7]] and a [[7/5]]. Aside from ~~22EDO~~, 34 with the val {{val| 34 54 79 96 }} and 56 with the val {{val| 56 89 130 158 }} are are interesting alternatives, with more accpetable fifths, and a tetrad which is more clearly a dominant seventh. As such, they are closer to the tuning of ~~12EDO~~ and of common practice Western music in general, while retaining the distictiveness of a sharp fifth.

Pajara is closely associated with 22 EDO (not to mention [[Paul Erlich]]) but other tunings are possible. The 1/2 octave period serves as both a [[10/7]] and a [[7/5]]. Aside from 22 EDO, 34 with the val {{val| 34 54 79 96 }} and 56 with the val {{val| 56 89 130 158 }} are are interesting alternatives, with more accpetable fifths, and a tetrad which is more clearly a dominant seventh. As such, they are closer to the tuning of 12 EDO and of common practice Western music in general, while retaining the distictiveness of a sharp fifth.

Pajara extends nicely to an 11-limit version, for which the 56 tuning can be used, but a good alternative is to make the major thirds pure by setting the fifth to be 706.843 cents. Now 99/98, 100/99, 176/175 and 896/891 are being tempered out.

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== Diaschismic ==

A simpler characterization than the one given by the normal comma list is that diaschismic adds [[126/125]] or [[5120/5103]] to the set of commas, and it can also be called 46&58. However described, diaschismic has a 1/2 period and a sharp fifth generator like pajara, but not so sharp, giving a more accurate but more complex temperament. [[~~58edo|58EDO~~]] provides an excellent tuning, but an alternative is to make [[7/4]] just by making the fifth 703.897 cents, as opposed to 703.448 cents for ~~58EDO~~.

A simpler characterization than the one given by the normal comma list is that diaschismic adds [[126/125]] or [[5120/5103]] to the set of commas, and it can also be called 46&58. However described, diaschismic has a 1/2 period and a sharp fifth generator like pajara, but not so sharp, giving a more accurate but more complex temperament. [[58 EDO]] provides an excellent tuning, but an alternative is to make [[7/4]] just by making the fifth 703.897 cents, as opposed to 703.448 cents for 58 EDO.

Diaschismic extends naturally to the 17-limit, for which the same tunings may be used, making it one of the most important of the higher limit rank two temperaments. Adding the 11-limit adds the commas 176/175, 896/891 and 441/440. The 13-limit yields 196/195, 351/350, and 364/363; the 17-limit adds 136/135, 221/220, and 442/441. If you want to explore higher limit harmonies, diaschismic is certainly one excellent way to do it; MOS of 34 notes and even more the 46 note MOS will encompass very great deal of it. Of course 46 or 58 equal provide alternatives which in many ways are similar, particularly in the case of 58.

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== Keen ==

Keen adds 875/864 as well as 2240/2187 to the set of commas. It may also be described as the 22&56 temperament. [[~~78edo|78EDO~~]] is a good tuning choice, and remains a good one in the 11-limit, where keen, {{multival| 2 -4 18 -12 … }}, is really more interesting, adding 100/99 and 385/384 to the commas.

Keen adds 875/864 as well as 2240/2187 to the set of commas. It may also be described as the 22&56 temperament. [[78 EDO]] is a good tuning choice, and remains a good one in the 11-limit, where keen, {{multival| 2 -4 18 -12 … }}, is really more interesting, adding 100/99 and 385/384 to the commas.

Subgroup: 2.3.5.7

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== Echidna ==

Echidna adds 1728/1715 to the commas and takes 9/7 as a generator. It may be called the 22&58 temperament. [[~~58edo|58EDO~~]] or [[~~80edo|80EDO~~]] make for good tunings, or their vals can be add to {{val| 138 219 321 388 }}.

Echidna adds 1728/1715 to the commas and takes 9/7 as a generator. It may be called the 22&58 temperament. [[58 EDO]] or [[80 EDO]] make for good tunings, or their vals can be add to {{val| 138 219 321 388 }}.

Echidna becomes more interesting when extended to be an 11-limit temperament by adding 176/175, 896/891 or 540/539 to the commas, where the same tunings can be used as before. It then is able to represent the entire 11-limit diamond to within about six cents of error, within a compass of 24 notes. The 28 note 2MOS gives scope for this, and the 36 note MOS much more.

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== Shrutar ==

Shrutar adds 245/243 to the commas, and also tempers out 6144/6125. It can also be described as 22&46. Its generator can be taken as either 36/35 or 35/24; the latter is interesting since along with 15/14 and 21/20, it connects opposite sides of a hexany. [[~~68edo|68EDO~~]] makes for a good tuning, but another and excellent choice is a generator of 14(1/7), making 7s just.

Shrutar adds 245/243 to the commas, and also tempers out 6144/6125. It can also be described as 22&46. Its generator can be taken as either 36/35 or 35/24; the latter is interesting since along with 15/14 and 21/20, it connects opposite sides of a hexany. [[68 EDO]] makes for a good tuning, but another and excellent choice is a generator of 14(1/7), making 7s just.

By adding 121/120 or 176/175 to the commas, shrutar can be extended to the 11-limit, which loses a bit of accuracy, but picks up low-complexity 11-limit harmony, making shrutar quite an interesting 11-limit system. 68, 114 or a 14(1/7) generator can again be used as tunings.

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-The 5-limit parent comma for the '''diaschismic family''' is 2048/2025, the [[diaschisma]]. Its monzo is {{monzo| 11 -4 -2 }}, and flipping that yields {{multival| 2 -4 -11 }} for the wedgie for 5-limit '''diaschismic''', or '''srutal''', temperament. This tells us the period is half an octave, the [[Wikipedia: Greatest common divisor|GCD]] of 2 and -4, and that the generator is a fifth. Three periods gives 1800 cents, and decreasing this by two fifths gives the major third. [[34edo|34EDO]] is a good tuning choice, with [[46edo|46EDO]], [[56edo|56EDO]], [[58edo|58EDO]] or [[80edo|80EDO]] being other possibilities. Both [[12edo|12EDO]] and [[22edo|22EDO]] support it, and retuning them to a MOS of diaschismic gives two scale possibilities.
+The 5-limit parent comma for the '''diaschismic family''' is 2048/2025, the [[diaschisma]]. Its monzo is {{monzo| 11 -4 -2 }}, and flipping that yields {{multival| 2 -4 -11 }} for the wedgie for 5-limit '''diaschismic''', or '''srutal''', temperament. This tells us the period is half an octave, the [[Wikipedia: Greatest common divisor|GCD]] of 2 and -4, and that the generator is a fifth. Three periods gives 1800 cents, and decreasing this by two fifths gives the major third. [[34 EDO]] is a good tuning choice, with [[46 EDO]], [[56 EDO]], [[58 EDO]] or [[80 EDO]] being other possibilities. Both [[12 EDO]] and [[22 EDO]] support it, and retuning them to a MOS of diaschismic gives two scale possibilities.
 == Srutal (12&amp;34, aka diaschismic) ==
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 {{main| Pajara }}
-Pajara is closely associated with 22EDO (not to mention [[Paul Erlich]]) but other tunings are possible. The 1/2 octave period serves as both a [[10/7]] and a [[7/5]]. Aside from 22EDO, 34 with the val {{val| 34 54 79 96 }} and 56 with the val {{val| 56 89 130 158 }} are are interesting alternatives, with more accpetable fifths, and a tetrad which is more clearly a dominant seventh. As such, they are closer to the tuning of 12EDO and of common practice Western music in general, while retaining the distictiveness of a sharp fifth.
+Pajara is closely associated with 22 EDO (not to mention [[Paul Erlich]]) but other tunings are possible. The 1/2 octave period serves as both a [[10/7]] and a [[7/5]]. Aside from 22 EDO, 34 with the val {{val| 34 54 79 96 }} and 56 with the val {{val| 56 89 130 158 }} are are interesting alternatives, with more accpetable fifths, and a tetrad which is more clearly a dominant seventh. As such, they are closer to the tuning of 12 EDO and of common practice Western music in general, while retaining the distictiveness of a sharp fifth.
 Pajara extends nicely to an 11-limit version, for which the 56 tuning can be used, but a good alternative is to make the major thirds pure by setting the fifth to be 706.843 cents. Now 99/98, 100/99, 176/175 and 896/891 are being tempered out.
@@ Line 260: / Line 260: @@
 == Diaschismic ==
-A simpler characterization than the one given by the normal comma list is that diaschismic adds [[126/125]] or [[5120/5103]] to the set of commas, and it can also be called 46&amp;58. However described, diaschismic has a 1/2 period and a sharp fifth generator like pajara, but not so sharp, giving a more accurate but more complex temperament. [[58edo|58EDO]] provides an excellent tuning, but an alternative is to make [[7/4]] just by making the fifth 703.897 cents, as opposed to 703.448 cents for 58EDO.
+A simpler characterization than the one given by the normal comma list is that diaschismic adds [[126/125]] or [[5120/5103]] to the set of commas, and it can also be called 46&amp;58. However described, diaschismic has a 1/2 period and a sharp fifth generator like pajara, but not so sharp, giving a more accurate but more complex temperament. [[58 EDO]] provides an excellent tuning, but an alternative is to make [[7/4]] just by making the fifth 703.897 cents, as opposed to 703.448 cents for 58 EDO.
 Diaschismic extends naturally to the 17-limit, for which the same tunings may be used, making it one of the most important of the higher limit rank two temperaments. Adding the 11-limit adds the commas 176/175, 896/891 and 441/440. The 13-limit yields 196/195, 351/350, and 364/363; the 17-limit adds 136/135, 221/220, and 442/441. If you want to explore higher limit harmonies, diaschismic is certainly one excellent way to do it; MOS of 34 notes and even more the 46 note MOS will encompass very great deal of it. Of course 46 or 58 equal provide alternatives which in many ways are similar, particularly in the case of 58.
@@ Line 318: / Line 318: @@
 == Keen ==
-Keen adds 875/864 as well as 2240/2187 to the set of commas. It may also be described as the 22&amp;56 temperament. [[78edo|78EDO]] is a good tuning choice, and remains a good one in the 11-limit, where keen, {{multival| 2 -4 18 -12 … }}, is really more interesting, adding 100/99 and 385/384 to the commas.
+Keen adds 875/864 as well as 2240/2187 to the set of commas. It may also be described as the 22&amp;56 temperament. [[78 EDO]] is a good tuning choice, and remains a good one in the 11-limit, where keen, {{multival| 2 -4 18 -12 … }}, is really more interesting, adding 100/99 and 385/384 to the commas.
 Subgroup: 2.3.5.7
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 == Echidna ==
-Echidna adds 1728/1715 to the commas and takes 9/7 as a generator. It may be called the 22&amp;58 temperament. [[58edo|58EDO]] or [[80edo|80EDO]] make for good tunings, or their vals can be add to {{val| 138 219 321 388 }}.
+Echidna adds 1728/1715 to the commas and takes 9/7 as a generator. It may be called the 22&amp;58 temperament. [[58 EDO]] or [[80 EDO]] make for good tunings, or their vals can be add to {{val| 138 219 321 388 }}.
 Echidna becomes more interesting when extended to be an 11-limit temperament by adding 176/175, 896/891 or 540/539 to the commas, where the same tunings can be used as before. It then is able to represent the entire 11-limit diamond to within about six cents of error, within a compass of 24 notes. The 28 note 2MOS gives scope for this, and the 36 note MOS much more.
@@ Line 522: / Line 522: @@
 == Shrutar ==
-Shrutar adds 245/243 to the commas, and also tempers out 6144/6125. It can also be described as 22&amp;46. Its generator can be taken as either 36/35 or 35/24; the latter is interesting since along with 15/14 and 21/20, it connects opposite sides of a hexany. [[68edo|68EDO]] makes for a good tuning, but another and excellent choice is a generator of 14<sup>(1/7)</sup>, making 7s just.
+Shrutar adds 245/243 to the commas, and also tempers out 6144/6125. It can also be described as 22&amp;46. Its generator can be taken as either 36/35 or 35/24; the latter is interesting since along with 15/14 and 21/20, it connects opposite sides of a hexany. [[68 EDO]] makes for a good tuning, but another and excellent choice is a generator of 14<sup>(1/7)</sup>, making 7s just.
 By adding 121/120 or 176/175 to the commas, shrutar can be extended to the 11-limit, which loses a bit of accuracy, but picks up low-complexity 11-limit harmony, making shrutar quite an interesting 11-limit system. 68, 114 or a 14<sup>(1/7)</sup> generator can again be used as tunings.