Biyatismic clan: Difference between revisions

Line 307:

==== Eros ====

Eros fairs impressively into the 23-limit as a rank 3 temperament; not only is it fairly simple (considering this is a subgroup as complex as the full 23-limit, with many challenges) but ~~all the generators are positive (or only 1 into the negatives in the case of the fifth) meaning it's even simpler than it might appear and~~ has the pleasing property of all harmonics ~~and subharmonics being "~~on the ~~same~~ side~~"; specifically:~~ -3 to 1 fifths ~~([[2L 3s]])~~ and -5 to 0 ~[[23/22]]'s will ~~get you every prime,~~ up to octave equivalence; you can think of this as a ~~5 by 6~~ grid ~~if you like and~~ is a recommendable place to start looking at its structure. Tempering the less accurate comma [[121/120~~|S11]]~~ can be seen as ~~a consequence~~ of tempering {[[441/440~~|S21~~]], [[484/483~~|S22~~]], [[529/528~~|S23~~]]} ~~so is very natural and given its properties certainly excusable~~. Therefore characteristic of any good tuning is the ~11 being the ~~most flat~~ prime, with other primes having strictly less than 5{{cent}} of error. This temperament was ~~first logged on x31eq~~ by [[Scott Dakota]].

Eros fairs impressively into the 23-limit as a rank-3 temperament; not only is it fairly simple (considering this is a subgroup as complex as the full 23-limit, with many challenges) but has the pleasing property that all the harmonics are on the negative side of the last generator. Specifically, -3 to 2 fifths and -5 to 0 [[~]][[23/22]]'s will provide odd harmonics 1–23 up to octave equivalence; you can think of this as a 6×6 grid, which is a recommendable place to start looking at its structure.

Tempering out the less accurate comma 121/120 can be seen as an implication of tempering out [[441/440]] ({{S|21}}), [[484/483]] ({{S|22}}), and [[529/528]] ({{S|23}}). Therefore, characteristic of any good tuning is the prime [[11/1|11]] being the flattest prime, with other primes having strictly less than 5{{cent}} of error.

This temperament was discovered by [[Scott Dakota]]. Note that the 17-limit extension requires the 29g val for 29edo, which has the sizes of 17/16 and 18/17 swapped.

Subgroup: 2.3.5.7.11.13

Line 324:

Line 328:

===== 17-limit =====

~~Note that this extension requires the 29g val for 29edo, which has the sizes of 17/16 and 18/17 swapped.~~

Subgroup: 2.3.5.7.11.13.17

@@ Line 307: / Line 307: @@
 ==== Eros ====
-Eros fairs impressively into the 23-limit as a rank 3 temperament; not only is it fairly simple (considering this is a subgroup as complex as the full 23-limit, with many challenges) but all the generators are positive (or only 1 into the negatives in the case of the fifth) meaning it's even simpler than it might appear and has the pleasing property of all harmonics and subharmonics being "on the same side"; specifically: -3 to 1 fifths ([[2L 3s]]) and -5 to 0 ~[[23/22]]'s will get you every prime, up to octave equivalence; you can think of this as a 5 by 6 grid if you like and is a recommendable place to start looking at its structure. Tempering the less accurate comma [[121/120|S11]] can be seen as a consequence of tempering {[[441/440|S21]], [[484/483|S22]], [[529/528|S23]]} so is very natural and given its properties certainly excusable. Therefore characteristic of any good tuning is the ~11 being the most flat prime, with other primes having strictly less than 5{{cent}} of error. This temperament was first logged on x31eq by [[Scott Dakota]].
+Eros fairs impressively into the 23-limit as a rank-3 temperament; not only is it fairly simple (considering this is a subgroup as complex as the full 23-limit, with many challenges) but has the pleasing property that all the harmonics are on the negative side of the last generator. Specifically, -3 to 2 fifths and -5 to 0 [[~]][[23/22]]'s will provide odd harmonics 1–23 up to octave equivalence; you can think of this as a 6×6 grid, which is a recommendable place to start looking at its structure.
+Tempering out the less accurate comma 121/120 can be seen as an implication of tempering out [[441/440]] ({{S|21}}), [[484/483]] ({{S|22}}), and [[529/528]] ({{S|23}}). Therefore, characteristic of any good tuning is the prime [[11/1|11]] being the flattest prime, with other primes having strictly less than 5{{cent}} of error.
+This temperament was discovered by [[Scott Dakota]]. Note that the 17-limit extension requires the 29g val for 29edo, which has the sizes of 17/16 and 18/17 swapped.
 Subgroup: 2.3.5.7.11.13
@@ Line 324: / Line 328: @@
 ===== 17-limit =====
-Note that this extension requires the 29g val for 29edo, which has the sizes of 17/16 and 18/17 swapped.
 Subgroup: 2.3.5.7.11.13.17