BPS: Difference between revisions

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=== Strong extensions ===

One harmonic that can be placed on the generator chain with some accuracy, compared to other primes, is the 19th, as by tempering out [[6561/6517]], or equivalently [[135/133]], [[19/9]] is equated to (9/7)3, or otherwise [[15/7]]. However, this mapping of 19 requires a generator close to just (in fact, it is exact ''flat'' of 22edt) and is therefore quite high-damage in sharper tunings.

While strong 11-limit extensions can be proposed, tempering out [[77/75]] in the flatter range (i.e. flat of [[13edt|3\13edt]]) and [[1375/1323]] in the sharper range, neither of these are of particular accuracy; more accurate extensions would be of considerably higher complexity. However, one could argue for the canonicity of the latter extension by being the no-twos retraction of 11-limit [[hedgehog]] temperament (which, as a member of the [[porcupine family]], makes more sense to consider with prime 11 in mind than without it).

In the 13-limit, sharp tunings can generally map the 13th harmonic by tempering out [[637/625]] and identifying (25/21)2 with [[13/9]], which is optimal near the 30edt tuning. For flat tunings, it is more accurate to temper out [[65/63]] instead.

One harmonic that can be placed on the generator chain with some accuracy, compared to other primes, is the 19th, as by tempering out [[6561/6517]], or equivalently [[135/133]], [[19/9]] is equated to (9/7)3, or otherwise [[15/7]]. However, this mapping of 19 requires a generator close to just (in fact, it is exact ''flat'' of 22edt) and is therefore feasible mainly on the flat side of the spectrum.

=== Prime 2 ===

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 === Strong extensions ===
-One harmonic that can be placed on the generator chain with some accuracy, compared to other primes, is the 19th, as by tempering out [[6561/6517]], or equivalently [[135/133]], [[19/9]] is equated to (9/7)<sup>3</sup>, or otherwise [[15/7]]. However, this mapping of 19 requires a generator close to just (in fact, it is exact ''flat'' of 22edt) and is therefore quite high-damage in sharper tunings.
 While strong 11-limit extensions can be proposed, tempering out [[77/75]] in the flatter range (i.e. flat of [[13edt|3\13edt]]) and [[1375/1323]] in the sharper range, neither of these are of particular accuracy; more accurate extensions would be of considerably higher complexity. However, one could argue for the canonicity of the latter extension by being the no-twos retraction of 11-limit [[hedgehog]] temperament (which, as a member of the [[porcupine family]], makes more sense to consider with prime 11 in mind than without it).
 In the 13-limit, sharp tunings can generally map the 13th harmonic by tempering out [[637/625]] and identifying (25/21)<sup>2</sup> with [[13/9]], which is optimal near the 30edt tuning. For flat tunings, it is more accurate to temper out [[65/63]] instead.
+One harmonic that can be placed on the generator chain with some accuracy, compared to other primes, is the 19th, as by tempering out [[6561/6517]], or equivalently [[135/133]], [[19/9]] is equated to (9/7)<sup>3</sup>, or otherwise [[15/7]]. However, this mapping of 19 requires a generator close to just (in fact, it is exact ''flat'' of 22edt) and is therefore feasible mainly on the flat side of the spectrum.
 === Prime 2 ===