BPS: Difference between revisions

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While strong 11-limit extensions can be proposed, tempering out [[77/75]] in the sharper range (i.e. sharp of [[13edt|3\13edt]]) and [[1375/1323]] in the flatter 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.

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~~.

One harmonic that can be placed on the generator chain with some accuracy, compared to other primes, is the 19th. Sharp of 13edt, it is best to temper out [[11907/11875]] and equate (25/21)2 to [[27/19]], thereby having the 19th harmonic 10 generators down. But on the flat side of the spectrum, it is less complex and more accurate flat of 13edt to temper out [[6561/6517]], or equivalently [[135/133]], so that [[19/9]] is equated to (9/7)3, or otherwise [[15/7]], though this mapping of 19 is exact ''flat'' of 22edt.

=== Prime 2 ===

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 While strong 11-limit extensions can be proposed, tempering out [[77/75]] in the sharper range (i.e. sharp of [[13edt|3\13edt]]) and [[1375/1323]] in the flatter 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.
+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.
+One harmonic that can be placed on the generator chain with some accuracy, compared to other primes, is the 19th. Sharp of 13edt, it is best to temper out [[11907/11875]] and equate (25/21)<sup>2</sup> to [[27/19]], thereby having the 19th harmonic 10 generators down. But on the flat side of the spectrum, it is less complex and more accurate flat of 13edt to temper out [[6561/6517]], or equivalently [[135/133]], so that [[19/9]] is equated to (9/7)<sup>3</sup>, or otherwise [[15/7]], though this mapping of 19 is exact ''flat'' of 22edt.
 === Prime 2 ===