BPS: Difference between revisions

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Another weak extension to add prime 17, known as ''Dubhe'', splits the 9/7 BPS generator in half, by tempering out [[2025/2023]] and equating two of [[17/15]] to 9/7. This produces [[8L 1s (3/1-equivalent)|8L 1s]] enneatonic and [[9L 8s (3/1-equivalent)|9L 8s]] chromatic MOS scales. Simple tunings include [[17edt]] and [[26edt]].

While strong 11-limit extensions can be proposed, tempering out [[77/75]] in the flat ~~range~~ and [[1375/1323]] in the ~~sharp~~ 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).

=== Strong extensions ===

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).

~~Sharp~~ tunings generally ~~possess a more convenient~~ 13th harmonic ~~than 11th,~~ by tempering out [[637/625]] and identifying (25/21)<sup>2</sup> with [[13/9]], which is optimal near the 30edt tuning. It is then very easy to insert in 19 by tempering [[247/245]], and identifying [[13/9]] with [[27/19]]~~, therefore placing the 19th harmonic 10 generators down; this extension is optimal near the 56edt tuning~~.

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 either case, it is then very easy to insert in 19 by tempering [[247/245]], and identifying [[13/9]] with [[27/19]].

=== Prime 2 ===

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 Another weak extension to add prime 17, known as ''Dubhe'', splits the 9/7 BPS generator in half, by tempering out [[2025/2023]] and equating two of [[17/15]] to 9/7. This produces [[8L 1s (3/1-equivalent)|8L 1s]] enneatonic and [[9L 8s (3/1-equivalent)|9L 8s]] chromatic MOS scales. Simple tunings include [[17edt]] and [[26edt]].
-While strong 11-limit extensions can be proposed, tempering out [[77/75]] in the flat range and [[1375/1323]] in the sharp 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).
+=== Strong extensions ===
+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).
-Sharp tunings generally possess a more convenient 13th harmonic than 11th, by tempering out [[637/625]] and identifying (25/21)<sup>2</sup> with [[13/9]], which is optimal near the 30edt tuning. It is then very easy to insert in 19 by tempering [[247/245]], and identifying [[13/9]] with [[27/19]], therefore placing the 19th harmonic 10 generators down; this extension is optimal near the 56edt tuning.
+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 either case, it is then very easy to insert in 19 by tempering [[247/245]], and identifying [[13/9]] with [[27/19]].
 === Prime 2 ===