User:Lucius Chiaraviglio/Musical Mad Science: Difference between revisions

#  It is noteworthy (more detail needed) that harmonics 3 and 23 are very stable over the tuning spectrum of this scale (at least for EDO values up into the mid double digits), although the 23rd harmonic is guaranteed to be sharp, meaning that at larger EDO values, increasingly fine divisions of the octave will cause the mapping to disagree with 10\19 and 9\17 (and thereby with 19\36), thus requiring an 'i' [[wart]].  The 7th harmonic is also reasonably stable, although it changes enough over the tuning spectrum to get rather bad at the extremes; the 5th harmonic is definitely not stable, and would need different extensions for at least the hard and soft halves of the tuning spectrum; commas including powers of 5 should be avoided in the core of the associated temperament, while commas including powers of 3 and 23 (and possibly 13 — need further checking to be sure, but this is looking less good than 3 and 23) seem like they would be good choices for the core.  (Coming in the future:  Checking this further.)

#  Tentatively assigning the generator as 23/16 ~ 13/9, tempering out [[208/207]].  But the problem is that — as can be seen in the table of harmonics below — the 13th harmonic is not stable enough for the entire 17L 2s tuning spectrum, even for the for the hard half of the tuning spectrum (closer to just 13/9, including having the best 3rd harmonic within the tuning spectrum).  Maybe splitting the tuning spectrum of 17L 2s into 2 or more temperaments is in order?  Maybe the 5th harmonic is stable enough for the soft half of the 17L 2s tuning spectrum (closer to just 23/16, but even closer to the just barely out-of-reach 36/25)?  And maybe the 7th and 17th harmonics are stable enough for the middle of the 17L 2s tuning spectrum?  (Coming in the future:  Checking this further; may need to insert some more supporting material above.)

=== Table of odd harmonics for various EDO values supporting 17L 2s ===

This table (actually a collection of tables for now) is for tracking trends in odd harmonics along the tuning spectrum of [[17L 2s]]; it is intended to match the organization of [[17L_2s#Scale_tree|the corresponding scale tree]], except for omitting the right-most column other than the top and bottom extremes:

(Need a way to combine the collection of tables into a single table for better readability.)