Huygens vs meanpop: Difference between revisions

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In meanpop, 11/8 is represented by the doubly diminished fifth, for example C–G𝄫. This is in the opposite direction along the circle of fifths – 13 fifths down. Meanpop is tuned best flat of 31edo, around 696 cents.

In the [[13-limit]], meanpop extends by [[105/104]], whereas meantone forks into fokkertone, grosstone, and meridetone.

In the [[13-limit]], meanpop extends by [[105/104]] (mapping 13/8 to the double-augmented fifth C–G𝄪, tempering out [[144/143]]); whereas meantone forks into fokkertone (also mapping 13/8 to the double-augmented fifth C–G𝄪); grosstone (mapping 13/8 to the double-diminished seventh C–B♭♭♭); and meridetone (mapping 13/8 to the quadruple-augmented fourth C–F𝄪𝄪).

Can huygens and meanpop be combined into a single temperament? Yes! It works wonderfully and that temperament is [[31edo]]. In 31edo the circle of fifths closes perfectly after 31 fifths, so E𝄪 and G𝄫 are the same note. (In other words, the interval of the ''quadruply diminished third'' is tuned to 0 cents, setting a minor third equal to four chromatic semitones. Expressed in tempered fifths and octave-reduced, this interval is the [[31-comma]] {{monzo| -49 31 }}, which is the 3-limit comma tempered out in 31edo.) This makes everything much simpler and results in [[121/120]] and [[243/242]] being tempered out, so that 12/11~11/10 is a true neutral second (exactly half of a minor third), and 11/9 is a true neutral third (exactly half of a perfect fifth). Keep in mind that neither of these things are true in either huygens or meanpop.

@@ Line 11: / Line 11: @@
 In meanpop, 11/8 is represented by the doubly diminished fifth, for example C–G𝄫. This is in the opposite direction along the circle of fifths – 13 fifths down. Meanpop is tuned best flat of 31edo, around 696 cents.
-In the [[13-limit]], meanpop extends by [[105/104]], whereas meantone forks into fokkertone, grosstone, and meridetone.
+In the [[13-limit]], meanpop extends by [[105/104]] (mapping 13/8 to the double-augmented fifth C–G𝄪, tempering out [[144/143]]); whereas meantone forks into fokkertone (also mapping 13/8 to the double-augmented fifth C–G𝄪); grosstone (mapping 13/8 to the double-diminished seventh C–B♭♭♭); and meridetone (mapping 13/8 to the quadruple-augmented fourth C–F𝄪𝄪).
 Can huygens and meanpop be combined into a single temperament? Yes! It works wonderfully and that temperament is [[31edo]]. In 31edo the circle of fifths closes perfectly after 31 fifths, so E𝄪 and G𝄫 are the same note. (In other words, the interval of the ''quadruply diminished third'' is tuned to 0 cents, setting a minor third equal to four chromatic semitones. Expressed in tempered fifths and octave-reduced, this interval is the [[31-comma]] {{monzo| -49 31 }}, which is the 3-limit comma tempered out in 31edo.) This makes everything much simpler and results in [[121/120]] and [[243/242]] being tempered out, so that 12/11~11/10 is a true neutral second (exactly half of a minor third), and 11/9 is a true neutral third (exactly half of a perfect fifth). Keep in mind that neither of these things are true in either huygens or meanpop.