Ringer scale: Difference between revisions

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== Perfect ringer scales ==

A perfect ringer ''n'' scale is one that by some val can map the first ''n'' odd harmonics to distinct numbers of steps up to [[octave equivalence]]. ~~It is conjectured that these~~ are the only perfect ringer scales:

A perfect ringer ''n'' scale is one that by some val can map the first ''n'' odd harmonics to distinct numbers of steps up to [[octave equivalence]]. These are the only perfect ringer scales:

'''Ringer 1:''' 1:2

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Notice how all of these do not skip any harmonics while representing the harmonic series ''completely'' up to some [[odd-limit]].

Such a scale will either contain a pair of intervals (n+2):n and [(3/2)n+3]:[(3/2)n] if ''n'' is even so long as (3/2)n+3 ≤ 2n and therefore n ≥ 6, or (n+3):(n+1) and [(3/2)(n+1)+3]:[(3/2)(n+1)] if ''n'' is odd so long as (3/2)(n+1)+3 = (3/2)(n+3) ≤ 2n and therefore n ≥ 9. Between these two conditions, it is apparent that ''n'' = 1, 2, 3, 4, 5, and 7 create the only constant structures representing the entire harmonic series from ''n'' to ''2n''.

Also note that there is a "Perfect Pseudoringer 9" if we allow a pair of harmonics to be swapped/out of order in order to preserve the constant structure property. It is not known how many "Perfect Pseudoringers" there are.

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 == Perfect ringer scales ==
-A perfect ringer ''n'' scale is one that by some val can map the first ''n'' odd harmonics to distinct numbers of steps up to [[octave equivalence]]. It is conjectured that these are the only perfect ringer scales:
+A perfect ringer ''n'' scale is one that by some val can map the first ''n'' odd harmonics to distinct numbers of steps up to [[octave equivalence]]. These are the only perfect ringer scales:
 '''Ringer 1:''' 1:2
@@ Line 21: / Line 21: @@
 Notice how all of these do not skip any harmonics while representing the harmonic series ''completely'' up to some [[odd-limit]].
+Such a scale will either contain a pair of intervals (n+2):n and [(3/2)n+3]:[(3/2)n] if ''n'' is even so long as (3/2)n+3 ≤ 2n and therefore n ≥ 6, or (n+3):(n+1) and [(3/2)(n+1)+3]:[(3/2)(n+1)] if ''n'' is odd so long as (3/2)(n+1)+3 = (3/2)(n+3) ≤ 2n and therefore n ≥ 9. Between these two conditions, it is apparent that ''n'' = 1, 2, 3, 4, 5, and 7 create the only constant structures representing the entire harmonic series from ''n'' to ''2n''.
 Also note that there is a "Perfect Pseudoringer 9" if we allow a pair of harmonics to be swapped/out of order in order to preserve the constant structure property. It is not known how many "Perfect Pseudoringers" there are.