Ringer scale: Difference between revisions

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A perfect Ringer ''n'' scale is one that by some val can map the first ''n'' odd harmonics (up to [[octave equivalence]]) to distinct numbers of steps. It is likely that only a small finite number of perfect Ringer scales exist. Here are the known ones so far (to be expanded as/if more are found):
A perfect Ringer ''n'' scale is one that by some val can map the first ''n'' odd harmonics (up to [[octave equivalence]]) to distinct numbers of steps. It is likely that only a small finite number of perfect Ringer scales exist. Here are the known ones so far (to be expanded as/if more are found):


Ringer 2: 2:3:4
('''Ringer 1:''' 1:2)


Ringer 3: 3:4:5:6
'''Ringer 2:''' 2:3:4


Ringer 4: 4:5:6:7:8
'''Ringer 3:''' 3:4:5:6


Ringer 5: 5:6:7:8:9:10
'''Ringer 4:''' 4:5:6:7:8


Ringer 7: 7:8:9:10:11:12:13:14
'''Ringer 5:''' 5:6:7:8:9:10
 
'''Ringer 7:''' 7:8:9:10:11:12:13:14


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