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 to distinct numbers of steps up to [[octave equivalence]]. These are the only perfect ringer scales:

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

'''Ringer 1:''' 1:2 (val: {{val| 1 }})

'''Ringer 2:''' 2:3:4

'''Ringer 2:''' 2:3:4 (val: {{val| 2 3 }})

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

'''Ringer 3:''' 3:4:5:6 (val: {{val| 3 5 7 }})

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

'''Ringer 4:''' 4:5:6:7:8 (val: {{val| 4 6 9 11 }})

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

'''Ringer 5:''' 5:6:7:8:9:10 (val: {{val| 5 8 12 14 }})

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

'''Ringer 7:''' 7:8:9:10:11:12:13:14 (val: {{val| 7 11 16 20 24 26 }})

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

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This section will detail known ringers for edos smaller than 100. Because [[wart]]s are limited when it comes to large primes, any primes past 43 are explicitly listed in the form [p, q, r, ...] rather than abbreviated (rather cryptically) as letters. A quick summary of all the warts up to 43 is:

b means 3 gets a next-best mapping, c means 5 gets a next-best mapping, d means 7 gets a next-best mapping and so on: e means 11, f means 13, g means 17, h means 19, i means 23, j means 29, k means 31, l means 37, m means 41, n means 43. (2 = a is not used as it must always be patent.)

b means 3 gets a next-best mapping, c means 5 gets a next-best mapping, d means 7 gets a next-best mapping and so on: e means 11, f means 13, g means 17, h means 19, i means 23, j means 29, k means 31, l means 37, m means 41, n means 43 and finally o means 47. (2 = a is not used as it must always be patent.)

There should be at least two forms listed. One will be in the form used for the example of Ringer 15. One will be in the minimum mode of the harmonic series that contains all harmonics. Both can be pasted directly into scale workshop using the enumerate chord feature or into other programs, but the latter form is useful in case a program does not support the other notation.

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75:76:77:78:79:80:81:82:83:84:85:86:87:88:89:90:91:92:93:94:96:'''97''':98:'''99''':100:102:'''103''':104:106:'''107''':108:'''109''':110:112:'''113''':114:116:'''[117''':118(q=59) OR 118:'''119]''':120:'''[121:122(rr=61)''' OR 122:'''123]''':124:126:128:'''129''':130:132:134:136:'''137''':138:140:142:144:146:148:'''149''':150

== See also ==

* [[Neji]]

[[Category:Just intonation scales]]

[[Category:Pages with proofs]]

@@ Line 8: / Line 8: @@
 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
+'''Ringer 1:''' 1:2 (val: {{val| 1 }})
-'''Ringer 2:''' 2:3:4
+'''Ringer 2:''' 2:3:4 (val: {{val| 2 3 }})
-'''Ringer 3:''' 3:4:5:6
+'''Ringer 3:''' 3:4:5:6 (val: {{val| 3 5 7 }})
-'''Ringer 4:''' 4:5:6:7:8
+'''Ringer 4:''' 4:5:6:7:8 (val: {{val| 4 6 9 11 }})
-'''Ringer 5:''' 5:6:7:8:9:10
+'''Ringer 5:''' 5:6:7:8:9:10 (val: {{val| 5 8 12 14 }})
-'''Ringer 7:''' 7:8:9:10:11:12:13:14
+'''Ringer 7:''' 7:8:9:10:11:12:13:14 (val: {{val| 7 11 16 20 24 26 }})
 Notice how all of these do not skip any harmonics while representing the harmonic series ''completely'' up to some [[odd-limit]].
@@ Line 75: / Line 75: @@
 This section will detail known ringers for edos smaller than 100. Because [[wart]]s are limited when it comes to large primes, any primes past 43 are explicitly listed in the form [p, q, r, ...] rather than abbreviated (rather cryptically) as letters. A quick summary of all the warts up to 43 is:
-b means 3 gets a next-best mapping, c means 5 gets a next-best mapping, d means 7 gets a next-best mapping and so on: e means 11, f means 13, g means 17, h means 19, i means 23, j means 29, k means 31, l means 37, m means 41, n means 43. (2 = a is not used as it must always be patent.)
+b means 3 gets a next-best mapping, c means 5 gets a next-best mapping, d means 7 gets a next-best mapping and so on: e means 11, f means 13, g means 17, h means 19, i means 23, j means 29, k means 31, l means 37, m means 41, n means 43 and finally o means 47. (2 = a is not used as it must always be patent.)
 There should be at least two forms listed. One will be in the form used for the example of Ringer 15. One will be in the minimum mode of the harmonic series that contains all harmonics. Both can be pasted directly into scale workshop using the enumerate chord feature or into other programs, but the latter form is useful in case a program does not support the other notation.
@@ Line 184: / Line 184: @@
 :76:77:78:79:80:81:82:83:84:85:86:87:88:89:90:91:92:93:94:96:'''97''':98:'''99''':100:102:'''103''':104:106:'''107''':108:'''109''':110:112:'''113''':114:116:'''[117''':118(q=59) OR 118:'''119]''':120:'''[121:122(rr=61)''' OR 122:'''123]''':124:126:128:'''129''':130:132:134:136:'''137''':138:140:142:144:146:148:'''149''':150
+== See also ==
+* [[Neji]]
 [[Category:Just intonation scales]]
 [[Category:Pages with proofs]]