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

'''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]].

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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=== Sketch of the proof ===

Consider an ''N''-note [[periodic scale]] with period ''P'' as being defined by a function <math>f: \mathbb{Z} \to \mathbb{Q}_0</math> with <math>f(Nk) = P^k.</math>

Consider an ''N''-note [[periodic scale]] with period ''P'' as being defined by a function <math>f: \mathbb{Z} \to \mathbb{Q}_{>0}</math> with <math>f(Nk) = P^k.</math>

By the construction of a ringer scale, we are given some [[val]] [[map]] <math>m : \mathbb{Q}_0 \to \mathbb{Z}</math> that satisfies <math>m(f(k+1)/f(k)) = 1</math> for all ''k'' in '''Z'''. (This can be checked by hand or by computer as we only need to check one period <i>P</i>s worth of 1-scalestep intervals.)

By the construction of a ringer scale, we are given some [[val]] [[map]] <math>m : \mathbb{Q}_{>0} \to \mathbb{Z}</math> that satisfies <math>m(f(k+1)/f(k)) = 1</math> for all ''k'' in '''Z'''. (This can be checked by hand or by computer as we only need to check one period <i>P</i>'s worth of 1-scalestep intervals.)

By induction this implies <math>m(f(k+s)/f(k)) = s</math> because the intervals from ''k'' to ''k''+1, from ''k''+1 to ''k''+2, ..., from ''k''+''s''-1 to ''k''+''s'' all multiply together. This also implies <math>m(f(k))=k,</math> proving ''f'' to be [[epimorphic]], therefore CS. {{qed}}

By induction this implies <math>m(f(k+s)/f(k)) = s</math> because the intervals from ''k'' to ''k''+1, from ''k''+1 to ''k''+2, ..., from ''k''+''s''-1 to ''k''+''s'' all multiply together. This also implies <math>m(f(k))=k,</math> proving ''f'' to be [[epimorphic]], therefore CS (see proof in the article [[epimorphic scale]]). {{qed}}

== Ringer scales ==

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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10:11:12:13:14:15:16:18:19:20

'''Ringer 10:''' 10:'''21/2''':11:12:13:14:15:16:17:18~~:20~~

'''Ringer 10:''' 9:10:'''21/2''':11:12:13:14:15:16:17:18

11:12:13:14:15:16:17:18:20:21:22

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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:~~Articles~~ with proofs]]

[[Category:Pages with proofs]]

@@ Line 6: / Line 6: @@
 == 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
+'''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]].
+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.
@@ Line 64: / Line 66: @@
 === Sketch of the proof ===
-Consider an ''N''-note [[periodic scale]] with period ''P'' as being defined by a function <math>f: \mathbb{Z} \to \mathbb{Q}_0</math> with <math>f(Nk) = P^k.</math>
+Consider an ''N''-note [[periodic scale]] with period ''P'' as being defined by a function <math>f: \mathbb{Z} \to \mathbb{Q}_{>0}</math> with <math>f(Nk) = P^k.</math>
-By the construction of a ringer scale, we are given some [[val]] [[map]] <math>m : \mathbb{Q}_0 \to \mathbb{Z}</math> that satisfies <math>m(f(k+1)/f(k)) = 1</math> for all ''k'' in '''Z'''. (This can be checked by hand or by computer as we only need to check one period <i>P</i>s worth of 1-scalestep intervals.)
+By the construction of a ringer scale, we are given some [[val]] [[map]] <math>m : \mathbb{Q}_{>0} \to \mathbb{Z}</math> that satisfies <math>m(f(k+1)/f(k)) = 1</math> for all ''k'' in '''Z'''. (This can be checked by hand or by computer as we only need to check one period <i>P</i>'s worth of 1-scalestep intervals.)
-By induction this implies <math>m(f(k+s)/f(k)) = s</math> because the intervals from ''k'' to ''k''+1, from ''k''+1 to ''k''+2, ..., from ''k''+''s''-1 to ''k''+''s'' all multiply together. This also implies <math>m(f(k))=k,</math> proving ''f'' to be [[epimorphic]], therefore CS. {{qed}}
+By induction this implies <math>m(f(k+s)/f(k)) = s</math> because the intervals from ''k'' to ''k''+1, from ''k''+1 to ''k''+2, ..., from ''k''+''s''-1 to ''k''+''s'' all multiply together. This also implies <math>m(f(k))=k,</math> proving ''f'' to be [[epimorphic]], therefore CS (see proof in the article [[epimorphic scale]]). {{qed}}
 == Ringer scales ==
 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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 :11:12:13:14:15:16:18:19:20
-'''Ringer 10:''' 10:'''21/2''':11:12:13:14:15:16:17:18:20
+'''Ringer 10:''' 9:10:'''21/2''':11:12:13:14:15:16:17:18
 :12:13:14:15:16:17:18:20:21:22
@@ Line 182: / 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:Articles with proofs]]
+[[Category:Pages with proofs]]