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.

== Origin of ringer scales ==

The name ''ringer'' was chosen by tuning theorist Scott Dakota (who discovered and raised awareness of the concept) to refer to the property of these scales to [[Harmonic_entropy#Background|"ring"]] extremely and about as much as might be possible for a [[JI]] scale because the [[odd-limit]] complexity of the intervals in such scales is near-minimal meaning they consume as much of the early harmonic series as possible. It is worth noting however that the appearance of a [[virtual fundamental]] depends strongly on which notes of the scale you play - an observation important to [[primodality]]. The concept of ringer scales was additionally further developed by tuning theorists Praveen Venkataramana and later [[user:Godtone]].

The name ''ringer'' was chosen by tuning theorist [[Scott Dakota]] (who discovered and raised awareness of the concept) to refer to the property of these scales to [[Harmonic_entropy#Background|"ring"]] extremely and about as much as might be possible for a [[JI]] scale because the [[odd-limit]] complexity of the intervals in such scales is near-minimal meaning they consume as much of the early harmonic series as possible. It is worth noting however that the appearance of a [[virtual fundamental]] depends strongly on which notes of the scale you play - an observation important to [[primodality]]. The concept of ringer scales was additionally further developed by tuning theorists Praveen Venkataramana and later [[user:Godtone]].

== Example: Ringer 15 ==

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The [[17-limit]] [[val]] that confirms this scale is CS is {{val|9 15 22 26 32 34 38}}, which written as [[wart]]s is 9bccdefgg. (Note that in this case, where there is two warts this corresponds to the patent val mapping for the prime already being sharp and being warted to be a step sharper. If we assume that every wart means "sharpen by one step from patent val" this val can be written rather curiously as 9bcdefg, which shows that this val is the one sharpening every applicable prime by one step above the [[patent val]] mapping.) One can confirm that the above is CS because if one traverses it step by step, every one-step interval is mapped to one EDOstep which by [[wikipedia:linearity|linearity]] (more precisely, [[epimorphic]]ity) [[#Proof of CS of by linearity|implies CS]]. Note that it is important to preserve the order of these intervals. 14:16 = 16/14 = 8/7 is mapped to one positive step, as is 16:15 = 15/16, as is 15:17 = 17/15. Similarly (or thus/by linearity), 14:15 = 15/14 is mapped to 2 steps, as is 16:17 = 17/16, as is 15:18 = 18/15 = 6/5.

== ~~Proof of~~ CS by ~~linearity of the epimorphic val~~ ==

== Proving CS by hand ==

Because the CS property means that every occurrence of an interval must occur with the same number of steps, it suffices to show that every one-step interval is mapped to one step by the [[val]] that the Ringer scale is constructed with. (This val shows that the Ringer scale is [[epimorphic]].)

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However, conversely, a scale being CS does not imply that such a val exists! In almost all observed practical cases if a scale is CS there is some val, but it is possible to construct scales where, for example, one 1-scalestep interval is equal to the product of more than one other 1-scalestep intervals; that is, if we have 1-scalestep intervals {''a'', ''b'', ''c'', ...} then we can choose ''ab'' as a 1-scalestep interval as long as ''ab'' doesn't occur as a 2-scalestep interval anywhere in the scale, which is why at least one extra 1-scalestep interval ''c'' is necessary to separate instances of ''a'' and ''b''. You can even choose ''b'' = ''a'' but you need to be careful to avoid CS-violating contradictions. For a concrete example, you can use {[[5/4]], [[9/8]], [[45/32]], ...} as 1-scalestep intervals to generate a nonlinear CS scale as long as [[45/32]] does not occur as a 2-scalestep interval anywhere in your scale.

~~=== 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>~~

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 (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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9:10:12:14:16:17:18

'''Ringer 8:''' 7:8:'''17/2''':9:10:11:12:13:14

'''Ringer 8d:''' 7:8:'''17/2''':9:10:11:12:13:14

9:10:11:12:13:14:16:17:18

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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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'''Ringer 12:''' 11:12:'''[25/2''':13(f) OR 13:'''27/2]''':14:15:16:17:18:19:20:21:22

14:15:16:17:18:19:20:21:22:24:[25:26 OR 26:27]:28

14:15:16:17:18:19:20:21:22:24:[25:26(f) OR 26:27]:28

'''Ringer 13g:''' 9:10:'''21/2''':11:'''23/2''':12:13:14:'''29/2''':15:16:'''33/2''':17:18

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'''Ringer 15:''' 13:14:'''29/2''':15:16:'''[33/2''':17(g) OR 17:'''35/2]''':18:19:20:21:22:23:24:25:26

18:19:20:21:22:23:24:25:26:28:29:30:32:[33:34 OR 34:35]:36

18:19:20:21:22:23:24:25:26:28:29:30:32:[33:34(g) OR 34:35]:36

'''Ringer 16:'''

13:'''55/4''':14:15:'''31/2''':16:17:18:'''[37/2''':19 OR 19:'''39/2(h)]''':20:21:22:23:24:25:26

28:30:31:32:34:36:[37:38 OR 38:39(h)]:40:42:44:46:48:50:52:55:56

'''Ringer 17cffg:''' 13:14:'''29/2''':15:16:'''33/2''':17:18:19:'''[77/4''' OR '''79/4]''':20:21:22:23:24:'''49/2''':25:26

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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.
 == Origin of ringer scales ==
-The name ''ringer'' was chosen by tuning theorist Scott Dakota (who discovered and raised awareness of the concept) to refer to the property of these scales to [[Harmonic_entropy#Background|"ring"]] extremely and about as much as might be possible for a [[JI]] scale because the [[odd-limit]] complexity of the intervals in such scales is near-minimal meaning they consume as much of the early harmonic series as possible. It is worth noting however that the appearance of a [[virtual fundamental]] depends strongly on which notes of the scale you play - an observation important to [[primodality]]. The concept of ringer scales was additionally further developed by tuning theorists Praveen Venkataramana and later [[user:Godtone]].
+The name ''ringer'' was chosen by tuning theorist [[Scott Dakota]] (who discovered and raised awareness of the concept) to refer to the property of these scales to [[Harmonic_entropy#Background|"ring"]] extremely and about as much as might be possible for a [[JI]] scale because the [[odd-limit]] complexity of the intervals in such scales is near-minimal meaning they consume as much of the early harmonic series as possible. It is worth noting however that the appearance of a [[virtual fundamental]] depends strongly on which notes of the scale you play - an observation important to [[primodality]]. The concept of ringer scales was additionally further developed by tuning theorists Praveen Venkataramana and later [[user:Godtone]].
 == Example: Ringer 15 ==
@@ Line 56: / Line 58: @@
 The [[17-limit]] [[val]] that confirms this scale is CS is  {{val|9 15 22 26 32 34 38}}, which written as [[wart]]s is 9bccdefgg. (Note that in this case, where there is two warts this corresponds to the patent val mapping for the prime already being sharp and being warted to be a step sharper. If we assume that every wart means "sharpen by one step from patent val" this val can be written rather curiously as 9bcdefg, which shows that this val is the one sharpening every applicable prime by one step above the [[patent val]] mapping.) One can confirm that the above is CS because if one traverses it step by step, every one-step interval is mapped to one EDOstep which by [[wikipedia:linearity|linearity]] (more precisely, [[epimorphic]]ity) [[#Proof of CS of by linearity|implies CS]]. Note that it is important to preserve the order of these intervals. 14:16 = 16/14 = 8/7 is mapped to one positive step, as is 16:15 = 15/16, as is 15:17 = 17/15. Similarly (or thus/by linearity), 14:15 = 15/14 is mapped to 2 steps, as is 16:17 = 17/16, as is 15:18 = 18/15 = 6/5.
-== Proof of CS by linearity of the epimorphic val ==
+== Proving CS by hand ==
 Because the CS property means that every occurrence of an interval must occur with the same number of steps, it suffices to show that every one-step interval is mapped to one step by the [[val]] that the Ringer scale is constructed with. (This val shows that the Ringer scale is [[epimorphic]].)
@@ Line 62: / Line 64: @@
 However, conversely, a scale being CS does not imply that such a val exists! In almost all observed practical cases if a scale is CS there is some val, but it is possible to construct scales where, for example, one 1-scalestep interval is equal to the product of more than one other 1-scalestep intervals; that is, if we have 1-scalestep intervals {''a'', ''b'', ''c'', ...} then we can choose ''ab'' as a 1-scalestep interval as long as ''ab'' doesn't occur as a 2-scalestep interval anywhere in the scale, which is why at least one extra 1-scalestep interval ''c'' is necessary to separate instances of ''a'' and ''b''. You can even choose ''b'' = ''a'' but you need to be careful to avoid CS-violating contradictions. For a concrete example, you can use {[[5/4]], [[9/8]], [[45/32]], ...} as 1-scalestep intervals to generate a nonlinear CS scale as long as [[45/32]] does not occur as a 2-scalestep interval anywhere in your scale.
-=== 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>
-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 (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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 :10:12:14:16:17:18
-'''Ringer 8:''' 7:8:'''17/2''':9:10:11:12:13:14
+'''Ringer 8d:''' 7:8:'''17/2''':9:10:11:12:13:14
 :10:11:12:13:14:16:17:18
@@ Line 93: / Line 88: @@
 :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 107: / Line 102: @@
 '''Ringer 12:''' 11:12:'''[25/2''':13(f) OR 13:'''27/2]''':14:15:16:17:18:19:20:21:22
-:15:16:17:18:19:20:21:22:24:[25:26 OR 26:27]:28
+:15:16:17:18:19:20:21:22:24:[25:26(f) OR 26:27]:28
 '''Ringer 13g:''' 9:10:'''21/2''':11:'''23/2''':12:13:14:'''29/2''':15:16:'''33/2''':17:18
@@ Line 121: / Line 116: @@
 '''Ringer 15:''' 13:14:'''29/2''':15:16:'''[33/2''':17(g) OR 17:'''35/2]''':18:19:20:21:22:23:24:25:26
-:19:20:21:22:23:24:25:26:28:29:30:32:[33:34 OR 34:35]:36
+:19:20:21:22:23:24:25:26:28:29:30:32:[33:34(g) OR 34:35]:36
+'''Ringer 16:'''
+:'''55/4''':14:15:'''31/2''':16:17:18:'''[37/2''':19 OR 19:'''39/2(h)]''':20:21:22:23:24:25:26
+:30:31:32:34:36:[37:38 OR 38:39(h)]:40:42:44:46:48:50:52:55:56
 '''Ringer 17cffg:''' 13:14:'''29/2''':15:16:'''33/2''':17:18:19:'''[77/4''' OR '''79/4]''':20:21:22:23:24:'''49/2''':25:26
@@ Line 183: / Line 183: @@
 :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]]
+{{navbox scale gallery}}
 [[Category:Just intonation scales]]
-[[Category:Articles with proofs]]
+[[Category:Pages with proofs]]