Ringer scale: Difference between revisions

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== 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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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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'''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:Pages with proofs]]

@@ Line 27: / Line 27: @@
 == 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 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 87: / Line 87: @@
 :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 109: / Line 109: @@
 '''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 123: / Line 123: @@
 '''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 185: / Line 190: @@
 :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:Pages with proofs]]