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80edo: Difference between revisions

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== Detemperaments ==
== Detemperaments ==
=== Ringer 80 ===
=== Ringer 80 ===
80edo is a great (essentially) no-limit system for conceptualising and internalising harmonic series interval categories/structures through '''Ringer 80''' which contains the entirety of the no-127's no-135's no-141's 145-odd-limit. An astounding ~84% of all intervals present are mapped consistently in Ringer 80. The Ringer 80 described below uses the best-performing val for 125-odd-limit consistency by a variety of metrics (squared error, sum of error, number of inconsistencies, number of inconsistencies if we require <25% error, etc.). To achieve the [[constant structure|CS]] property, the primes 31, 47, 53, 61, 67, 73, 79, 107, 109 are sharpened by 1 step compared to their flat patent val mapping (i.e. are mapped to their second-best mapping); all other primes are of the patent val. It is maybe worth noting that the least intuitive of these warts for prime 73 corresponds to getting the interval [[73/63]] to be mapped consistently, which is not insignificant because 80edo has an accurate enough approximation that it is a giant circle of 73/63's, among other such circles. Warting prime 73 also allows the introduction of the 145th harmonic which adds a lot of low-complexity and consistent intervals; specifically, all intervals made in ratio with the 145th harmonic that simplify are mapped consistently.  
80edo is a great essentially no-limit system for conceptualising and internalising harmonic series interval categories/structures through '''Ringer 80''' which contains the entirety of the no-127's no-135's no-141's 145-odd-limit. An astounding ~84% of all intervals present are mapped consistently in Ringer 80. The Ringer 80 described below uses the best-performing val for 125-odd-limit consistency by a variety of metrics (squared error, sum of error, number of inconsistencies, number of inconsistencies if we require <25% error, etc.). The primes 31, 47, 53, 61, 67, 73, 79, 107, 109 are sharpened by one step compared to their flat patent val mapping (i.e. are mapped to their second-best mapping); all other primes are of the patent val. It is maybe worth noting that the least intuitive of these warts for prime 73 corresponds to getting the interval [[73/63]] to be mapped consistently, which is not insignificant because 80edo has an accurate enough approximation that it is practically a giant circle of 73/63's, among other such circles. Warting prime 73 also allows the introduction of the 145th harmonic which adds a lot of low-complexity and consistent intervals; specifically, all intervals made in ratio with the 145th harmonic that simplify are mapped consistently.  


This scale has a few remarkable properties. Firstly, all the intervals that are inconsistent are mapped – at worst – to their second-best mapping, meaning you will never have a categorical/interval mapping exceeding 15 cents of error. Although this property is not as rare as it may sound it is still musically useful. Secondly, all of the "filler harmonics" beyond the 125-odd-limit fit in an obvious way; note how there are no warts beyond the 113-prime-limit (which the 125-odd-limit corresponds to due to the sizeable record prime gap from 113 to 127), meaning all composite harmonics were either already part of the 113-prime-limit or if prime the primes were patent val and sharp-tending. "In an obvious way" also means that every superparticular (''n'' + 1)/''n'' in the 125-odd-limit that was mapped to 2 steps is split into (2''n'' + 2)/(2''n'' + 1) and (2''n'' + 1)/(2''n''), retaining the lowest possible complexity and most elegant possible structure for a [[ringer scale]]. Finally, note that while composite odd harmonics start going missing after the 125th harmonic, prime harmonics are very much not lacking. This scale exists inside the no-127's no-151's no-163's 179-prime-limit, meaning that ''all primes up to and including 179'' are present excluding only those three, making it full of prime flavour on top of its capability for representing high compositeness due to the 125-odd-limit corresponding to a record prime gap. Note that prime 127 cannot be included because to match the increasing trend of sharpness it would need to be warted, leading to 128/127 being tempered out.
This scale has a few remarkable properties. Firstly, all the intervals that are inconsistent are mapped – at worst – to their second-best mapping, meaning you will never have a categorical/interval mapping exceeding 15 cents of error. Although this property is not as rare as it may sound it is still musically useful. Secondly, all of the "filler harmonics" beyond the 125-odd-limit fit in an obvious way; note how there are no warts beyond the 113-prime-limit (which the 125-odd-limit corresponds to due to the sizeable record prime gap from 113 to 127), meaning all composite harmonics were either already part of the 113-prime-limit or if prime the primes were patent val and sharp-tending. "In an obvious way" also means that every superparticular (''n'' + 1)/''n'' in the 125-odd-limit that was mapped to 2 steps is split into (2''n'' + 2)/(2''n'' + 1) and (2''n'' + 1)/(2''n''), retaining the lowest possible complexity and most elegant possible structure for a [[ringer scale]]. Finally, note that while composite odd harmonics start going missing after the 125th harmonic, prime harmonics are very much not lacking. This scale exists inside the no-127's no-151's no-163's 179-prime-limit, meaning that ''all primes up to and including 179'' are present excluding only those three, making it full of prime flavour on top of its capability for representing high compositeness due to the 125-odd-limit corresponding to a record prime gap. Note that prime 127 cannot be included because to match the increasing trend of sharpness it would need to be warted, leading to 128/127 being tempered out.


<pre>
Mode 63 of the harmonic series (corresponding to 125-odd-limit) with added odds from mode 63 × 2 = 126 in square brackets:
Mode 63 of the harmonic series (corresponding to 125-odd-limit) with added odds from mode 63 × 2 = 126 in square brackets:
63:64:[129]:65:[131]:66:[133]:67:68:[137]:69:[139]:70:71:[143]:72:[145]:73:74:[149]
 
75:76:[153]:77:78:[157]:79:80:[161]:81:82:83:[167]:84:85:86:[173]:87:88:89
: 63:64:[129]:65:[131]:66:[133]:67:68:[137]:69:[139]:70:71:[143]:72:[145]:73:74:[149]
[179]:90:91:92:[185]:93:94:95:96:97:[195]:98:99:100:101:102:103:104:[209]:105
: 75:76:[153]:77:78:[157]:79:80:[161]:81:82:83:[167]:84:85:86:[173]:87:88:89
106:107:108:109:110:111:112:113:114:115:116:117:118:119:120:121:122:123:124:125(:126)
: [179]:90:91:92:[185]:93:94:95:96:97:[195]:98:99:100:101:102:103:104:[209]:105
: 106:107:108:109:110:111:112:113:114:115:116:117:118:119:120:121:122:123:124:125(:126)
 
The above is split into 20 harmonics per line a.k.a. ~300¢ worth of harmonic content per line.
The above is split into 20 harmonics per line a.k.a. ~300¢ worth of harmonic content per line.


In lowest terms as a /105 scale corresponding to a primodal /53 scale, among other possible interpretations:
In lowest terms as a /105 scale corresponding to a primodal /53 scale, among other possible interpretations:
105:106:107:108:109:110:111:112:113:114:115:116:117:118:119:120:121:122:123:124:125:126:128:129:130:131:132:133:134:136:137:138:139:140:142:143:144:145:146:148:149:150:152:153:154:156:157:158:160:161:162:164:166:167:168:170:172:173:174:176:178:179:180:182:184:185:186:188:190:192:194:195:196:198:200:202:204:206:208:209:210
 
: 105:106:107:108:109:110:111:112:113:114:115:116:117:118:119:120:121:122:123:124:125:126:128:129:130:131:132:133:134:136:137:138:139:140:142:143:144:145:146:148:149:150:152:153:154:156:157:158:160:161:162:164:166:167:168:170:172:173:174:176:178:179:180:182:184:185:186:188:190:192:194:195:196:198:200:202:204:206:208:209:210
 
This form is useful for copy-pasting into tools that accept colon-separated harmonic series chord enumerations as scales.
This form is useful for copy-pasting into tools that accept colon-separated harmonic series chord enumerations as scales.


As reduced, rooted intervals (16 intervals per line):
As reduced, rooted intervals (16 intervals per line):
129/128, 65/64, 131/128, 33/32, 133/128, 67/64, 17/16, 137/128, 69/64, 139/128, 35/32, 71/64, 143/128, 9/8, 145/128, 73/64,
 
37/32, 75/64, 19/16, 153/128, 77/64, 39/32, 157/128, 79/64, 5/4, 161/128, 81/64, 41/32, 83/64, 167/128, 21/16, 85/64,
: 129/128, 65/64, 131/128, 33/32, 133/128, 67/64, 17/16, 137/128, 69/64, 139/128, 35/32, 71/64, 143/128, 9/8, 145/128, 73/64,
43/32, 173/128, 87/64, 11/8, 89/64, 179/128, 45/32, 91/64, 23/16, 185/128, 93/64, 47/32, 95/64, 3/2, 97/64, 195/128,
: 37/32, 75/64, 19/16, 153/128, 77/64, 39/32, 157/128, 79/64, 5/4, 161/128, 81/64, 41/32, 83/64, 167/128, 21/16, 85/64,
49/32, 99/64, 25/16, 101/64, 51/32, 103/64, 13/8, 209/128, 105/64, 53/32, 107/64, 27/16, 109/64, 55/32, 111/64, 7/4,
: 43/32, 173/128, 87/64, 11/8, 89/64, 179/128, 45/32, 91/64, 23/16, 185/128, 93/64, 47/32, 95/64, 3/2, 97/64, 195/128,
113/64, 57/32, 115/64, 29/16, 117/64, 59/32, 119/64, 15/8, 121/64, 61/32, 123/64, 31/16, 125/64, 63/32, 2/1
: 49/32, 99/64, 25/16, 101/64, 51/32, 103/64, 13/8, 209/128, 105/64, 53/32, 107/64, 27/16, 109/64, 55/32, 111/64, 7/4,
</pre>
: 113/64, 57/32, 115/64, 29/16, 117/64, 59/32, 119/64, 15/8, 121/64, 61/32, 123/64, 31/16, 125/64, 63/32, 2/1


== Music ==
== Music ==
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