36edo: Difference between revisions

Line 14:

The 36edo patent val, like 12, tempers out 81/80, 128/125 and 648/625 in the 5-limit. It departs from 12 in the 7-limit, tempering out 686/675, and as a no-fives temperament, 1029/1024 and 118098/117649. The no-fives temperament tempering out 1029/1024, [[slendric]], is well supported by 36edo, its generator of ~8/7 represented by 7 steps of 36edo. In the 11-limit, the patent val tempers out 56/55, 245/242 and 540/539, and is the [[optimal patent val]] for the rank four temperament tempering out 56/55, as well as the rank three temperament [[Didymus rank three family|melpomene]] tempering out 81/80 and 56/55. In the 13-limit, it tempers out 78/77, in the 17-limit 51/50, and in the 19-limit 76/75, 91/90 and 96/95.

As a 5-limit temperament, the patent val for 36edo is [[Wedgies and Multivals|contorted]], meaning there are notes of it which cannot be reached from the unison using only 5-limit intervals. A curious alternative val for the 5-limit is ~~<~~36 65 116| which is not contorted. It is also a meantone val, in the sense that 81/80 is tempered out. However, the "comma" {{monzo|29 0 -9}} is also tempered out, and the "fifth", 29\36, is actually approximately 7/4, whereas the "major third", 44\36, is actually approximately 7/3. Any 5-limit musical piece or scale which is a [[transversal]] for a meantone piece or scale will be converted to a no-fives piece tempering out 1029/1024 in place of 81/80 by applying this val.

As a 5-limit temperament, the patent val for 36edo is [[Wedgies and Multivals|contorted]], meaning there are notes of it which cannot be reached from the unison using only 5-limit intervals. A curious alternative val for the 5-limit is {{val| 36 65 116 }}, which is not contorted. It is also a meantone val, in the sense that 81/80 is tempered out. However, the "comma" {{monzo|29 0 -9}} is also tempered out, and the "fifth", 29\36, is actually approximately 7/4, whereas the "major third", 44\36, is actually approximately 7/3. Any 5-limit musical piece or scale which is a [[transversal]] for a meantone piece or scale will be converted to a no-fives piece tempering out 1029/1024 in place of 81/80 by applying this val.

Another 5-limit alternative val is {{~~monzo~~|36 57 83}} (36c-edo), which is similar to the patent val but has 5/4 mapped to the 367-cent submajor third rather than the major third. This mapping supports very sharp [[porcupine]] temperament using 5\36 as a generator.

Another 5-limit alternative val is {{val| 36 57 83 }} (36c-edo), which is similar to the patent val but has 5/4 mapped to the 367-cent submajor third rather than the major third. This mapping supports very sharp [[porcupine]] temperament using 5\36 as a generator.

Heinz Bohlen proposed 36edo as a suitable temperament for approximating his 833-cents scale. The 13th harmonic (octave reduced) is so closely mapped on [[acoustic phi]] that 36edo could be treated as a 2.3.7.ϕ.17 temperament.

@@ Line 14: / Line 14: @@
 The 36edo patent val, like 12, tempers out 81/80, 128/125 and 648/625 in the 5-limit. It departs from 12 in the 7-limit, tempering out 686/675, and as a no-fives temperament, 1029/1024 and 118098/117649. The no-fives temperament tempering out 1029/1024, [[slendric]], is well supported by 36edo, its generator of ~8/7 represented by 7 steps of 36edo. In the 11-limit, the patent val tempers out 56/55, 245/242 and 540/539, and is the [[optimal patent val]] for the rank four temperament tempering out 56/55, as well as the rank three temperament [[Didymus rank three family|melpomene]] tempering out 81/80 and 56/55. In the 13-limit, it tempers out 78/77, in the 17-limit 51/50, and in the 19-limit 76/75, 91/90 and 96/95.
-As a 5-limit temperament, the patent val for 36edo is [[Wedgies and Multivals|contorted]], meaning there are notes of it which cannot be reached from the unison using only 5-limit intervals. A curious alternative val for the 5-limit is &lt;36 65 116| which is not contorted. It is also a meantone val, in the sense that 81/80 is tempered out. However, the "comma" {{monzo|29 0 -9}} is also tempered out, and the "fifth", 29\36, is actually approximately 7/4, whereas the "major third", 44\36, is actually approximately 7/3. Any 5-limit musical piece or scale which is a [[transversal]] for a meantone piece or scale will be converted to a no-fives piece tempering out 1029/1024 in place of 81/80 by applying this val.
+As a 5-limit temperament, the patent val for 36edo is [[Wedgies and Multivals|contorted]], meaning there are notes of it which cannot be reached from the unison using only 5-limit intervals. A curious alternative val for the 5-limit is {{val| 36 65 116 }}, which is not contorted. It is also a meantone val, in the sense that 81/80 is tempered out. However, the "comma" {{monzo|29 0 -9}} is also tempered out, and the "fifth", 29\36, is actually approximately 7/4, whereas the "major third", 44\36, is actually approximately 7/3. Any 5-limit musical piece or scale which is a [[transversal]] for a meantone piece or scale will be converted to a no-fives piece tempering out 1029/1024 in place of 81/80 by applying this val.
-Another 5-limit alternative val is {{monzo|36 57 83}} (36c-edo), which is similar to the patent val but has 5/4 mapped to the 367-cent submajor third rather than the major third. This mapping supports very sharp [[porcupine]] temperament using 5\36 as a generator.
+Another 5-limit alternative val is {{val| 36 57 83 }} (36c-edo), which is similar to the patent val but has 5/4 mapped to the 367-cent submajor third rather than the major third. This mapping supports very sharp [[porcupine]] temperament using 5\36 as a generator.
 Heinz Bohlen proposed 36edo as a suitable temperament for approximating his 833-cents scale. The 13th harmonic (octave reduced) is so closely mapped on [[acoustic phi]] that 36edo could be treated as a 2.3.7.ϕ.17 temperament.