Ringer scale: Difference between revisions
made the odd-limit rotation compatible with scale workshop's "enumerate chord" feature |
m made the odd-limit rotation compatible with scale workshop's "enumerate chord" feature and changed the description correspondingly; also fixed a broken link |
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As definitions can be confusing, it can help to work through an example. For [[15edo]], we can look at how many harmonics it can map without tempering the intervals between any of them. In other words, what is the largest (meaning lowest [[odd-limit]]) [[superparticular]] interval that it tempers? This can be checked with code (an interesting exercise) or can be checked by hand. The answer is [[28/27]], meaning that the 27th harmonic cannot be included unless we choose a different mapping. If we change the mapping of prime 3 to second best (using the 15b val) then [[18/17]] is tempered instead. Changing the mapping of 17 to untemper [[18/17]] would not help as that would cause [[17/16]] to be tempered, meaning we have to keep the patent val mapping for 3 to maximize odd limit. | As definitions can be confusing, it can help to work through an example. For [[15edo]], we can look at how many harmonics it can map without tempering the intervals between any of them. In other words, what is the largest (meaning lowest [[odd-limit]]) [[superparticular]] interval that it tempers? This can be checked with code (an interesting exercise) or can be checked by hand. The answer is [[28/27]], meaning that the 27th harmonic cannot be included unless we choose a different mapping. If we change the mapping of prime 3 to second best (using the 15b val) then [[18/17]] is tempered instead. Changing the mapping of 17 to untemper [[18/17]] would not help as that would cause [[17/16]] to be tempered, meaning we have to keep the patent val mapping for 3 to maximize odd limit. | ||
If we change the mapping of prime 7 to second best (using the 15d val), which is our only other option for trying to untemper [[28/27]], then we get [[21/20]] tempered instead. Note that of the primes present in the [[prime factorization]] of [[21/20]], 2 is fixed, 3 we deduced must be fixed to patent and 7 is what we are changing, leaving only changing the mapping of prime 5 as a way out, which if we then use the 15cd val we get [[15/14]] tempered with no options left [[#Problem of warts|unless we use try to use a third-, fourth-, etc. mapping for primes]], which we will assume for simplicity does not lead anywhere. If we change the mapping of both (using the 15bd val) we also get [[15/14]] tempered. Therefore it seems like the patent val gives the best performance, so we will continue the construction of Ringer 15 assuming that the 25-odd-limit is the highest it is capable of. The corresponding [[harmonic mode|mode]] of the harmonic series is mode 13, giving us: | If we change the mapping of prime 7 to second best (using the 15d val), which is our only other option for trying to untemper [[28/27]], then we get [[21/20]] tempered instead. Note that of the primes present in the [[Wikipedia:prime factorization#Prime decomposition|prime factorization]] of [[21/20]], 2 is fixed, 3 we deduced must be fixed to patent and 7 is what we are changing, leaving only changing the mapping of prime 5 as a way out, which if we then use the 15cd val we get [[15/14]] tempered with no options left [[#Problem of warts|unless we use try to use a third-, fourth-, etc. mapping for primes]], which we will assume for simplicity does not lead anywhere. If we change the mapping of both (using the 15bd val) we also get [[15/14]] tempered. Therefore it seems like the patent val gives the best performance, so we will continue the construction of Ringer 15 assuming that the 25-odd-limit is the highest it is capable of. The corresponding [[harmonic mode|mode]] of the harmonic series is mode 13, giving us: | ||
13:14:15:16:17:18:19:20:21:22:23:24:25:26 | 13:14:15:16:17:18:19:20:21:22:23:24:25:26 | ||
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Then we can notice we are missing two notes (as 26-13=13) to make it a 15-note scale and thus a Ringer 15 scale, so we need to add two odd harmonics above 27 to complete it. Here there are multiple choices based on the [[val]] used and one's preference. The patent val way to complete the scale, which seems to be the lowest complexity and thus arguably the canonical one, is: | Then we can notice we are missing two notes (as 26-13=13) to make it a 15-note scale and thus a Ringer 15 scale, so we need to add two odd harmonics above 27 to complete it. Here there are multiple choices based on the [[val]] used and one's preference. The patent val way to complete the scale, which seems to be the lowest complexity and thus arguably the canonical one, is: | ||
13:14: | 13:14:29/2:15:16:17:35/2:18:19:20:21:22:23:24:25:26 | ||
Where the | Where the n/2 notation means that we are adding an odd harmonic that is imbetween those two harmonics in some higher [[harmonic mode]] (mode of the harmonic series). For example, mode 5 is 5:6:7:8:9:10 so because 6+7=13, we have the 13th harmonic appearing in mode 5*2=10 of the harmonic series between 6*2=12 and 7*2=14, so relative to mode 5 its as if the 13th harmonic is the 13/2 = 6.5th harmonic in the context of 5:6:6.5:7:8:9:10 = 5:6:13/2:7:8:9:10. (In other words the /2 serves to make the harmonic appear in the same octave as the rest.) | ||
Another Ringer 15 scale, if one prefers to get a 33rd harmonic instead of a 35th, is: | Another Ringer 15 scale, if one prefers to get a 33rd harmonic instead of a 35th, is: | ||
13:14: | 13:14:29/2:15:16:33/2:17:18:19:20:21:22:23:24:25:26 | ||
This uses the 15g val meaning prime 17 is mapped to the second-best mapping in [[15edo]]. | This uses the 15g val meaning prime 17 is mapped to the second-best mapping in [[15edo]]. | ||