TOP tuning: Difference between revisions

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For an example of how this works, in the 5 and 7 limits, the TOP comma for magic temperament is 3125/3072; in the 11-limit, |0 -11 15 0 -5>; in the 13 limit, |0 0 46 0 -19 -11>. Putting these in Graham's app will show how closely these are associated with magic. In many cases, the association is even more emphatic.

=TOP with "Inconsistent" Rational Tuning Extensions=

It can sometimes be useful to look not just at "indirect" prime-based mappings, but also add extra "direct" mappings for important rationals -- deliberately inconsistent with the indirect ones -- for which the indirect mapping is subpar.

A good example of this is in 16-EDO, which has a perfectly good 9/8 at 225 cents, but which does not agree with the mapping of 3/2 at 675 cents. In this instance, the associated "2.3.5.9" sval would be <math>\langle 16\, 25\, 37\, 51|</math>, where it is seen that the mapping of 51 steps for 9 is "inconsistent" with the mapping of 25 steps for 3.

Note that there is no mapping for 3 at all which will map 9/1 to 51 steps, since 51 is an odd number, so it is useful to have both mappings: the regular 9/1, for use in chords such as the "[[Mavila]]" major 9 chord of 0-375-675-1050-1350, so that the 1350 cent 9/4 is a stack of two ~675 cent 3/2's, and the tempered 4:7:9 at 0-975-1425, which need not have any 3/2 at all.

It so happens that for some full prime-limit temperament, the TOP tuning remains optimal even if we use "inconsistent" mappings for any composite rational - or even ''every'' rational - as long as we are willing to go with the restriction that such mappings only be used if they are tuned better than the regular consistent ones. We will call tuning maps that obey this restriction '''admissible.'''

As an example, if our tuning map has the "consistent" 9/8 tuned to 204, but where we have an extra "inconsistent" 9/8 mapping that is tuned to 230 cents - that would be an '''inadmissible''' tuning map, because we have added an extraneous extra 9/8 tuning that is worse than the original. In such situations we would throw away the extra inconsistent 9/8 entirely as it serves no purpose, and only use the consistent mapping.

Given this restriction, the proof is easy: for any rational number, any "inconsistent" tuning must have a weighted error that is no worse than the "consistent" tuning, which in turn is no worse than the worst weighted prime error. However, there is no such thing as an "inconsistent prime mapping" - the mapping of a prime must always be consistent with itself! As a result, the worst-weighted error of the entire temperament cannot be changed by improving the errors of individual composite rationals - it will always be found at the worst-weighted prime, which will never change in this way.

As a result, the tuning that minimizes the max Tenney-weighted error on the primes is the same tuning that minimizes the max Tenney-weighted error on all rationals, even if those rationals are inconsistently adjusted to get a better tuning than the consistent ones.

Note that the above proof is only for full-prime limits: for arbitrary subgroups, some care is needed to extend the above argument, as it is possible (for instance) to work in the 2.5.9 subgroup without mapping 3 at all. In this situation, it is no longer the case that we have an extra mapping for 9/1 that is "inconsistent" with the mapping for 3/1, because there is no mapping for 3/1 at all, so 9/1 needs to be treated as thought it were a "prime." While a more thorough treatment of inconsistent mappings on arbitrary subgroups is needed, it is easy to see that for subgroups with a basis consisting only of prime powers, the same argument is easily shown to hold, with the worst weighted error being found at a prime power rather than a prime.

[[Category:benedetti]]

[[Category:erlich]]

[[Category:tenney]]

[[Category:tuning_technique]]

@@ Line 30: / Line 30: @@
 For an example of how this works, in the 5 and 7 limits, the TOP comma for magic temperament is 3125/3072; in the 11-limit, |0 -11 15 0 -5&gt;; in the 13 limit, |0 0 46 0 -19 -11&gt;. Putting these in Graham's app will show how closely these are associated with magic. In many cases, the association is even more emphatic.
+=TOP with "Inconsistent" Rational Tuning Extensions=
+It can sometimes be useful to look not just at "indirect" prime-based mappings, but also add extra "direct" mappings for important rationals -- deliberately inconsistent with the indirect ones -- for which the indirect mapping is subpar.
+A good example of this is in 16-EDO, which has a perfectly good 9/8 at 225 cents, but which does not agree with the mapping of 3/2 at 675 cents. In this instance, the associated "2.3.5.9" sval would be <math>\langle 16\, 25\, 37\, 51|</math>, where it is seen that the mapping of 51 steps for 9 is "inconsistent" with the mapping of 25 steps for 3.
+Note that there is no mapping for 3 at all which will map 9/1 to 51 steps, since 51 is an odd number, so it is useful to have both mappings: the regular 9/1, for use in chords such as the "[[Mavila]]" major 9 chord of 0-375-675-1050-1350, so that the 1350 cent 9/4 is a stack of two ~675 cent 3/2's, and the tempered 4:7:9 at 0-975-1425, which need not have any 3/2 at all.
+It so happens that for some full prime-limit temperament, the TOP tuning remains optimal even if we use "inconsistent" mappings for any composite rational - or even ''every'' rational - as long as we are willing to go with the restriction that such mappings only be used if they are tuned better than the regular consistent ones. We will call tuning maps that obey this restriction '''admissible.'''
+As an example, if our tuning map has the "consistent" 9/8 tuned to 204, but where we have an extra "inconsistent" 9/8 mapping that is tuned to 230 cents - that would be an '''inadmissible''' tuning map, because we have added an extraneous extra 9/8 tuning that is worse than the original. In such situations we would throw away the extra inconsistent 9/8 entirely as it serves no purpose, and only use the consistent mapping.
+Given this restriction, the proof is easy: for any rational number, any "inconsistent" tuning must have a weighted error that is no worse than the "consistent" tuning, which in turn is no worse than the worst weighted prime error. However, there is no such thing as an "inconsistent prime mapping" - the mapping of a prime must always be consistent with itself! As a result, the worst-weighted error of the entire temperament cannot be changed by improving the errors of individual composite rationals - it will always be found at the worst-weighted prime, which will never change in this way.
+As a result, the tuning that minimizes the max Tenney-weighted error on the primes is the same tuning that minimizes the max Tenney-weighted error on all rationals, even if those rationals are inconsistently adjusted to get a better tuning than the consistent ones.
+Note that the above proof is only for full-prime limits: for arbitrary subgroups, some care is needed to extend the above argument, as it is possible (for instance) to work in the 2.5.9 subgroup without mapping 3 at all. In this situation, it is no longer the case that we have an extra mapping for 9/1 that is "inconsistent" with the mapping for 3/1, because there is no mapping for 3/1 at all, so 9/1 needs to be treated as thought it were a "prime." While a more thorough treatment of inconsistent mappings on arbitrary subgroups is needed, it is easy to see that for subgroups with a basis consisting only of prime powers, the same argument is easily shown to hold, with the worst weighted error being found at a prime power rather than a prime.
 [[Category:benedetti]]
 [[Category:erlich]]
 [[Category:tenney]]
 [[Category:tuning_technique]]