TOP tuning: Difference between revisions

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Note that there is no mapping for 3 which would 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 temperaments, 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.'''

It so happens that for some full prime-limit temperaments, 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 map]]s 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.

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 Note that there is no mapping for 3 which would 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 temperaments, 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.'''
+It so happens that for some full prime-limit temperaments, 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 map]]s 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.