TOP tuning: Difference between revisions

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A good example of this is 16-EDO, in which 9/8 is mapped to 225 cents, while 3/2 is mapped to 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.

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 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.'''

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 A good example of this is 16-EDO, in which 9/8 is mapped to 225 cents, while 3/2 is mapped to 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.
+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 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.'''