TAMNAMS/Appendix: Difference between revisions
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## If ''x'' and ''y'' share a common factor ''k'', where ''k'' is greater than 1, then recursively call this algorithm to find the scale for {{nowrap|(''x''/''k'')L (''y''/''k'')s}}; the final scale will be {{nowrap|(''x''/''k'')L (''y''/''k'')s}} duplicated ''k'' times. | ## If ''x'' and ''y'' share a common factor ''k'', where ''k'' is greater than 1, then recursively call this algorithm to find the scale for {{nowrap|(''x''/''k'')L (''y''/''k'')s}}; the final scale will be {{nowrap|(''x''/''k'')L (''y''/''k'')s}} duplicated ''k'' times. | ||
## If ''x'' and ''y'' don't share a common factor that is greater than 1 (if ''x'' and ''y'' are coprime), then: | ## If ''x'' and ''y'' don't share a common factor that is greater than 1 (if ''x'' and ''y'' are coprime), then: | ||
### Let {{nowrap|'' | ### Let {{nowrap|''m''<sub>1</sub> {{=}} min(''x'', ''y'')}} and {{nowrap|''m''<sub>2</sub> {{=}} max(''x'', ''y'')}}. | ||
### Let {{nowrap|''z'' {{=}} '' | ### Let {{nowrap|''z'' {{=}} ''m''<sub>2</sub> mod ''m''<sub>1</sub>}} and {{nowrap|''w'' {{=}} ''m''<sub>1</sub> − ''z''}}. | ||
### Let ''prescale'' be the mos string for ''z''L ''w''s. Recursively call this algorithm to find the scale for ''z''L ''w''s; the final scale will be based on this. | ### Let ''prescale'' be the mos string for ''z''L ''w''s. Recursively call this algorithm to find the scale for ''z''L ''w''s; the final scale will be based on this. | ||
### If {{nowrap|''x'' < ''y''}}, reverse the order of characters in the prescale. This is only needed if there are more L's than s's in the final scale. | ### If {{nowrap|''x'' < ''y''}}, reverse the order of characters in the prescale. This is only needed if there are more L's than s's in the final scale. | ||
### To produce the final scale, the L's and s's of the prescale must be replaced with substrings consisting of L's and s's. Let {{nowrap|''u'' {{=}} ⌈'' | ### To produce the final scale, the L's and s's of the prescale must be replaced with substrings consisting of L's and s's. Let {{nowrap|''u'' {{=}} ⌈''m''<sub>2</sub>/''m''<sub>1</sub>⌉}} and {{nowrap|''v'' {{=}} ⌊''m''<sub>2</sub>/''m''<sub>1</sub>⌋}}.<ref group="note" name="floorceiling">⌈ ⌉ denotes the ceiling function and ⌊ ⌋ denotes the floor function.</ref> | ||
#### If {{nowrap|''x'' > ''y''}}, every instance of an L in ''prescale'' is replaced with one L and ''u'' s's, and every s replaced with one L and ''v'' s's. This produces the final scale in its brightest mode. | #### If {{nowrap|''x'' > ''y''}}, every instance of an L in ''prescale'' is replaced with one L and ''u'' s's, and every s replaced with one L and ''v'' s's. This produces the final scale in its brightest mode. | ||
#### If {{nowrap|''x'' < ''y''}}, every instance of an L in ''prescale'' is replaced with ''u'' L's and one s, and every s replaced with ''v'' L's and one s. This produces the final scale in its brightest mode. | #### If {{nowrap|''x'' < ''y''}}, every instance of an L in ''prescale'' is replaced with ''u'' L's and one s, and every s replaced with ''v'' L's and one s. This produces the final scale in its brightest mode. | ||
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## If ''x'' and ''y'' share a common factor ''k'', where ''k'' is greater than 1, then recursively call this algorithm to find the generator and complement for {{nowrap|(''x''/''k'')L (''y''/''k'')s}}; the intervals returned this way will apply to the period rather than the octave. | ## If ''x'' and ''y'' share a common factor ''k'', where ''k'' is greater than 1, then recursively call this algorithm to find the generator and complement for {{nowrap|(''x''/''k'')L (''y''/''k'')s}}; the intervals returned this way will apply to the period rather than the octave. | ||
## If ''x'' and ''y'' don't share a common factor that is greater than 1 (if ''x'' and ''y'' are coprime), then: | ## If ''x'' and ''y'' don't share a common factor that is greater than 1 (if ''x'' and ''y'' are coprime), then: | ||
### Let {{nowrap|'' | ### Let {{nowrap|''m''<sub>1</sub> {{=}} min(''x'', ''y'')}} and {{nowrap|''m''<sub>2</sub> {{=}} max(''x'', ''y'')}}. | ||
### Let {{nowrap|''z'' {{=}} '' | ### Let {{nowrap|''z'' {{=}} ''m''<sub>2</sub> mod ''m''<sub>1</sub>}} and {{nowrap|''w'' {{=}} ''m1 − z''}}. | ||
### Let ''gen'' be the scale's generator and ''comp'' be the generator's octave complement for the mos ''z''L ''w''s. Recursively call this algorithm to find these intervals for ''z''L ''w''s; the final scale's generator and complement will be based on this. | ### Let ''gen'' be the scale's generator and ''comp'' be the generator's octave complement for the mos ''z''L ''w''s. Recursively call this algorithm to find these intervals for ''z''L ''w''s; the final scale's generator and complement will be based on this. | ||
### If {{nowrap|''x'' < ''y''}}, reverse the order of characters in ''gen'' and ''comp'', then swap ''gen'' and ''comp''. This is only needed if there are more L's than s's in the scale. | ### If {{nowrap|''x'' < ''y''}}, reverse the order of characters in ''gen'' and ''comp'', then swap ''gen'' and ''comp''. This is only needed if there are more L's than s's in the scale. | ||
### To produce the scale's generator and complement, the L's and s's of both intervals must be replaced with substrings consisting of L's and s's. Let {{nowrap|''u'' {{=}} ⌈'' | ### To produce the scale's generator and complement, the L's and s's of both intervals must be replaced with substrings consisting of L's and s's. Let {{nowrap|''u'' {{=}} ⌈''m''<sub>2</sub>/''m''<sub>1</sub>⌉}} and {{nowrap|''v'' {{=}} ⌊''m''<sub>2</sub>/''m''<sub>1</sub>⌋}}.<ref group="note" name="floorceiling" /> | ||
#### If {{nowrap|''x'' > ''y''}}, every instance of an L in both intervals is replaced with one L and ''u'' s's, and every s replaced with one L and ''v'' s's. This produces the final scale's generator and complement. | #### If {{nowrap|''x'' > ''y''}}, every instance of an L in both intervals is replaced with one L and ''u'' s's, and every s replaced with one L and ''v'' s's. This produces the final scale's generator and complement. | ||
#### If {{nowrap|''x'' < ''y''}}, every instance of an L in both intervals is replaced with ''u'' L's and one s, and every s replaced with ''v'' L's and one s. This produces the final scale's generator and complement. | #### If {{nowrap|''x'' < ''y''}}, every instance of an L in both intervals is replaced with ''u'' L's and one s, and every s replaced with ''v'' L's and one s. This produces the final scale's generator and complement. | ||
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[[Dicot family#Dichotic|Dichotic]] is the 7&10 temperament in the 11-limit with commas including 25/24, 45/44, 55/54, 56/55, 64/63. This is an extension of the 5-limit exotemperament [[dicot]] which tempers 25/24, equating 5/4 and 6/5 into a neutral third sized interval, which is the generator. | [[Dicot family#Dichotic|Dichotic]] is the 7&10 temperament in the 11-limit with commas including 25/24, 45/44, 55/54, 56/55, 64/63. This is an extension of the 5-limit exotemperament [[dicot]] which tempers 25/24, equating 5/4 and 6/5 into a neutral third sized interval, which is the generator. | ||
This temperament is supported by {{Optimal ET sequence| 7, 10, and 17 }} equal divisions, with non-patent val tunings including (but not limited to) {{nowrap|7 + 7 {{=}} 14cd|10 + 10 {{=}} 20e|17 + 7 {{=}} 24cd|and 17 + 10 {{=}}27ce}}. | This temperament is supported by {{Optimal ET sequence| 7, 10, and 17 }} equal divisions, with non-patent val tunings including (but not limited to) {{nowrap|7 + 7 {{=}} 14cd|10 + 10 {{=}} 20e|17 + 7 {{=}} 24cd|and 17 + 10 {{=}} 27ce}}. | ||
==== Armotonic (7L 2s) ==== | ==== Armotonic (7L 2s) ==== | ||
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==== On the term ''diatonic'' ==== | ==== On the term ''diatonic'' ==== | ||
Although the term ''diatonic'' has accrued a variety of exact meanings over time, both within and outside the contexts of xenharmonic music theory, in the context of TAMNAMS and moment-of-symmetry scales, the term ''diatonic'' exclusively refers to 5L 2s. | Although the term ''diatonic'' has accrued a variety of exact meanings over time, both within and outside the contexts of xenharmonic music theory, in the context of TAMNAMS and moment-of-symmetry scales, the term ''diatonic'' exclusively refers to 5L 2s. | ||
== Notes == | |||
<references group="note" /> | |||
[[Category:TAMNAMS]] | [[Category:TAMNAMS]] | ||