TAMNAMS/Appendix: Difference between revisions

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### 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.

### 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'' {{=}} ~~&lceil;~~''m''2/''m''1~~&rceil;~~}} and {{nowrap|''v'' {{=}} ~~&lfloor;~~''m''2/''m''1~~&rfloor;~~}}.<ref group="note" name="floorceiling">~~&lceil;~~ ~~&rceil;~~ denotes the ceiling function and ~~&lfloor;~~ ~~&rfloor;~~ denotes the floor function.</ref>

### 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'' {{=}} {{ceil|''m''2/''m''1}}}} and {{nowrap|''v'' {{=}} {{floor|''m''2/''m''1}}}}.<ref group="note" name="floorceiling">{{ceil| }} denotes the ceiling function and {{floor| }} 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 ''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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### 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.

### 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'' {{=}} ~~&lceil;~~''m''2/''m''1~~&rceil;~~}} and {{nowrap|''v'' {{=}} ~~&lfloor;~~''m''2/''m''1~~&rfloor;~~}}.<ref group="note" name="floorceiling" />

### 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'' {{=}} {{ceil|''m''2/''m''1}}}} and {{nowrap|''v'' {{=}} {{floor|''m''2/''m''1}}}}.<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 ''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.

@@ Line 192: / Line 192: @@
 ### Let ''prescale'' be the mos string for ''z''L&nbsp;''w''s. Recursively call this algorithm to find the scale for ''z''L&nbsp;''w''s; the final scale will be based on this.
 ### If {{nowrap|''x'' &lt; ''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'' {{=}} &lceil;''m''<sub>2</sub>/''m''<sub>1</sub>&rceil;}} and {{nowrap|''v'' {{=}} &lfloor;''m''<sub>2</sub>/''m''<sub>1</sub>&rfloor;}}.<ref group="note" name="floorceiling">&lceil;&nbsp;&rceil; denotes the ceiling function and &lfloor;&nbsp;&rfloor; denotes the floor function.</ref>
+### 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'' {{=}} {{ceil|''m''<sub>2</sub>/''m''<sub>1</sub>}}}} and {{nowrap|''v'' {{=}} {{floor|''m''<sub>2</sub>/''m''<sub>1</sub>}}}}.<ref group="note" name="floorceiling">{{ceil|&nbsp;}} denotes the ceiling function and {{floor|&nbsp;}} denotes the floor function.</ref>
 #### If {{nowrap|''x'' &gt; ''y''}}, every instance of an L in ''prescale'' is replaced with one L and ''u''&nbsp;s's, and every s replaced with one L and ''v''&nbsp;s's. This produces the final scale in its brightest mode.
 #### If {{nowrap|''x'' &lt; ''y''}}, every instance of an L in ''prescale'' is replaced with ''u''&nbsp;L's and one s, and every s replaced with ''v''&nbsp;L's and one s. This produces the final scale in its brightest mode.
@@ Line 271: / Line 271: @@
 ### Let ''gen'' be the scale's generator and ''comp'' be the generator's octave complement for the mos ''z''L&nbsp;''w''s. Recursively call this algorithm to find these intervals for ''z''L&nbsp;''w''s; the final scale's generator and complement will be based on this.
 ### If {{nowrap|''x'' &lt; ''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'' {{=}} &lceil;''m''<sub>2</sub>/''m''<sub>1</sub>&rceil;}} and {{nowrap|''v'' {{=}} &lfloor;''m''<sub>2</sub>/''m''<sub>1</sub>&rfloor;}}.<ref group="note" name="floorceiling" />
+### 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'' {{=}} {{ceil|''m''<sub>2</sub>/''m''<sub>1</sub>}}}} and {{nowrap|''v'' {{=}} {{floor|''m''<sub>2</sub>/''m''<sub>1</sub>}}}}.<ref group="note" name="floorceiling" />
 #### If {{nowrap|''x'' &gt; ''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'' &lt; ''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.