TAMNAMS/Appendix: Difference between revisions

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=== Terminology and final notes ===

A ratio of {{nowrap|L/s {{=}} ''k''/1}} can be called ''k-hard'' and a ratio of {{nowrap|L/s {{=}} ''k''/(''k'' ~~−~~ 1)}} can analogously be called ''k-soft'', so the simplest ultrasoft tuning is 5-soft or ''pentasoft'', the simplest hyperhard tuning is 5-hard or ''pentahard'', the simplest clustered tuning is 7-hard or ''heptahard'', 8-hard is ''octahard'', 9-hard is ''nonahard'', and finally, the characteristic simple ultrahard tuning is 6-hard or ''extrahard'', as previously discussed, which can be seen to be similar to ''hexahard''—hopefully helping with memorisation.

A ratio of {{nowrap|L/s {{=}} ''k''/1}} can be called ''k-hard'' and a ratio of {{nowrap|L/s {{=}} ''k''/(''k'' − 1)}} can analogously be called ''k-soft'', so the simplest ultrasoft tuning is 5-soft or ''pentasoft'', the simplest hyperhard tuning is 5-hard or ''pentahard'', the simplest clustered tuning is 7-hard or ''heptahard'', 8-hard is ''octahard'', 9-hard is ''nonahard'', and finally, the characteristic simple ultrahard tuning is 6-hard or ''extrahard'', as previously discussed, which can be seen to be similar to ''hexahard''—hopefully helping with memorisation.

A perhaps useful (or otherwise mildly amusing) mnemonic is ''2-soft is too soft to be hard and 2-hard is too hard to be soft'', representing that {{nowrap|2-soft {{=}} 2-hard {{=}} 2/1 {{=}} '''basic'''}}.

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## If ''x'' and ''y'' don't share a common factor that is greater than 1 (if ''x'' and ''y'' are coprime), then:

### Let {{nowrap|''m''1 {{=}} min(''x'', ''y'')}} and {{nowrap|''m''2 {{=}} max(''x'', ''y'')}}.

### Let {{nowrap|''z'' {{=}} ''m''2 mod ''m''1}} and {{nowrap|''w'' {{=}} ''m''1 ~~−~~ ''z''}}.

### Let {{nowrap|''z'' {{=}} ''m''2 mod ''m''1}} and {{nowrap|''w'' {{=}} ''m''1 − ''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.

### 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.

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# If either ''x'' or ''y'' is equal to 1 (base cases):

## If both ''x'' and ''y'' are equal to 1, then the generator is "L" and its complement is "s".

## If only ''x'' is equal to 1, then the generator is "L" followed by {{nowrap|''y'' ~~−~~ 1}} s's, and the complement is "s".

## If only ''x'' is equal to 1, then the generator is "L" followed by {{nowrap|''y'' − 1}} s's, and the complement is "s".

## If only ''y'' is equal to 1, then the generator is "L" and the complement is {{nowrap|''x'' ~~−~~ 1}} L's followed by "s".

## If only ''y'' is equal to 1, then the generator is "L" and the complement is {{nowrap|''x'' − 1}} L's followed by "s".

# If neither ''x'' nor ''y'' is equal to 1 (recursive cases):

## Let ''k'' be the greatest common factor of ''x'' and ''y''.

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## If ''x'' and ''y'' don't share a common factor that is greater than 1 (if ''x'' and ''y'' are coprime), then:

### Let {{nowrap|''m''1 {{=}} min(''x'', ''y'')}} and {{nowrap|''m''2 {{=}} max(''x'', ''y'')}}.

### Let {{nowrap|''z'' {{=}} ''m''2 mod ''m''1}} and {{nowrap|''w'' {{=}} ''m1 ~~−~~ z''}}.

### Let {{nowrap|''z'' {{=}} ''m''2 mod ''m''1}} 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.

### 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.

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 === Terminology and final notes ===
-A ratio of {{nowrap|L/s {{=}} ''k''/1}} can be called ''k-hard'' and a ratio of {{nowrap|L/s {{=}} ''k''/(''k'' &minus; 1)}} can analogously be called ''k-soft'', so the simplest ultrasoft tuning is 5-soft or ''pentasoft'', the simplest hyperhard tuning is 5-hard or ''pentahard'', the simplest clustered tuning is 7-hard or ''heptahard'', 8-hard is ''octahard'', 9-hard is ''nonahard'', and finally, the characteristic simple ultrahard tuning is 6-hard or ''extrahard'', as previously discussed, which can be seen to be similar to ''hexahard''&mdash;hopefully helping with memorisation.
+A ratio of {{nowrap|L/s {{=}} ''k''/1}} can be called ''k-hard'' and a ratio of {{nowrap|L/s {{=}} ''k''/(''k'' − 1)}} can analogously be called ''k-soft'', so the simplest ultrasoft tuning is 5-soft or ''pentasoft'', the simplest hyperhard tuning is 5-hard or ''pentahard'', the simplest clustered tuning is 7-hard or ''heptahard'', 8-hard is ''octahard'', 9-hard is ''nonahard'', and finally, the characteristic simple ultrahard tuning is 6-hard or ''extrahard'', as previously discussed, which can be seen to be similar to ''hexahard''&mdash;hopefully helping with memorisation.
 A perhaps useful (or otherwise mildly amusing) mnemonic is ''2-soft is too soft to be hard and 2-hard is too hard to be soft'', representing that {{nowrap|2-soft {{=}} 2-hard {{=}} 2/1 {{=}} '''basic'''}}.
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 ## If ''x'' and ''y'' don't share a common factor that is greater than 1 (if ''x'' and ''y'' are coprime), then:
 ### Let {{nowrap|''m''<sub>1</sub> {{=}} min(''x'', ''y'')}} and {{nowrap|''m''<sub>2</sub> {{=}} max(''x'', ''y'')}}.
-### Let {{nowrap|''z'' {{=}} ''m''<sub>2</sub> mod ''m''<sub>1</sub>}} and {{nowrap|''w'' {{=}} ''m''<sub>1</sub> &minus; ''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&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.
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 # If either ''x'' or ''y'' is equal to 1 (base cases):
 ## If both ''x'' and ''y'' are equal to 1, then the generator is "L" and its complement is "s".
-## If only ''x'' is equal to 1, then the generator is "L" followed by {{nowrap|''y'' &minus; 1}} s's, and the complement is "s".
+## If only ''x'' is equal to 1, then the generator is "L" followed by {{nowrap|''y'' − 1}} s's, and the complement is "s".
-## If only ''y'' is equal to 1, then the generator is "L" and the complement is {{nowrap|''x'' &minus; 1}} L's followed by "s".
+## If only ''y'' is equal to 1, then the generator is "L" and the complement is {{nowrap|''x'' − 1}} L's followed by "s".
 # If neither ''x'' nor ''y'' is equal to 1 (recursive cases):
 ## Let ''k'' be the greatest common factor of ''x'' and ''y''.
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 ## If ''x'' and ''y'' don't share a common factor that is greater than 1 (if ''x'' and ''y'' are coprime), then:
 ### Let {{nowrap|''m''<sub>1</sub> {{=}} min(''x'', ''y'')}} and {{nowrap|''m''<sub>2</sub> {{=}} max(''x'', ''y'')}}.
-### Let {{nowrap|''z'' {{=}} ''m''<sub>2</sub> mod ''m''<sub>1</sub>}} and {{nowrap|''w'' {{=}} ''m1 &minus; 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&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.