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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'''}}. | 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'''}}. | ||
Note that often the central spectrum will be sufficient for exploring a mos pattern-period combination, and the extended spectrum is intended more for (literally) edge cases where it may be useful. Often if a temperament interpretation doesn't seem to show up for a mos pattern-period combination, it just means the temperament needs a more complex mos pattern to narrow down the generator range. An example of this phenomena is the highly complex mos pattern of [[ | Note that often the central spectrum will be sufficient for exploring a mos pattern-period combination, and the extended spectrum is intended more for (literally) edge cases where it may be useful. Often if a temperament interpretation doesn't seem to show up for a mos pattern-period combination, it just means the temperament needs a more complex mos pattern to narrow down the generator range. An example of this phenomena is the highly complex mos pattern of [[12L 17s]] represents near-Pythagorean tunings well due to having a generator of a fourth or a fifth bounded between those of [[12edo]] and those of [[29edo]], which are roughly equally off but in opposite directions, and many important near-Pythagorean systems show up in just the ratios of the central spectrum alone. | ||
== Reasoning for mos interval names == | == Reasoning for mos interval names == | ||
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# If either ''x'' or ''y'' is equal to 1 (base cases): | # If either ''x'' or ''y'' is equal to 1 (base cases): | ||
## If both ''x'' and ''y'' are equal to 1, then the final scale is "Ls". | ## If both ''x'' and ''y'' are equal to 1, then the final scale is "Ls". | ||
## If only ''x'' is equal to 1, then the final scale is L followed by ''y'' s's. | ## If only ''x'' is equal to 1, then the final scale is L followed by ''y'' s's. | ||
## If only ''y'' is equal to 1, then the final scale is ''x'' L's followed by s. | ## If only ''y'' is equal to 1, then the final scale is ''x'' L's followed by s. | ||
# If neither ''x'' nor ''y'' is equal to 1 (recursive cases): | # If neither ''x'' nor ''y'' is equal to 1 (recursive cases): | ||
## Let ''k'' be the greatest common factor of ''x'' and ''y''. | ## Let ''k'' be the greatest common factor of ''x'' and ''y''. | ||
## 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 | ## 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 (''x''/''k'')L (''y''/''k'')s; the final scale will be (''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|''m''<sub>1</sub> {{=}} min(''x'', ''y'')}} and {{nowrap|''m''<sub>2</sub> {{=}} max(''x'', ''y'')}}. | ### Let {{nowrap|''m''<sub>1</sub> {{=}} min(''x'', ''y'')}} and {{nowrap|''m''<sub>2</sub> {{=}} max(''x'', ''y'')}}. | ||
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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. | ### 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'' {{=}} {{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| }} 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 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. | ||
Using 3L 4s as an example, this is LsLsLss (brightest) and ssLsLsL (darkest). To find the large sizes of each ''k''-mosstep, consider the first ''k'' mossteps that make up the mos pattern for the brightest mode. Repeat this process with the mos pattern for the darkest mode to find each ''k''-mosstep's small size. To make these sizes more clear, we can denote the mos intervals as a sum of large and small steps {{nowrap|''i''L + ''j''s}}, where ''i'' and ''j'' are the number of L's and s's in the interval's step pattern; this is to say that the order of L's and s's doesn't matter, rather the amount of each step size. The large and small sizes should differ by replacing one L in the large size with an s. | Using 3L 4s as an example, this is LsLsLss (brightest) and ssLsLsL (darkest). To find the large sizes of each ''k''-mosstep, consider the first ''k'' mossteps that make up the mos pattern for the brightest mode. Repeat this process with the mos pattern for the darkest mode to find each ''k''-mosstep's small size. To make these sizes more clear, we can denote the mos intervals as a sum of large and small steps {{nowrap|''i''L + ''j''s}}, where ''i'' and ''j'' are the number of L's and s's in the interval's step pattern; this is to say that the order of L's and s's doesn't matter, rather the amount of each step size. The large and small sizes should differ by replacing one L in the large size with an s. | ||
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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. | ### 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'' {{=}} {{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'' > ''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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! Can be non-octave? !! Etymology | ! Can be non-octave? !! Etymology | ||
|- | |- | ||
| rowspan="2" | [[1L 1s]] || trivial || triv- || trv | | rowspan="2" | [[1L 1s]] || trivial || triv- || trv | ||
| Yes || The simplest valid mos pattern. | | Yes || The simplest valid mos pattern. | ||
|- | |- | ||
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! Can be non-octave? !! Etymology | ! Can be non-octave? !! Etymology | ||
|- | |- | ||
| [[1L 2s]] || antrial || atri- || at | | [[1L 2s]] || antrial || atri- || at | ||
| Yes || Opposite pattern of 2L 1s, with broader range. Shortening of ''anti-trial''. | | Yes || Opposite pattern of 2L 1s, with broader range. Shortening of ''anti-trial''. | ||
|- | |- | ||
| [[2L 1s]] || trial || tri- || t | | [[2L 1s]] || trial || tri- || t | ||
| Yes || From tri- for 3. | | Yes || From tri- for 3. | ||
|- | |- | ||
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! Can be non-octave? !! Etymology | ! Can be non-octave? !! Etymology | ||
|- | |- | ||
| [[1L 3s]] || antetric || atetra- || att | | [[1L 3s]] || antetric || atetra- || att | ||
| Yes || Opposite pattern of 3L 1s, with broader range. Shortening of ''anti-tetric''. | | Yes || Opposite pattern of 3L 1s, with broader range. Shortening of ''anti-tetric''. | ||
|- | |- | ||
| [[2L 2s]] || biwood || biwd- || bw | | [[2L 2s]] || biwood || biwd- || bw | ||
| No (octave-only) || Blackwood[10] and whitewood[14] generalized to 2 periods. | | No (octave-only) || Blackwood[10] and whitewood[14] generalized to 2 periods. | ||
|- | |- | ||
| [[3L 1s]] || tetric || tetra- || tt | | [[3L 1s]] || tetric || tetra- || tt | ||
| Yes || From tetra- for 4. | | Yes || From tetra- for 4. | ||
|- | |- | ||
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! Can be non-octave? !! Etymology | ! Can be non-octave? !! Etymology | ||
|- | |- | ||
| [[1L 4s]] || pedal || ped- || pd | | [[1L 4s]] || pedal || ped- || pd | ||
| Yes || From Latin ''ped'', for ''foot''; one big toe and four small toes. | | Yes || From Latin ''ped'', for ''foot''; one big toe and four small toes. | ||
|- | |- | ||
| [[2L 3s]] || pentic || pent- || pt | | [[2L 3s]] || pentic || pent- || pt | ||
| Yes || Common pentatonic; from penta- for 5. | | Yes || Common pentatonic; from penta- for 5. | ||
|- | |- | ||
| [[3L 2s]] || antipentic || apent- || apt | | [[3L 2s]] || antipentic || apent- || apt | ||
| Yes || Opposite pattern of 2L 3s. | | Yes || Opposite pattern of 2L 3s. | ||
|- | |- | ||
| [[4L 1s]] || manual || manu- || mnu | | [[4L 1s]] || manual || manu- || mnu | ||
| Yes || From Latin ''manus'', for ''hand''; one thumb and four longer fingers. | | Yes || From Latin ''manus'', for ''hand''; one thumb and four longer fingers. | ||
|} | |} | ||
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! Name | ! Name | ||
|- | |- | ||
| rowspan="5" | ''2L 2s'' | | rowspan="5" | ''2L 2s'' | ||
| rowspan="5" | biwood<br />''(formerly unnamed)'' | | rowspan="5" | biwood<br />''(formerly unnamed)'' | ||
| rowspan="2" | 4L 2s | | rowspan="2" | 4L 2s | ||
| rowspan="2" | citric<br />''(formerly lemon)'' | | rowspan="2" | citric<br />''(formerly lemon)'' | ||
| 4L 6s | | 4L 6s | ||
| lime<br />''(formerly dipentic)'' | | lime<br />''(formerly dipentic)'' | ||
| | | | ||
| | | | ||
|- | |- | ||
| 6L 4s | | 6L 4s | ||
| lemon<br />''(formerly antidipentic)'' | | lemon<br />''(formerly antidipentic)'' | ||
| | | | ||
| | | | ||
|- | |- | ||
| rowspan="3" | 2L 4s | | rowspan="3" | 2L 4s | ||
| rowspan="3" | malic<br />''(formerly antilemon)'' | | rowspan="3" | malic<br />''(formerly antilemon)'' | ||
| 6L 2s | | 6L 2s | ||
| ekic<br />''(formerly echidnoid)'' | | ekic<br />''(formerly echidnoid)'' | ||
| | | | ||
| | | | ||
|- | |- | ||
| rowspan="2" | 2L 6s | | rowspan="2" | 2L 6s | ||
| rowspan="2" | subaric<br />''(formerly antiechidnoid)'' | | rowspan="2" | subaric<br />''(formerly antiechidnoid)'' | ||
| 8L 2s | | 8L 2s | ||
| taric<br />''(formerly antidimanic)'' | | taric<br />''(formerly antidimanic)'' | ||
|- | |- | ||
| 2L 8s | | 2L 8s | ||
| jaric<br />''(formerly dimanic)'' | | jaric<br />''(formerly dimanic)'' | ||
|} | |} | ||