TAMNAMS/Appendix: Difference between revisions

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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'''}}.

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|~~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.

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 ==

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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.

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! 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.

|-

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! 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''.

|-

| [[2L 1s]] || trial || tri- || t

| [[2L 1s]] || trial || tri- || t

| Yes || From tri- for 3.

|-

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! 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''.

|-

| [[2L 2s]] || biwood || biwd- || bw

| [[2L 2s]] || biwood || biwd- || bw

| 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.

|-

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! 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.

|-

| [[2L 3s]] || pentic || pent- || pt

| [[2L 3s]] || pentic || pent- || pt

| Yes || Common pentatonic; from penta- for 5.

|-

| [[3L 2s]] || antipentic || apent- || apt

| [[3L 2s]] || antipentic || apent- || apt

| 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.

|}

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! Name

|-

| rowspan="5" | ''2L 2s''

| rowspan="5" | ''2L 2s''

| rowspan="5" | biwood ''(formerly unnamed)''

| rowspan="2" | 4L 2s

| rowspan="2" | 4L 2s

| rowspan="2" | citric ''(formerly lemon)''

| 4L 6s

| 4L 6s

| lime ''(formerly dipentic)''

|

|-

| 6L 4s

| 6L 4s

| lemon ''(formerly antidipentic)''

|

|-

| rowspan="3" | 2L 4s

| rowspan="3" | 2L 4s

| rowspan="3" | malic ''(formerly antilemon)''

| 6L 2s

| 6L 2s

| ekic ''(formerly echidnoid)''

|

|-

| rowspan="2" | 2L 6s

| rowspan="2" | 2L 6s

| rowspan="2" | subaric ''(formerly antiechidnoid)''

| 8L 2s

| 8L 2s

| taric ''(formerly antidimanic)''

|-

| 2L 8s

| 2L 8s

| jaric ''(formerly dimanic)''

|}

@@ Line 167: / Line 167: @@
 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 [[12L 17s|12L&nbsp;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.
+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&nbsp;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 ==
@@ 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.
@@ Line 363: / Line 363: @@
 ! Can be non-octave? !! Etymology
 |-
-| rowspan="2" | [[1L 1s]] || trivial || triv- || trv
+| rowspan="2" | [[1L&nbsp;1s]] || trivial || triv- || trv
 | Yes || The simplest valid mos pattern.
 |-
@@ Line 377: / Line 377: @@
 ! Can be non-octave? !! Etymology
 |-
-| [[1L 2s]] || antrial || atri- || at
+| [[1L&nbsp;2s]] || antrial || atri- || at
 | Yes || Opposite pattern of 2L&nbsp;1s, with broader range. Shortening of ''anti-trial''.
 |-
-| [[2L 1s]] || trial || tri- || t
+| [[2L&nbsp;1s]] || trial || tri- || t
 | Yes || From tri- for 3.
 |-
@@ Line 388: / Line 388: @@
 ! Can be non-octave? !! Etymology
 |-
-| [[1L 3s]] || antetric || atetra- || att
+| [[1L&nbsp;3s]] || antetric || atetra- || att
 | Yes || Opposite pattern of 3L&nbsp;1s, with broader range. Shortening of ''anti-tetric''.
 |-
-| [[2L 2s]] || biwood || biwd- || bw
+| [[2L&nbsp;2s]] || biwood || biwd- || bw
 | No (octave-only) || Blackwood[10] and whitewood[14] generalized to 2 periods.
 |-
-| [[3L 1s]] || tetric || tetra- || tt
+| [[3L&nbsp;1s]] || tetric || tetra- || tt
 | Yes || From tetra- for 4.
 |-
@@ Line 402: / Line 402: @@
 ! Can be non-octave? !! Etymology
 |-
-| [[1L 4s]] || pedal || ped- || pd
+| [[1L&nbsp;4s]] || pedal || ped- || pd
 | Yes || From Latin ''ped'', for ''foot''; one big toe and four small toes.
 |-
-| [[2L 3s]] || pentic || pent- || pt
+| [[2L&nbsp;3s]] || pentic || pent- || pt
 | Yes || Common pentatonic; from penta- for 5.
 |-
-| [[3L 2s]] || antipentic || apent- || apt
+| [[3L&nbsp;2s]] || antipentic || apent- || apt
 | Yes || Opposite pattern of 2L&nbsp;3s.
 |-
-| [[4L 1s]] || manual || manu- || mnu
+| [[4L&nbsp;1s]] || manual || manu- || mnu
 | Yes || From Latin ''manus'', for ''hand''; one thumb and four longer fingers.
 |}
@@ Line 469: / Line 469: @@
 ! Name
 |-
-| rowspan="5" | ''2L 2s''
+| rowspan="5" | ''2L&nbsp;2s''
 | rowspan="5" | biwood<br />''(formerly unnamed)''
-| rowspan="2" | 4L 2s
+| rowspan="2" | 4L&nbsp;2s
 | rowspan="2" | citric<br />''(formerly lemon)''
-| 4L 6s
+| 4L&nbsp;6s
 | lime<br />''(formerly dipentic)''
 |
 |
 |-
-| 6L 4s
+| 6L&nbsp;4s
 | lemon<br />''(formerly antidipentic)''
 |
 |
 |-
-| rowspan="3" | 2L 4s
+| rowspan="3" | 2L&nbsp;4s
 | rowspan="3" | malic<br />''(formerly antilemon)''
-| 6L 2s
+| 6L&nbsp;2s
 | ekic<br />''(formerly echidnoid)''
 |
 |
 |-
-| rowspan="2" | 2L 6s
+| rowspan="2" | 2L&nbsp;6s
 | rowspan="2" | subaric<br />''(formerly antiechidnoid)''
-| 8L 2s
+| 8L&nbsp;2s
 | taric<br />''(formerly antidimanic)''
 |-
-| 2L 8s
+| 2L&nbsp;8s
 | jaric<br />''(formerly dimanic)''
 |}