TAMNAMS/Appendix - Revision history

ArrowHead294 at 15:39, 17 April 2025

2025-04-17T15:39:21Z

← Older revision		Revision as of 15:39, 17 April 2025
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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'' {{=}} ~~&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;~~ ~~&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''<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.
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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'' {{=}} ~~&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'' > ''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.

ArrowHead294 at 15:37, 17 April 2025

2025-04-17T15:37:23Z

← Older revision		Revision as of 15:37, 17 April 2025
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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 [[~~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 ==		== Reasoning for mos interval names ==
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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)''
	\|}		\|}

ArrowHead294: /* Reasoning for mos interval names */

2025-02-27T15:55:12Z

Reasoning for mos interval names

ArrowHead294: Not needed

2025-02-26T19:11:33Z

Not needed

← Older revision		Revision as of 19:11, 26 February 2025
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	{{breadcrumb}}		{{breadcrumb}}
	This page is an appendix to '''[[TAMNAMS]]'''.		This page is an appendix to '''TAMNAMS'''.

	== Reasoning for step ratio names ==		== Reasoning for step ratio names ==

ArrowHead294 at 19:11, 26 February 2025

2025-02-26T19:11:20Z

← Older revision		Revision as of 19:11, 26 February 2025
Line 1:		Line 1:
	{{breadcrumb}}		{{breadcrumb}}
	This page is an appendix to '''TAMNAMS'''.		This page is an appendix to '''[[TAMNAMS]]'''.

	== Reasoning for step ratio names ==		== Reasoning for step ratio names ==

ArrowHead294: /* Reasonings for specific names */

2025-02-26T19:10:33Z

Reasonings for specific names

← Older revision		Revision as of 19:10, 26 February 2025
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	==== Machinoid (5L 1s)====		==== Machinoid (5L 1s)====
	[[Machine]] is the 5&6 temperament in the 2.9.7.11 subgroup with a comma list of 64/63 and 99/98.		[[Machine]] is the {{nowrap\|5 & 6}} temperament in the 2.9.7.11 subgroup with a comma list of 64/63 and 99/98.

	This temperament is supported by {{Optimal ET sequence\| 5, 6, 11, 12, 16, 17, 22, 23, 27, 28 and 33 }} equal divisions, many of which correspond to both simple tunings ({{nowrap\|L:s {{=}} 2:1}}, 3:1, 3:2, etc) and degenerate tunings ({{nowrap\|L:s {{=}} 1:1}} or 1:0) for 5L 1s. Non-patent val tunings include {{nowrap\|5 + 5 {{=}} 10e\|5 + 10e + 12 {{=}} 21be\|and 5 + 5 + 5 + 5 + 6 {{=}} 26qe}}; these are mentioned here for demonstrating virtual completeness of the tuning range, as is 33edo to show 11edo's strength as a tuning.		This temperament is supported by {{Optimal ET sequence\| 5, 6, 11, 12, 16, 17, 22, 23, 27, 28 and 33 }} equal divisions, many of which correspond to both simple tunings ({{nowrap\|L:s {{=}} 2:1}}, 3:1, 3:2, etc) and degenerate tunings ({{nowrap\|L:s {{=}} 1:1}} or 1:0) for 5L 1s. Non-patent val tunings include {{nowrap\|5 + 5 {{=}} 10e\|5 + 10e + 12 {{=}} 21be\|and 5 + 5 + 5 + 5 + 6 {{=}} 26qe}}; these are mentioned here for demonstrating virtual completeness of the tuning range, as is 33edo to show 11edo's strength as a tuning.

	==== Sephiroid (3L 7s) ====		==== Sephiroid (3L 7s) ====
	[[Sephiroth]] is the 3&10 temperament in the 2.5.11.13.17.21 subgroup with commas including 65/64, 85/84, 105/104, 169/168, 170/169, 221/220, 273/272, 275/273.		[[Sephiroth]] is the {{nowrap\|3 & 10}} temperament in the 2.5.11.13.17.21 subgroup with commas including 65/64, 85/84, 105/104, 169/168, 170/169, 221/220, 273/272, 275/273.

	This temperament is supported by {{Optimal ET sequence\| 3, 10, 13, 16, 23, and 26 }} equal divisions, with non-patent val tunings including 6eg, 7e, 19eg, 20e, 29g, 32egq, 33ce, 36c. Like with that of 5L 1s, these represent both simple and degenerate tunings for 3L 7s. Extreme tunings, such as 7e, may lie outside the mos's step ratio spectrum, although such tunings are generally not considered good tunings.		This temperament is supported by {{Optimal ET sequence\| 3, 10, 13, 16, 23, and 26 }} equal divisions, with non-patent val tunings including 6eg, 7e, 19eg, 20e, 29g, 32egq, 33ce, 36c. Like with that of 5L 1s, these represent both simple and degenerate tunings for 3L 7s. Extreme tunings, such as 7e, may lie outside the mos's step ratio spectrum, although such tunings are generally not considered good tunings.

	==== Dicoid (7L 3s) ====		==== Dicoid (7L 3s) ====
	[[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 {{nowrap\|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}}.

ArrowHead294 at 19:09, 26 February 2025

2025-02-26T19:09:18Z

ArrowHead294: /* Extended spectrum */ Compress table

2025-02-26T19:08:22Z

Extended spectrum: Compress table

← Older revision		Revision as of 19:08, 26 February 2025
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	=== Extended spectrum ===		=== Extended spectrum ===
	{\| class="wikitable"		{\| class="wikitable" style="text-align: center;"
	\|+ style="font-size: 105%;" \| Extended spectrum of step ratio ranges and specific step ratios		\|+ style="font-size: 105%;" \| Extended spectrum of step ratio ranges and specific step ratios
	\|-		\|-
	! colspan="3" \| Central ranges		! colspan="3" \| Central ranges
	! colspan="2" \| Extended ranges		! colspan="2" \| Extended ranges
	! Specific step ratios		! Specific<br />step ratios
	! Notes		! Notes
	\|-		\|-
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	\|		\|
	\| colspan="2" \|		\| colspan="2" \|
	\| '''1:1 (equalized)'''		\| '''1:1<br />(equalized)'''
	\| Trivial/pathological		\| Trivial/pathological
	\|-		\|-
	\| rowspan="9" \| 1:1 to 2:1 (soft-of-basic)		\| rowspan="9" \| 1:1 to 2:1<br />(soft-of-basic)
	\| colspan="2" rowspan="3" \| 1:1 to 4:3 (ultrasoft)		\| colspan="2" rowspan="3" \| 1:1 to 4:3<br />(ultrasoft)
	\| colspan="2" \| 1:1 to 6:5 (pseudoequalized)		\| colspan="2" \| 1:1 to 6:5<br />(pseudoequalized)
	\|		\|
	\|		\|
	\|-		\|-
	\| colspan="2" \|		\| colspan="2" \|
	\| '''6:5 (semiequalized)'''		\| '''6:5<br />(semiequalized)'''
	\|		\|
	\|-		\|-
	\| colspan="2" \| 6:5 to 4:3 (ultrasoft)		\| colspan="2" \| 6:5 to 4:3<br />(ultrasoft)
	\|		\|
	\|		\|
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	\|		\|
	\| colspan="2" \|		\| colspan="2" \|
	\| '''4:3 (supersoft)'''		\| '''4:3<br />(supersoft)'''
	\| rowspan="13" \| Nonextreme range, as detailed by central spectrum		\| rowspan="13" \| Nonextreme range, as detailed<br />by central spectrum
	\|-		\|-
	\| colspan="2" \| 4:3 to 3:2 (parasoft)		\| colspan="2" \| 4:3 to 3:2<br />(parasoft)
	\| colspan="2" \| 4:3 to 3:2 (parasoft)		\| colspan="2" \| 4:3 to 3:2<br />(parasoft)
	\|		\|
	\|-		\|-
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	\|		\|
	\| colspan="2" \|		\| colspan="2" \|
	\| '''3:2 (soft)'''		\| '''3:2<br />(soft)'''
	\|-		\|-
	\| rowspan="3" \| 3:2 to 2:1 (hyposoft)		\| rowspan="3" \| 3:2 to 2:1<br />(hyposoft)
	\| 3:2 to 5:3 (quasisoft)		\| 3:2 to 5:3<br />(quasisoft)
	\| colspan="2" \| 3:2 to 5:3 (quasisoft)		\| colspan="2" \| 3:2 to 5:3<br />(quasisoft)
	\|		\|
	\|-		\|-
	\|		\|
	\| colspan="2" \|		\| colspan="2" \|
	\| '''5:3 (semisoft)'''		\| '''5:3<br />(semisoft)'''
	\|-		\|-
	\| 5:3 to 2:1 (minisoft)		\| 5:3 to 2:1<br />(minisoft)
	\| colspan="2" \| 5:3 to 2:1 (minisoft)		\| colspan="2" \| 5:3 to 2:1<br />(minisoft)
	\|		\|
	\|-		\|-
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	\|		\|
	\| colspan="2" \|		\| colspan="2" \|
	\| '''2:1 (basic)'''		\| '''2:1<br />(basic)'''
	\|-		\|-
	\| rowspan="11" \| 2:1 to 1:0 (hard-of-basic)		\| rowspan="11" \| 2:1 to 1:0<br />(hard-of-basic)
	\| rowspan="3" \| 2:1 to 3:1 (hypohard)		\| rowspan="3" \| 2:1 to 3:1<br />(hypohard)
	\| 2:1 to 5:2 (minihard)		\| 2:1 to 5:2<br />(minihard)
	\| colspan="2" \| 2:1 to 5:2 (minihard)		\| colspan="2" \| 2:1 to 5:2<br />(minihard)
	\|		\|
	\|-		\|-
	\|		\|
	\| colspan="2" \|		\| colspan="2" \|
	\| '''5:2 (semihard)'''		\| '''5:2<br />(semihard)'''
	\|-		\|-
	\| 5:2 to 3:1 (quasihard)		\| 5:2 to 3:1<br />(quasihard)
	\| colspan="2" \| 5:2 to 3:1 (quasihard)		\| colspan="2" \| 5:2 to 3:1<br />(quasihard)
	\|		\|
	\|-		\|-
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	\|		\|
	\| colspan="2" \|		\| colspan="2" \|
	\| '''3:1 (hard)'''		\| '''3:1<br />(hard)'''
	\|-		\|-
	\| colspan="2" \| 3:1 to 4:1 (parahard)		\| colspan="2" \| 3:1 to 4:1<br />(parahard)
	\| colspan="2" \| 3:1 to 4:1 (parahard)		\| colspan="2" \| 3:1 to 4:1<br />(parahard)
	\|		\|
	\|-		\|-
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	\|		\|
	\| colspan="2" \|		\| colspan="2" \|
	\| '''4:1 (superhard)'''		\| '''4:1<br />(superhard)'''
	\|-		\|-
	\| colspan="2" rowspan="5" \| 4:1 to 1:0 (ultrahard)		\| colspan="2" rowspan="5" \| 4:1 to 1:0<br />(ultrahard)
	\| rowspan="3" \| 4:1 to 10:1 (ultrahard)		\| rowspan="3" \| 4:1 to 10:1<br />(ultrahard)
	\| 4:1 to 6:1 (hyperhard)		\| 4:1 to 6:1<br />(hyperhard)
	\|		\|
	\|		\|
	\|-		\|-
	\|		\|
	\| '''6:1 (extrahard)'''		\| '''6:1<br />(extrahard)'''
	\|		\|
	\|-		\|-
	\| 6:1 to 10:1 (clustered)		\| 6:1 to 10:1<br />(clustered)
	\|		\|
	\|		\|
	\|-		\|-
	\| colspan="2" \|		\| colspan="2" \|
	\| '''10:1 (semicollapsed)'''		\| '''10:1<br />(semicollapsed)'''
	\|		\|
	\|-		\|-
	\| colspan="2" \| 10:1 to 1:0 (pseudocollapsed)		\| colspan="2" \| 10:1 to 1:0<br />(pseudocollapsed)
	\|		\|
	\|		\|
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	\|		\|
	\| colspan="2" \|		\| colspan="2" \|
	\| '''1:0 (collapsed)'''		\| '''1:0<br />(collapsed)'''
	\| Trivial/pathological		\| Trivial/pathological
	\|}		\|}

ArrowHead294: Move note to footnotes section and make subscripts

2024-11-18T14:37:39Z

Move note to footnotes section and make subscripts

Ganaram inukshuk: don't assume people already know what the symbols for ceil and floor are

2024-11-18T05:23:29Z

don't assume people already know what the symbols for ceil and floor are

← Older revision		Revision as of 05:23, 18 November 2024
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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'' {{=}} &lceil;''m2''/''m1''&rceil;}} and {{nowrap\|''v'' {{=}} &lfloor;''m2''/''m1''&rfloor;}}.		### 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;''m2''/''m1''&rceil;}} and {{nowrap\|''v'' {{=}} &lfloor;''m2''/''m1''&rfloor;}}. (&lceil; &rceil; denotes the ceiling function and &lfloor; &rfloor; denotes the floor function.)
	#### 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.

← Older revision		Revision as of 15:39, 17 April 2025
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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'' {{=}} ~~&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;~~ ~~&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''<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.
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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'' {{=}} ~~&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'' > ''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.

← Older revision		Revision as of 05:23, 18 November 2024
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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'' {{=}} &lceil;''m2''/''m1''&rceil;}} and {{nowrap\|''v'' {{=}} &lfloor;''m2''/''m1''&rfloor;}}.		### 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;''m2''/''m1''&rceil;}} and {{nowrap\|''v'' {{=}} &lfloor;''m2''/''m1''&rfloor;}}. (&lceil; &rceil; denotes the ceiling function and &lfloor; &rfloor; denotes the floor function.)
	#### 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.

← Older revision		Revision as of 15:55, 27 February 2025
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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 ~~{{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 (''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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	### 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'' {{=}} &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; &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'' {{=}} &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; &rceil; denotes the ceiling function and &lfloor; &rfloor; 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.

← Older revision		Revision as of 19:09, 26 February 2025
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	=== Terminology and final notes ===		=== 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'''}}.		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:		## 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'')}}.
	### Let {{nowrap\|''z'' {{=}} ''m''<sub>2</sub> mod ''m''<sub>1</sub>}} and {{nowrap\|''w'' {{=}} ''m''<sub>1</sub> ~~−~~ ''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.
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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 generator is "L" and its complement is "s".		## 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):		# 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''.
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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:		## 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'')}}.
	### Let {{nowrap\|''z'' {{=}} ''m''<sub>2</sub> mod ''m''<sub>1</sub>}} and {{nowrap\|''w'' {{=}} ''m1 ~~−~~ 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.

← Older revision		Revision as of 14:37, 18 November 2024
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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\|''m1'' {{=}} min(''x'', ''y'')}} and {{nowrap\|''m2'' {{=}} max(''x'', ''y'')}}.		### Let {{nowrap\|''m''<sub>1</sub> {{=}} min(''x'', ''y'')}} and {{nowrap\|''m''<sub>2</sub> {{=}} max(''x'', ''y'')}}.
	### Let {{nowrap\|''z'' {{=}} ''m2'' mod ''m1''}} and {{nowrap\|''w'' {{=}} ''m1'' − ''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'' {{=}} &lceil;''m2''/''m1''&rceil;}} and {{nowrap\|''v'' {{=}} &lfloor;''m2''/''m1''&rfloor;}}.		### 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; &rceil; denotes the ceiling function and &lfloor; &rfloor; 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\|''m1'' {{=}} min(''x'', ''y'')}} and {{nowrap\|''m2'' {{=}} max(''x'', ''y'')}}.		### Let {{nowrap\|''m''<sub>1</sub> {{=}} min(''x'', ''y'')}} and {{nowrap\|''m''<sub>2</sub> {{=}} max(''x'', ''y'')}}.
	### Let {{nowrap\|''z'' {{=}} ''m2'' mod ''m1''}} and {{nowrap\|''w'' {{=}} ''m1 − 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'' {{=}} &lceil;''m2''/''m1''&rceil;}} and {{nowrap\|''v'' {{=}} &lfloor;''m2''/''m1''&rfloor;}}. ~~(&lceil; &rceil; denotes the ceiling function and &lfloor; &rfloor; denotes the floor function.)~~		### 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" />
	#### 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]]