User:Lucius Chiaraviglio/Musical Mad Science - Revision history

Lucius Chiaraviglio: /* Musical Mad Science Musings on Diatonicized Third-Tone Sub-Chromaticism */ No neat one size fits all mapping for third-tones

2025-09-06T07:55:09Z

Musical Mad Science Musings on Diatonicized Third-Tone Sub-Chromaticism: No neat one size fits all mapping for third-tones

← Older revision		Revision as of 07:55, 6 September 2025
Line 134:		Line 134:
	## For those EDOs having a less flat fifth (but also including 19), the extension is actually much simpler, needing only 11 bright generators to get ~[[7/4]] — optimal ET sequence: 17c, 19, 36, 55 of these, only 36 qualifies for Mothra. The comma for this is a spectrum of less complicated (but still very complicated) commas from \|47 0 0 1 0 0 0 0 -11⟩ ~ \|-74 -11 11 1 0 0 0 0 0 0 0 0 0 0 0 11⟩, of which \|3 -4 4 1 0 0 0 0 -7 0 0 0 0 0 0 4⟩ has the closest 53-limit just intonation value to 0 (2.35850949135{{c}}, made using 7 instances of 23/16 and 4 instances of 384/265).		## For those EDOs having a less flat fifth (but also including 19), the extension is actually much simpler, needing only 11 bright generators to get ~[[7/4]] — optimal ET sequence: 17c, 19, 36, 55 of these, only 36 qualifies for Mothra. The comma for this is a spectrum of less complicated (but still very complicated) commas from \|47 0 0 1 0 0 0 0 -11⟩ ~ \|-74 -11 11 1 0 0 0 0 0 0 0 0 0 0 0 11⟩, of which \|3 -4 4 1 0 0 0 0 -7 0 0 0 0 0 0 4⟩ has the closest 53-limit just intonation value to 0 (2.35850949135{{c}}, made using 7 instances of 23/16 and 4 instances of 384/265).
	## This leaves out 129edo, which we don't want to miss because it has a very accurate 7th harmonic; for 129edo, if we want a strong extension, we need -44 bright generators (which is +44 dark generators) — optimal ET sequence: 55, 74, 129; however, this is overly complex for all 3 members, since 55 and 74 also belong to much simpler strong extensions, while 129 qualifies for Mothra. For 129edo, this means that we can proceed by -3 bright generators (+3 dark generators), octave-reduce, and divide by 3, which simplifies to +1 dark generator and +1/3 octave; furthermore, this also works for the other EDO sizes divisible by 3, which suggests the name Alpha-Mothra (since these are both tricot and alpha-tricot, which simplifies to triploid alpha-dark_generator); optimal ET sequence: 36, 93, 129. The comma for the simplified form is a spectrum of merely highly complicated commas from [[4173281/4194304]] (\|3 -4 4 1 0 0 0 0 -7 0 0 0 0 0 0 4⟩) ~ [[18966528/18609625]] (\|11 3 -3 3 0 0 0 0 0 0 0 0 0 0 0 -3⟩), of which [[544341/542720]] (\|-11 1 -1 3 0 0 0 0 2 0 0 0 0 0 0 -1⟩) has the closest 53-limit just intonation value to 0 (5.1631554689{{c}}, smaller than the [[1029/1024\|Gamelisma]] for which it substitutes).		## This leaves out 129edo, which we don't want to miss because it has a very accurate 7th harmonic; for 129edo, if we want a strong extension, we need -44 bright generators (which is +44 dark generators) — optimal ET sequence: 55, 74, 129; however, this is overly complex for all 3 members, since 55 and 74 also belong to much simpler strong extensions, while 129 qualifies for Mothra. For 129edo, this means that we can proceed by -3 bright generators (+3 dark generators), octave-reduce, and divide by 3, which simplifies to +1 dark generator and +1/3 octave; furthermore, this also works for the other EDO sizes divisible by 3, which suggests the name Alpha-Mothra (since these are both tricot and alpha-tricot, which simplifies to triploid alpha-dark_generator); optimal ET sequence: 36, 93, 129. The comma for the simplified form is a spectrum of merely highly complicated commas from [[4173281/4194304]] (\|3 -4 4 1 0 0 0 0 -7 0 0 0 0 0 0 4⟩) ~ [[18966528/18609625]] (\|11 3 -3 3 0 0 0 0 0 0 0 0 0 0 0 -3⟩), of which [[544341/542720]] (\|-11 1 -1 3 0 0 0 0 2 0 0 0 0 0 0 -1⟩) has the closest 53-limit just intonation value to 0 (5.1631554689{{c}}, smaller than the [[1029/1024\|Gamelisma]] for which it substitutes).
	# Embedded within 17L 2s is the standard diatonic scale [[5L 2s]], of which each large step consists of 3 large steps of 17L 2s, and each small step (the standard diatonic semitone) consists of 1 large step + 1 small step of 17L 2s. In the parent of 17L 2s, which is [[2L 15s]], the large step of 2L 15s; is the standard diatonic semitone of 5L 2s.		# Embedded within 17L 2s is the standard diatonic scale [[5L 2s]], of which each large step consists of 3 large steps of 17L 2s, and each small step (the standard diatonic semitone) consists of 1 large step + 1 small step of 17L 2s. In the parent of 17L 2s, which is [[2L 15s]], the large step of 2L 15s; is the standard diatonic semitone of 5L 2s. This points the way to a rank-3 elevation of this temperament.
	# The large step of 17L 2s maps to the septimal third-tone ~[[28/27]], at least in 19edo, 55edo, 36edo, 53edo, and 17edo.		# The large step of 17L 2s maps to the septimal third-tone ~[[28/27]], at least in 19edo, 55edo, 36edo, 53edo, and 17edo; however, this mapping stops working fairly quickly, for instance failing for 74edo and 93edo, for which it is necessary to use the large tridecimal third-tone ~[[26/25]] instead (which, however, fails for 36edo and 17edo as patent val and 17c), which means that no universal or near-universal third-tone mapping will suffice other than that obtained by stacking and octave-reducing pairs of bright generators.
	# The dark and bright generators of 17L 2s (and 2L 15s) consist of stacks of 7 and 8 (respectively) ~~septimal~~ third-tones + 1 standard diatonic semitone; this maps correctly at least for 19edo, 55edo, 36edo, 53edo, and 17edo. Need to choose which mapping of diatonic semitone to use to get the widest range of EDOs for which this maps correctly. These potential mappings for the bright generator have not yet been added to the table (actually group of tables) below of odd harmonics for various EDO values supporting 17L 2s.		# The dark and bright generators of 17L 2s (and 2L 15s) consist of stacks of 7 and 8 (respectively) third-tones + 1 standard diatonic semitone; this maps correctly at least for 19edo, 55edo, 36edo, 53edo, and 17edo. Need to choose which mapping of diatonic semitone to use to get the widest range of EDOs for which this maps correctly. These potential mappings for the bright generator have not yet been added to the table (actually group of tables) below of odd harmonics for various EDO values supporting 17L 2s.

	Added: [[User:Lucius Chiaraviglio\|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio\|talk]]) 08:20, 4 April 2025 (UTC)<br>		Added: [[User:Lucius Chiaraviglio\|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio\|talk]]) 08:20, 4 April 2025 (UTC)<br>
	Last modified: [[User:Lucius Chiaraviglio\|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio\|talk]]) 22:06, ~~23 August~~ 2025 (UTC)		Last modified: [[User:Lucius Chiaraviglio\|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio\|talk]]) 07:55, 6 September 2025 (UTC)

	=== Table of odd harmonics for various EDO values supporting 17L 2s ===		=== Table of odd harmonics for various EDO values supporting 17L 2s ===

Lucius Chiaraviglio: /* Musical Mad Science Musings on Diatonicized Third-Tone Sub-Chromaticism */ Mapping the generators of 17L 2s (and its parent 2L 2s) to septimal third tones stacked with the standard diatonic semitone

2025-08-23T22:06:47Z

Musical Mad Science Musings on Diatonicized Third-Tone Sub-Chromaticism: Mapping the generators of 17L 2s (and its parent 2L 2s) to septimal third tones stacked with the standard diatonic semitone

← Older revision		Revision as of 22:06, 23 August 2025
Line 134:		Line 134:
	## For those EDOs having a less flat fifth (but also including 19), the extension is actually much simpler, needing only 11 bright generators to get ~[[7/4]] — optimal ET sequence: 17c, 19, 36, 55 of these, only 36 qualifies for Mothra. The comma for this is a spectrum of less complicated (but still very complicated) commas from \|47 0 0 1 0 0 0 0 -11⟩ ~ \|-74 -11 11 1 0 0 0 0 0 0 0 0 0 0 0 11⟩, of which \|3 -4 4 1 0 0 0 0 -7 0 0 0 0 0 0 4⟩ has the closest 53-limit just intonation value to 0 (2.35850949135{{c}}, made using 7 instances of 23/16 and 4 instances of 384/265).		## For those EDOs having a less flat fifth (but also including 19), the extension is actually much simpler, needing only 11 bright generators to get ~[[7/4]] — optimal ET sequence: 17c, 19, 36, 55 of these, only 36 qualifies for Mothra. The comma for this is a spectrum of less complicated (but still very complicated) commas from \|47 0 0 1 0 0 0 0 -11⟩ ~ \|-74 -11 11 1 0 0 0 0 0 0 0 0 0 0 0 11⟩, of which \|3 -4 4 1 0 0 0 0 -7 0 0 0 0 0 0 4⟩ has the closest 53-limit just intonation value to 0 (2.35850949135{{c}}, made using 7 instances of 23/16 and 4 instances of 384/265).
	## This leaves out 129edo, which we don't want to miss because it has a very accurate 7th harmonic; for 129edo, if we want a strong extension, we need -44 bright generators (which is +44 dark generators) — optimal ET sequence: 55, 74, 129; however, this is overly complex for all 3 members, since 55 and 74 also belong to much simpler strong extensions, while 129 qualifies for Mothra. For 129edo, this means that we can proceed by -3 bright generators (+3 dark generators), octave-reduce, and divide by 3, which simplifies to +1 dark generator and +1/3 octave; furthermore, this also works for the other EDO sizes divisible by 3, which suggests the name Alpha-Mothra (since these are both tricot and alpha-tricot, which simplifies to triploid alpha-dark_generator); optimal ET sequence: 36, 93, 129. The comma for the simplified form is a spectrum of merely highly complicated commas from [[4173281/4194304]] (\|3 -4 4 1 0 0 0 0 -7 0 0 0 0 0 0 4⟩) ~ [[18966528/18609625]] (\|11 3 -3 3 0 0 0 0 0 0 0 0 0 0 0 -3⟩), of which [[544341/542720]] (\|-11 1 -1 3 0 0 0 0 2 0 0 0 0 0 0 -1⟩) has the closest 53-limit just intonation value to 0 (5.1631554689{{c}}, smaller than the [[1029/1024\|Gamelisma]] for which it substitutes).		## This leaves out 129edo, which we don't want to miss because it has a very accurate 7th harmonic; for 129edo, if we want a strong extension, we need -44 bright generators (which is +44 dark generators) — optimal ET sequence: 55, 74, 129; however, this is overly complex for all 3 members, since 55 and 74 also belong to much simpler strong extensions, while 129 qualifies for Mothra. For 129edo, this means that we can proceed by -3 bright generators (+3 dark generators), octave-reduce, and divide by 3, which simplifies to +1 dark generator and +1/3 octave; furthermore, this also works for the other EDO sizes divisible by 3, which suggests the name Alpha-Mothra (since these are both tricot and alpha-tricot, which simplifies to triploid alpha-dark_generator); optimal ET sequence: 36, 93, 129. The comma for the simplified form is a spectrum of merely highly complicated commas from [[4173281/4194304]] (\|3 -4 4 1 0 0 0 0 -7 0 0 0 0 0 0 4⟩) ~ [[18966528/18609625]] (\|11 3 -3 3 0 0 0 0 0 0 0 0 0 0 0 -3⟩), of which [[544341/542720]] (\|-11 1 -1 3 0 0 0 0 2 0 0 0 0 0 0 -1⟩) has the closest 53-limit just intonation value to 0 (5.1631554689{{c}}, smaller than the [[1029/1024\|Gamelisma]] for which it substitutes).
			# Embedded within 17L 2s is the standard diatonic scale [[5L 2s]], of which each large step consists of 3 large steps of 17L 2s, and each small step (the standard diatonic semitone) consists of 1 large step + 1 small step of 17L 2s. In the parent of 17L 2s, which is [[2L 15s]], the large step of 2L 15s; is the standard diatonic semitone of 5L 2s.
			# The large step of 17L 2s maps to the septimal third-tone ~[[28/27]], at least in 19edo, 55edo, 36edo, 53edo, and 17edo.
			# The dark and bright generators of 17L 2s (and 2L 15s) consist of stacks of 7 and 8 (respectively) septimal third-tones + 1 standard diatonic semitone; this maps correctly at least for 19edo, 55edo, 36edo, 53edo, and 17edo. Need to choose which mapping of diatonic semitone to use to get the widest range of EDOs for which this maps correctly. These potential mappings for the bright generator have not yet been added to the table (actually group of tables) below of odd harmonics for various EDO values supporting 17L 2s.

	Added: [[User:Lucius Chiaraviglio\|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio\|talk]]) 08:20, 4 April 2025 (UTC)<br>		Added: [[User:Lucius Chiaraviglio\|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio\|talk]]) 08:20, 4 April 2025 (UTC)<br>
	Last modified: [[User:Lucius Chiaraviglio\|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio\|talk]]) 08:07, ~~18 June~~ 2025 (UTC)		Last modified: [[User:Lucius Chiaraviglio\|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio\|talk]]) 22:06, 23 August 2025 (UTC)

	=== Table of odd harmonics for various EDO values supporting 17L 2s ===		=== Table of odd harmonics for various EDO values supporting 17L 2s ===

Lucius Chiaraviglio: /* Various Lumatone mappings */ Insert Shaping Consonance with Harmonic Timbre after this, starting with ''The Physics of Dissonance'' by minutephysics (2025)

2025-08-01T23:06:50Z

Various Lumatone mappings: Insert Shaping Consonance with Harmonic Timbre after this, starting with ''The Physics of Dissonance'' by minutephysics (2025)

← Older revision		Revision as of 23:06, 1 August 2025
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	Moved to [[User:Lucius Chiaraviglio/Keyboard Layout Lab\|Keyboard Layout lab]]: [[User:Lucius Chiaraviglio\|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio\|talk]]) 06:55, 28 March 2025 (UTC)		Moved to [[User:Lucius Chiaraviglio/Keyboard Layout Lab\|Keyboard Layout lab]]: [[User:Lucius Chiaraviglio\|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio\|talk]]) 06:55, 28 March 2025 (UTC)

			== Shaping Consonance with Harmonic Timbre ==

			[https://www.youtube.com/watch?v=tCsl6ZcY9ag&t=1s ''The Physics of Dissonance''] by minutephysics (2025) is an excellent video on how changes in harmonic timbre, including inharmonic partials, can change what counts as consonant or dissonant. Not sure where to put this yet, so putting it here to make sure it doesn't get lost.

			And some more is going to need to go here in the future.

			Added: [[User:Lucius Chiaraviglio\|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio\|talk]]) 23:06, 1 August 2025 (UTC)

	== Musical Mad Science Musings on Diatonicized Chromaticism ==		== Musical Mad Science Musings on Diatonicized Chromaticism ==

Lucius Chiaraviglio: /* Musical Mad Science Musings on Diatonicized Third-Tone Sub-Chromaticism */ Add mapping of ~62/43 for far right column of 17L 2s scale tree

2025-06-23T08:00:28Z

Musical Mad Science Musings on Diatonicized Third-Tone Sub-Chromaticism: Add mapping of ~62/43 for far right column of 17L 2s scale tree

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Lucius Chiaraviglio: Fix interval notation in mapping tables (cleanup in 11L 2s and 17L 2s)

2025-06-21T09:25:56Z

Fix interval notation in mapping tables (cleanup in 11L&nbsp2s and 17L 2s)

Lucius Chiaraviglio: Fix interval notation in mapping tables (last half of 17L 2s)

2025-06-21T09:19:27Z

Fix interval notation in mapping tables (last half of 17L 2s)

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Lucius Chiaraviglio: Fix interval notation in mapping tables (all of 11L 2s; and first half of 17L 2s)

2025-06-21T08:49:58Z

Fix interval notation in mapping tables (all of 11L&nbsp2s; and first half of 17L 2s)

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Lucius Chiaraviglio: /* Table of odd harmonics for various EDO values supporting 17L 2s */ Fix typo in 320edo BrightGen

2025-06-19T06:09:44Z

Table of odd harmonics for various EDO values supporting 17L 2s: Fix typo in 320edo BrightGen

← Older revision		Revision as of 06:09, 19 June 2025
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	{{Harmonics in equal\|303\|intervals=odd\|prec=2\|columns=28\|title=[[303edo]] (L=17, s=7, BrightGen is 160; patent ~13/9 = _; patent ~23/16 = _; patent ~49/34 = _; patent ~62/43 = _; patent ~75/52 = _; patent ~384/265 = _; patent ~59049/40960 = _)}}		{{Harmonics in equal\|303\|intervals=odd\|prec=2\|columns=28\|title=[[303edo]] (L=17, s=7, BrightGen is 160; patent ~13/9 = _; patent ~23/16 = _; patent ~49/34 = _; patent ~62/43 = _; patent ~75/52 = _; patent ~384/265 = _; patent ~59049/40960 = _)}}
	{{Harmonics in equal\|89\|intervals=odd\|prec=2\|columns=28\|title=[[89edo]] (L=5, s=2, BrightGen is 47; patent ~13/9 = 47; patent ~23/16 = 47; patent ~49/34 = 47; patent ~62/43 = 47; ''patent ~75/52 = 48''; patent ~384/265 = 47; ''patent ~59049/40960 = 46'') — Semihard 17L 2s}}		{{Harmonics in equal\|89\|intervals=odd\|prec=2\|columns=28\|title=[[89edo]] (L=5, s=2, BrightGen is 47; patent ~13/9 = 47; patent ~23/16 = 47; patent ~49/34 = 47; patent ~62/43 = 47; ''patent ~75/52 = 48''; patent ~384/265 = 47; ''patent ~59049/40960 = 46'') — Semihard 17L 2s}}
	{{Harmonics in equal\|320\|intervals=odd\|prec=2\|columns=28\|title=[[320edo]] (L=18, s=7, BrightGen is ~~160~~; patent ~13/9 = _; patent ~23/16 = _; patent ~49/34 = _; patent ~62/43 = _; patent ~75/52 = _; patent ~384/265 = _; patent ~59049/40960 = _)}}		{{Harmonics in equal\|320\|intervals=odd\|prec=2\|columns=28\|title=[[320edo]] (L=18, s=7, BrightGen is 169; patent ~13/9 = _; patent ~23/16 = _; patent ~49/34 = _; patent ~62/43 = _; patent ~75/52 = _; patent ~384/265 = _; patent ~59049/40960 = _)}}
	{{Harmonics in equal\|231\|intervals=odd\|prec=2\|columns=28\|title=[[231edo]] (L=13, s=5, BrightGen is 122; ''patent ~13/9 = 123''; ''patent ~23/16 = 121''; ''patent ~49/34 = 121''; patent ~62/43 = 122; ''patent ~75/52 = 121''; ''patent ~384/265 = 124''; ''patent ~59049/40960 = 121'')}}		{{Harmonics in equal\|231\|intervals=odd\|prec=2\|columns=28\|title=[[231edo]] (L=13, s=5, BrightGen is 122; ''patent ~13/9 = 123''; ''patent ~23/16 = 121''; ''patent ~49/34 = 121''; patent ~62/43 = 122; ''patent ~75/52 = 121''; ''patent ~384/265 = 124''; ''patent ~59049/40960 = 121'')}}
	{{Harmonics in equal\|373\|intervals=odd\|prec=2\|columns=28\|title=[[373edo]] (L=21, s=8, BrightGen is 197; patent ~13/9 = _; patent ~23/16 = _; patent ~49/34 = _; patent ~62/43 = _; patent ~75/52 = _; patent ~384/265 = _; patent ~59049/40960 = _)}}		{{Harmonics in equal\|373\|intervals=odd\|prec=2\|columns=28\|title=[[373edo]] (L=21, s=8, BrightGen is 197; patent ~13/9 = _; patent ~23/16 = _; patent ~49/34 = _; patent ~62/43 = _; patent ~75/52 = _; patent ~384/265 = _; patent ~59049/40960 = _)}}
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	Added: [[User:Lucius Chiaraviglio\|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio\|talk]]) 07:42, 8 April 2025 (UTC)<br>		Added: [[User:Lucius Chiaraviglio\|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio\|talk]]) 07:42, 8 April 2025 (UTC)<br>
	Last modified: [[User:Lucius Chiaraviglio\|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio\|talk]]) 06:05, 19 June 2025 (UTC)		Last modified: [[User:Lucius Chiaraviglio\|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio\|talk]]) 06:09, 19 June 2025 (UTC)

Lucius Chiaraviglio: /* Table of odd harmonics for various EDO values supporting 17L 2s */ Continue inserting EDO values from far right column of 11L 2s scale tree (done; need to start filling in mapping values)

2025-06-19T06:05:12Z

Table of odd harmonics for various EDO values supporting 17L 2s: Continue inserting EDO values from far right column of 11L 2s scale tree (done; need to start filling in mapping values)

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Lucius Chiaraviglio: /* Table of odd harmonics for various EDO values supporting 17L 2s */ Insert EDO values from the far right column of the 17L 2s scale tree, with provision for generator mappings (not yet filled in) (safety save)

2025-06-19T05:41:04Z

Table of odd harmonics for various EDO values supporting 17L 2s: Insert EDO values from the far right column of the 17L 2s scale tree, with provision for generator mappings (not yet filled in) (safety save)

Show changes

← Older revision		Revision as of 06:09, 19 June 2025
Line 175:		Line 175:
	{{Harmonics in equal\|303\|intervals=odd\|prec=2\|columns=28\|title=[[303edo]] (L=17, s=7, BrightGen is 160; patent ~13/9 = _; patent ~23/16 = _; patent ~49/34 = _; patent ~62/43 = _; patent ~75/52 = _; patent ~384/265 = _; patent ~59049/40960 = _)}}		{{Harmonics in equal\|303\|intervals=odd\|prec=2\|columns=28\|title=[[303edo]] (L=17, s=7, BrightGen is 160; patent ~13/9 = _; patent ~23/16 = _; patent ~49/34 = _; patent ~62/43 = _; patent ~75/52 = _; patent ~384/265 = _; patent ~59049/40960 = _)}}
	{{Harmonics in equal\|89\|intervals=odd\|prec=2\|columns=28\|title=[[89edo]] (L=5, s=2, BrightGen is 47; patent ~13/9 = 47; patent ~23/16 = 47; patent ~49/34 = 47; patent ~62/43 = 47; ''patent ~75/52 = 48''; patent ~384/265 = 47; ''patent ~59049/40960 = 46'') — Semihard 17L 2s}}		{{Harmonics in equal\|89\|intervals=odd\|prec=2\|columns=28\|title=[[89edo]] (L=5, s=2, BrightGen is 47; patent ~13/9 = 47; patent ~23/16 = 47; patent ~49/34 = 47; patent ~62/43 = 47; ''patent ~75/52 = 48''; patent ~384/265 = 47; ''patent ~59049/40960 = 46'') — Semihard 17L 2s}}
	{{Harmonics in equal\|320\|intervals=odd\|prec=2\|columns=28\|title=[[320edo]] (L=18, s=7, BrightGen is ~~160~~; patent ~13/9 = _; patent ~23/16 = _; patent ~49/34 = _; patent ~62/43 = _; patent ~75/52 = _; patent ~384/265 = _; patent ~59049/40960 = _)}}		{{Harmonics in equal\|320\|intervals=odd\|prec=2\|columns=28\|title=[[320edo]] (L=18, s=7, BrightGen is 169; patent ~13/9 = _; patent ~23/16 = _; patent ~49/34 = _; patent ~62/43 = _; patent ~75/52 = _; patent ~384/265 = _; patent ~59049/40960 = _)}}
	{{Harmonics in equal\|231\|intervals=odd\|prec=2\|columns=28\|title=[[231edo]] (L=13, s=5, BrightGen is 122; ''patent ~13/9 = 123''; ''patent ~23/16 = 121''; ''patent ~49/34 = 121''; patent ~62/43 = 122; ''patent ~75/52 = 121''; ''patent ~384/265 = 124''; ''patent ~59049/40960 = 121'')}}		{{Harmonics in equal\|231\|intervals=odd\|prec=2\|columns=28\|title=[[231edo]] (L=13, s=5, BrightGen is 122; ''patent ~13/9 = 123''; ''patent ~23/16 = 121''; ''patent ~49/34 = 121''; patent ~62/43 = 122; ''patent ~75/52 = 121''; ''patent ~384/265 = 124''; ''patent ~59049/40960 = 121'')}}
	{{Harmonics in equal\|373\|intervals=odd\|prec=2\|columns=28\|title=[[373edo]] (L=21, s=8, BrightGen is 197; patent ~13/9 = _; patent ~23/16 = _; patent ~49/34 = _; patent ~62/43 = _; patent ~75/52 = _; patent ~384/265 = _; patent ~59049/40960 = _)}}		{{Harmonics in equal\|373\|intervals=odd\|prec=2\|columns=28\|title=[[373edo]] (L=21, s=8, BrightGen is 197; patent ~13/9 = _; patent ~23/16 = _; patent ~49/34 = _; patent ~62/43 = _; patent ~75/52 = _; patent ~384/265 = _; patent ~59049/40960 = _)}}
Line 205:		Line 205:

	Added: [[User:Lucius Chiaraviglio\|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio\|talk]]) 07:42, 8 April 2025 (UTC)<br>		Added: [[User:Lucius Chiaraviglio\|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio\|talk]]) 07:42, 8 April 2025 (UTC)<br>
	Last modified: [[User:Lucius Chiaraviglio\|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio\|talk]]) 06:05, 19 June 2025 (UTC)		Last modified: [[User:Lucius Chiaraviglio\|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio\|talk]]) 06:09, 19 June 2025 (UTC)

← Older revision		Revision as of 09:25, 21 June 2025
Line 80:		Line 80:
	{{Harmonics in equal\|98\|intervals=odd\|prec=2\|columns=28\|title=[[98edo]] (L=8, s=5, ~16/11 = 53\98)}}		{{Harmonics in equal\|98\|intervals=odd\|prec=2\|columns=28\|title=[[98edo]] (L=8, s=5, ~16/11 = 53\98)}}
	{{Harmonics in equal\|159\|intervals=odd\|prec=2\|columns=28\|title=[[159edo]] (L=13, s=8, ~16/11 = 86\159)}}		{{Harmonics in equal\|159\|intervals=odd\|prec=2\|columns=28\|title=[[159edo]] (L=13, s=8, ~16/11 = 86\159)}}
	{{Harmonics in equal\|61\|intervals=odd\|prec=2\|columns=28\|title=[[61edo]] (L=5, s=3, ~16/11 = 33) — Semisoft 11L 2s}}		{{Harmonics in equal\|61\|intervals=odd\|prec=2\|columns=28\|title=[[61edo]] (L=5, s=3, ~16/11 = 33\61) — Semisoft 11L 2s}}
	{{Harmonics in equal\|146\|intervals=odd\|prec=2\|columns=28\|title=[[146edo]] (L=12, s=7, ~16/11 = 79\146)}}		{{Harmonics in equal\|146\|intervals=odd\|prec=2\|columns=28\|title=[[146edo]] (L=12, s=7, ~16/11 = 79\146)}}
	{{Harmonics in equal\|85\|intervals=odd\|prec=2\|columns=28\|title=[[85edo]] (L=7, s=4, ~16/11 = 46\85)}}		{{Harmonics in equal\|85\|intervals=odd\|prec=2\|columns=28\|title=[[85edo]] (L=7, s=4, ~16/11 = 46\85)}}
Line 109:		Line 109:

	Added: [[User:Lucius Chiaraviglio\|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio\|talk]]) 07:00, 9 April 2025 (UTC)<br>		Added: [[User:Lucius Chiaraviglio\|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio\|talk]]) 07:00, 9 April 2025 (UTC)<br>
	Last modified: [[User:Lucius Chiaraviglio\|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio\|talk]]) 08:49, 21 June 2025 (UTC)		Last modified: [[User:Lucius Chiaraviglio\|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio\|talk]]) 09:25, 21 June 2025 (UTC)

	== Musical Mad Science Musings on Diatonicized Third-Tone Sub-Chromaticism ==		== Musical Mad Science Musings on Diatonicized Third-Tone Sub-Chromaticism ==
Line 135:		Line 135:

	{{Harmonics in equal\|19\|intervals=odd\|prec=2\|columns=28\|title=[[19edo]] (L=1, s=1, BrightGen is 10\19; patent ~[[13/9]] = 10\19; patent ~[[23/16]] = 10\19; ''patent ~[[49/34]] = 9\19''; patent ~[[62/43]] = 10\19; patent ~[[75/52]] = 10\19; patent ~[[384/265]] = 10\19; ''patent ~[[59049/40960]] = 9\19'') — Equalized 17L 2s}}		{{Harmonics in equal\|19\|intervals=odd\|prec=2\|columns=28\|title=[[19edo]] (L=1, s=1, BrightGen is 10\19; patent ~[[13/9]] = 10\19; patent ~[[23/16]] = 10\19; ''patent ~[[49/34]] = 9\19''; patent ~[[62/43]] = 10\19; patent ~[[75/52]] = 10\19; patent ~[[384/265]] = 10\19; ''patent ~[[59049/40960]] = 9\19'') — Equalized 17L 2s}}
	{{Harmonics in equal\|131\|intervals=odd\|prec=2\|columns=28\|title=[[131edo]] (L=7, s=6, BrightGen is 69\131; patent ~13/9 = _\131; patent ~23/16 = _~~\131\131~~\131; patent ~49/34 = _\131; patent ~62/43 = _~~\131~~\131; patent ~75/52 = _\131; patent ~384/265 = _~~\131~~\131; patent ~59049/40960 = _\131)}}		{{Harmonics in equal\|131\|intervals=odd\|prec=2\|columns=28\|title=[[131edo]] (L=7, s=6, BrightGen is 69\131; patent ~13/9 = _\131; patent ~23/16 = _\131; patent ~49/34 = _\131; patent ~62/43 = _\131; patent ~75/52 = _\131; patent ~384/265 = _\131; patent ~59049/40960 = _\131)}}
	{{Harmonics in equal\|112\|intervals=odd\|prec=2\|columns=28\|title=[[112edo]] (L=6, s=5, BrightGen is 59\112; ''patent ~13/9 = 58\112; b val ~13/9 = 60\112''; patent ~23/16 = 59\112; ''patent ~49/34 = 58\112''; patent ~62/43 = 59\112; ''patent ~75/52 = 60\112'', 'b' or 'f' wart usable; ''patent ~384/265 = 60\112; b val ~384/255 = 61\112''; ''patent ~59049/40960 = 64\112; b val ~59049/40960 = 54\112'')}}		{{Harmonics in equal\|112\|intervals=odd\|prec=2\|columns=28\|title=[[112edo]] (L=6, s=5, BrightGen is 59\112; ''patent ~13/9 = 58\112; b val ~13/9 = 60\112''; patent ~23/16 = 59\112; ''patent ~49/34 = 58\112''; patent ~62/43 = 59\112; ''patent ~75/52 = 60\112'', 'b' or 'f' wart usable; ''patent ~384/265 = 60\112; b val ~384/255 = 61\112''; ''patent ~59049/40960 = 64\112; b val ~59049/40960 = 54\112'')}}
	{{Harmonics in equal\|205\|intervals=odd\|prec=2\|columns=28\|title=[[205edo]] (L=11, s=9, BrightGen is 108\205; patent ~13/9 = _\205; patent ~23/16 = _\205; patent ~49/34 = _\205; patent ~62/43 = _\205; patent ~75/52 = _\205; patent ~384/265 = _\205\205; patent ~59049/40960 = _\205)}}		{{Harmonics in equal\|205\|intervals=odd\|prec=2\|columns=28\|title=[[205edo]] (L=11, s=9, BrightGen is 108\205; patent ~13/9 = _\205; patent ~23/16 = _\205; patent ~49/34 = _\205; patent ~62/43 = _\205; patent ~75/52 = _\205; patent ~384/265 = _\205\205; patent ~59049/40960 = _\205)}}
Line 205:		Line 205:

	Added: [[User:Lucius Chiaraviglio\|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio\|talk]]) 07:42, 8 April 2025 (UTC)<br>		Added: [[User:Lucius Chiaraviglio\|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio\|talk]]) 07:42, 8 April 2025 (UTC)<br>
	Last modified: [[User:Lucius Chiaraviglio\|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio\|talk]]) 06:09, 19 June 2025 (UTC)		Last modified: [[User:Lucius Chiaraviglio\|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio\|talk]]) 09:25, 21 June 2025 (UTC)