Low harmonic entropy linear temperaments - Revision history

Sintel: fix helmholtz link

2025-04-16T15:46:16Z

fix helmholtz link

← Older revision		Revision as of 15:46, 16 April 2025
Line 29:		Line 29:
	<ul><li>[[5edo\|5edo]] (coarse)</li><li>[[7edo\|7edo]] (coarse)</li><li>[[12edo\|12edo]] (coarse/medium)</li></ul>Temperaments where 4/3 has complexity 1 all have the same structure:		<ul><li>[[5edo\|5edo]] (coarse)</li><li>[[7edo\|7edo]] (coarse)</li><li>[[12edo\|12edo]] (coarse/medium)</li></ul>Temperaments where 4/3 has complexity 1 all have the same structure:

	<ul><li>[[Meantone\|Meantone]] (all-around)</li><li>[[Superpyth\|Superpyth]] (all-around)</li><li>[[Mavila\|Mavila]] (coarse)</li><li>[[Helmholtz\|Helmholtz]]/[[Garibaldi\|garibaldi]] (fine)</li></ul>Temperaments where 4/3 has complexity 2:		<ul><li>[[Meantone\|Meantone]] (all-around)</li><li>[[Superpyth\|Superpyth]] (all-around)</li><li>[[Mavila\|Mavila]] (coarse)</li><li>[[Helmholtz tempermanet\|Helmholtz]]/[[Garibaldi\|garibaldi]] (fine)</li></ul>Temperaments where 4/3 has complexity 2:

	<ul><li>[[Semaphore_and_Godzilla\|Semaphore / godzilla]] (all-around)</li><li>[[Neutral_third_scales\|Neutral third scales]] (mohajira, neutrominant, beatles..., all-around)</li><li>[[Srutal\|Srutal]]/[[pajara\|pajara]] (all-around)</li></ul>Temperaments where 4/3 has complexity 3:		<ul><li>[[Semaphore_and_Godzilla\|Semaphore / godzilla]] (all-around)</li><li>[[Neutral_third_scales\|Neutral third scales]] (mohajira, neutrominant, beatles..., all-around)</li><li>[[Srutal\|Srutal]]/[[pajara\|pajara]] (all-around)</li></ul>Temperaments where 4/3 has complexity 3:

BudjarnLambeth: Add see also; fix category formatting

2025-01-10T06:48:38Z

Add see also; fix category formatting

← Older revision		Revision as of 06:48, 10 January 2025
Line 48:		Line 48:

	<ul><li>Augmented (indistinguishable in practice from 12-EDO subsets)</li><li>Roulette (index-2 subtemperament of meantone)</li><li>Injera (pajara is simply better; injera just appears as a little shoulder on its side in a plot of average HE)</li></ul>		<ul><li>Augmented (indistinguishable in practice from 12-EDO subsets)</li><li>Roulette (index-2 subtemperament of meantone)</li><li>Injera (pajara is simply better; injera just appears as a little shoulder on its side in a plot of average HE)</li></ul>

			== See also ==
			* [[Harmonic entropy of just intervals]]

	[[Category:Consonance and dissonance]]		[[Category:Consonance and dissonance]]
	[[Category:entropy]]		[[Category:Harmonic entropy]]
	~~[[Category:harmonic_entropy~~]]
	[[Category:Lists of temperaments]]		[[Category:Lists of temperaments]]

ArrowHead294 at 01:59, 27 October 2024

2024-10-27T01:59:08Z

BudjarnLambeth: Updated name from maqamic to neutrominant

2024-01-27T05:03:54Z

Updated name from maqamic to neutrominant

← Older revision		Revision as of 05:03, 27 January 2024
Line 31:		Line 31:
	<ul><li>[[Meantone\|Meantone]] (all-around)</li><li>[[Superpyth\|Superpyth]] (all-around)</li><li>[[Mavila\|Mavila]] (coarse)</li><li>[[Helmholtz\|Helmholtz]]/[[Garibaldi\|garibaldi]] (fine)</li></ul>Temperaments where 4/3 has complexity 2:		<ul><li>[[Meantone\|Meantone]] (all-around)</li><li>[[Superpyth\|Superpyth]] (all-around)</li><li>[[Mavila\|Mavila]] (coarse)</li><li>[[Helmholtz\|Helmholtz]]/[[Garibaldi\|garibaldi]] (fine)</li></ul>Temperaments where 4/3 has complexity 2:

	<ul><li>[[Semaphore_and_Godzilla\|Semaphore / godzilla]] (all-around)</li><li>[[Neutral_third_scales\|Neutral third scales]] (mohajira, ~~maqamic~~, beatles..., all-around)</li><li>[[Srutal\|Srutal]]/[[pajara\|pajara]] (all-around)</li></ul>Temperaments where 4/3 has complexity 3:		<ul><li>[[Semaphore_and_Godzilla\|Semaphore / godzilla]] (all-around)</li><li>[[Neutral_third_scales\|Neutral third scales]] (mohajira, neutrominant, beatles..., all-around)</li><li>[[Srutal\|Srutal]]/[[pajara\|pajara]] (all-around)</li></ul>Temperaments where 4/3 has complexity 3:

	<ul><li>[[Porcupine\|Porcupine]] (all-around)</li><li>[[Slendric\|Slendric]] (all-around; quite accurate)</li><li>[[Liese\|Liese]]/[[Triton\|triton]] (fine)</li></ul>Temperaments where 4/3 has higher complexity:		<ul><li>[[Porcupine\|Porcupine]] (all-around)</li><li>[[Slendric\|Slendric]] (all-around; quite accurate)</li><li>[[Liese\|Liese]]/[[Triton\|triton]] (fine)</li></ul>Temperaments where 4/3 has higher complexity:

Fredg999 category edits: Removing from Category:Sonance using Cat-a-lot

2023-03-04T18:39:31Z

Removing from Category:Sonance using Cat-a-lot

← Older revision		Revision as of 18:39, 4 March 2023
Line 50:		Line 50:
	[[Category:entropy]]		[[Category:entropy]]
	[[Category:harmonic_entropy]]		[[Category:harmonic_entropy]]
	~~[[Category:sonance]]~~
	[[Category:Lists of temperaments]]		[[Category:Lists of temperaments]]

Fredg999 category edits: Removing from Category:Dissonance using Cat-a-lot

2023-03-04T18:39:15Z

Removing from Category:Dissonance using Cat-a-lot

← Older revision		Revision as of 18:39, 4 March 2023
Line 48:		Line 48:

	<ul><li>Augmented (indistinguishable in practice from 12-EDO subsets)</li><li>Roulette (index-2 subtemperament of meantone)</li><li>Injera (pajara is simply better; injera just appears as a little shoulder on its side in a plot of average HE)</li></ul> [[Category:Consonance and dissonance]]		<ul><li>Augmented (indistinguishable in practice from 12-EDO subsets)</li><li>Roulette (index-2 subtemperament of meantone)</li><li>Injera (pajara is simply better; injera just appears as a little shoulder on its side in a plot of average HE)</li></ul> [[Category:Consonance and dissonance]]
	~~[[Category:dissonance]]~~
	[[Category:entropy]]		[[Category:entropy]]
	[[Category:harmonic_entropy]]		[[Category:harmonic_entropy]]
	[[Category:sonance]]		[[Category:sonance]]
	[[Category:Lists of temperaments]]		[[Category:Lists of temperaments]]

Fredg999 category edits: Moving from Category:Consonance to Category:Consonance and dissonance using Cat-a-lot

2023-03-04T18:38:55Z

Moving from Category:Consonance to Category:Consonance and dissonance using Cat-a-lot

← Older revision		Revision as of 18:38, 4 March 2023
Line 47:		Line 47:
	The following temperaments were not included in the list, because they don't stand out as good independent temperaments:		The following temperaments were not included in the list, because they don't stand out as good independent temperaments:

	<ul><li>Augmented (indistinguishable in practice from 12-EDO subsets)</li><li>Roulette (index-2 subtemperament of meantone)</li><li>Injera (pajara is simply better; injera just appears as a little shoulder on its side in a plot of average HE)</li></ul> [[Category:~~consonance~~]]		<ul><li>Augmented (indistinguishable in practice from 12-EDO subsets)</li><li>Roulette (index-2 subtemperament of meantone)</li><li>Injera (pajara is simply better; injera just appears as a little shoulder on its side in a plot of average HE)</li></ul> [[Category:Consonance and dissonance]]
	[[Category:dissonance]]		[[Category:dissonance]]
	[[Category:entropy]]		[[Category:entropy]]

Fredg999 category edits: Moving from Category:Theory to Category:Lists of temperaments using Cat-a-lot

2023-03-02T04:37:03Z

Moving from Category:Theory to Category:Lists of temperaments using Cat-a-lot

← Older revision		Revision as of 04:37, 2 March 2023
Line 52:		Line 52:
	[[Category:harmonic_entropy]]		[[Category:harmonic_entropy]]
	[[Category:sonance]]		[[Category:sonance]]
	[[Category:~~theory~~]]		[[Category:Lists of temperaments]]

Wikispaces>FREEZE at 00:00, 17 July 2018

2018-07-17T00:00:00Z

Show changes

Wikispaces>mbattaglia1: Imported revision 337248976 - Original comment:

2012-05-19T04:29:05Z

**Imported revision 337248976 - Original comment: **

← Older revision		Revision as of 15:46, 16 April 2025
Line 29:		Line 29:
	<ul><li>[[5edo\|5edo]] (coarse)</li><li>[[7edo\|7edo]] (coarse)</li><li>[[12edo\|12edo]] (coarse/medium)</li></ul>Temperaments where 4/3 has complexity 1 all have the same structure:		<ul><li>[[5edo\|5edo]] (coarse)</li><li>[[7edo\|7edo]] (coarse)</li><li>[[12edo\|12edo]] (coarse/medium)</li></ul>Temperaments where 4/3 has complexity 1 all have the same structure:

	<ul><li>[[Meantone\|Meantone]] (all-around)</li><li>[[Superpyth\|Superpyth]] (all-around)</li><li>[[Mavila\|Mavila]] (coarse)</li><li>[[Helmholtz\|Helmholtz]]/[[Garibaldi\|garibaldi]] (fine)</li></ul>Temperaments where 4/3 has complexity 2:		<ul><li>[[Meantone\|Meantone]] (all-around)</li><li>[[Superpyth\|Superpyth]] (all-around)</li><li>[[Mavila\|Mavila]] (coarse)</li><li>[[Helmholtz tempermanet\|Helmholtz]]/[[Garibaldi\|garibaldi]] (fine)</li></ul>Temperaments where 4/3 has complexity 2:

	<ul><li>[[Semaphore_and_Godzilla\|Semaphore / godzilla]] (all-around)</li><li>[[Neutral_third_scales\|Neutral third scales]] (mohajira, neutrominant, beatles..., all-around)</li><li>[[Srutal\|Srutal]]/[[pajara\|pajara]] (all-around)</li></ul>Temperaments where 4/3 has complexity 3:		<ul><li>[[Semaphore_and_Godzilla\|Semaphore / godzilla]] (all-around)</li><li>[[Neutral_third_scales\|Neutral third scales]] (mohajira, neutrominant, beatles..., all-around)</li><li>[[Srutal\|Srutal]]/[[pajara\|pajara]] (all-around)</li></ul>Temperaments where 4/3 has complexity 3:

← Older revision		Revision as of 05:03, 27 January 2024
Line 31:		Line 31:
	<ul><li>[[Meantone\|Meantone]] (all-around)</li><li>[[Superpyth\|Superpyth]] (all-around)</li><li>[[Mavila\|Mavila]] (coarse)</li><li>[[Helmholtz\|Helmholtz]]/[[Garibaldi\|garibaldi]] (fine)</li></ul>Temperaments where 4/3 has complexity 2:		<ul><li>[[Meantone\|Meantone]] (all-around)</li><li>[[Superpyth\|Superpyth]] (all-around)</li><li>[[Mavila\|Mavila]] (coarse)</li><li>[[Helmholtz\|Helmholtz]]/[[Garibaldi\|garibaldi]] (fine)</li></ul>Temperaments where 4/3 has complexity 2:

	<ul><li>[[Semaphore_and_Godzilla\|Semaphore / godzilla]] (all-around)</li><li>[[Neutral_third_scales\|Neutral third scales]] (mohajira, ~~maqamic~~, beatles..., all-around)</li><li>[[Srutal\|Srutal]]/[[pajara\|pajara]] (all-around)</li></ul>Temperaments where 4/3 has complexity 3:		<ul><li>[[Semaphore_and_Godzilla\|Semaphore / godzilla]] (all-around)</li><li>[[Neutral_third_scales\|Neutral third scales]] (mohajira, neutrominant, beatles..., all-around)</li><li>[[Srutal\|Srutal]]/[[pajara\|pajara]] (all-around)</li></ul>Temperaments where 4/3 has complexity 3:

	<ul><li>[[Porcupine\|Porcupine]] (all-around)</li><li>[[Slendric\|Slendric]] (all-around; quite accurate)</li><li>[[Liese\|Liese]]/[[Triton\|triton]] (fine)</li></ul>Temperaments where 4/3 has higher complexity:		<ul><li>[[Porcupine\|Porcupine]] (all-around)</li><li>[[Slendric\|Slendric]] (all-around; quite accurate)</li><li>[[Liese\|Liese]]/[[Triton\|triton]] (fine)</li></ul>Temperaments where 4/3 has higher complexity:

← Older revision		Revision as of 01:59, 27 October 2024
Line 1:		Line 1:
	If you do a survey of [[MOS\|MOS]]es and look for the ones that have the lowest typical [[Harmonic_Entropy\|harmonic entropy]] of an interval (where "typical" means average, but you throw away the highest and lowest values first), you get an interesting list of reasonably low-complexity yet accurate temperaments, which accords well with lists obtained by starting from temperaments rather than MOS. The results are different according to the "sigma" of the harmonic entropy function you use (coarse versus fine), but some temperaments appear for a wide range of sigma values; this might be compared to the situation with [[Cangwu_badness\|cangwu badness]].		If you do a survey of [[MOS\|MOS]]es and look for the ones that have the lowest typical [[Harmonic_Entropy\|harmonic entropy]] of an interval (where "typical" means average, but you throw away the highest and lowest values first), you get an interesting list of reasonably low-complexity yet accurate temperaments, which accords well with lists obtained by starting from temperaments rather than MOS. The results are different according to the "sigma" of the harmonic entropy function you use (coarse versus fine), but some temperaments appear for a wide range of sigma values; this might be compared to the situation with [[Cangwu_badness\|cangwu badness]].

	[[:File:coarse-period1-4-thru-11-trimmed-10-70.pdf\|~~coarse~~, octave period]]		[[:File:coarse-period1-4-thru-11-trimmed-10-70.pdf\|Coarse, octave period]]

	[[:File:coarse-period2-4-thru-12-trimmed-10-70.pdf\|~~coarse~~, half-octave period]]		[[:File:coarse-period2-4-thru-12-trimmed-10-70.pdf\|Coarse, half-octave period]]

	[[:File:medium-period1-5-thru-12-trimmed-10-70.pdf\|~~medium~~, octave period]]		[[:File:medium-period1-5-thru-12-trimmed-10-70.pdf\|Medium, octave period]]

	[[:File:medium-period2-6-thru-14-trimmed-10-70.pdf\|~~medium~~, half-octave period]]		[[:File:medium-period2-6-thru-14-trimmed-10-70.pdf\|Medium, half-octave period]]

	[[:File:fine-period1-7-thru-13-trimmed-10-70.pdf\|~~fine~~, octave period]]		[[:File:fine-period1-7-thru-13-trimmed-10-70.pdf\|Fine, octave period]]

	[[:File:fine-period2-6-thru-14-trimmed-10-70.pdf\|~~fine~~, half-octave period]]		[[:File:fine-period2-6-thru-14-trimmed-10-70.pdf\|Fine, half-octave period]]

	[[:File:extra-fine-period1-7-thru-13-trimmed-10-70.pdf\|~~extra~~ fine, octave period]]		[[:File:extra-fine-period1-7-thru-13-trimmed-10-70.pdf\|Extra fine, octave period]]

	[[:File:extra-fine-period2-8-thru-18-trimmed-10-70.pdf\|~~extra~~ fine, half-octave period]]		[[:File:extra-fine-period2-8-thru-18-trimmed-10-70.pdf\|Extra fine, half-octave period]]

	[[:File:extra-fine-period3-9-thru-18-trimmed-10-70.pdf\|~~extra~~ fine, third-octave period]]		[[:File:extra-fine-period3-9-thru-18-trimmed-10-70.pdf\|Extra fine, third-octave period]]

	It makes sense to organize the results by the complexity of 4/3, because 4/3 (or its octave equivalent 3/2) has by far the lowest harmonic entropy of any interval within an octave, and the results are accordingly dominated by temperaments with lots of good 4/3s.		It makes sense to organize the results by the complexity of 4/3, because 4/3 (or its octave equivalent 3/2) has by far the lowest harmonic entropy of any interval within an octave, and the results are accordingly dominated by temperaments with lots of good 4/3s.
Line 47:		Line 47:
	The following temperaments were not included in the list, because they don't stand out as good independent temperaments:		The following temperaments were not included in the list, because they don't stand out as good independent temperaments:

	<ul><li>Augmented (indistinguishable in practice from 12-EDO subsets)</li><li>Roulette (index-2 subtemperament of meantone)</li><li>Injera (pajara is simply better; injera just appears as a little shoulder on its side in a plot of average HE)</li></ul> [[Category:Consonance and dissonance]]		<ul><li>Augmented (indistinguishable in practice from 12-EDO subsets)</li><li>Roulette (index-2 subtemperament of meantone)</li><li>Injera (pajara is simply better; injera just appears as a little shoulder on its side in a plot of average HE)</li></ul>

			[[Category:Consonance and dissonance]]
	[[Category:entropy]]		[[Category:entropy]]
	[[Category:harmonic_entropy]]		[[Category:harmonic_entropy]]
	[[Category:Lists of temperaments]]		[[Category:Lists of temperaments]]

← Older revision		Revision as of 04:29, 19 May 2012
Line 1:		Line 1:
	<h2>IMPORTED REVISION FROM WIKISPACES</h2>		<h2>IMPORTED REVISION FROM WIKISPACES</h2>
	This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>		This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
	: This revision was by author [[User:mbattaglia1\|mbattaglia1]] and made on <tt>2012-05-19 04:28:49 UTC</tt>.<br>		: This revision was by author [[User:mbattaglia1\|mbattaglia1]] and made on <tt>2012-05-19 04:29:05 UTC</tt>.<br>
	: The original revision id was <tt>~~337248952~~</tt>.<br>		: The original revision id was <tt>337248976</tt>.<br>
	: The revision comment was: <tt></tt><br>		: The revision comment was: <tt></tt><br>
	The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>		The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
Line 50:		Line 50:

	Finally, a temperament in which 3 has two different mappings:		Finally, a temperament in which 3 has two different mappings:
	* [[Pseudo-~~semiphore~~]] (medium)		* [[Pseudo-semaphore]] (medium)

	All-around runners up:		All-around runners up:
Line 86:		Line 86:
	<ul><li><a class="wiki_link" href="/Magic">Magic</a> (5, fine)</li><li><a class="wiki_link" href="/Orwell">Orwell</a> (7, extra fine)</li><li><a class="wiki_link" href="/Myna">Myna</a> (10, extra fine)</li><li><a class="wiki_link" href="/Hanson">Hanson</a>/<a class="wiki_link" href="/catakleismic">catakleismic</a> (6, extra fine)</li><li><a class="wiki_link" href="/Miracle">Miracle</a> (6, super duper fine)</li><li><a class="wiki_link" href="/Valentine">Valentine</a> (9, super duper fine)</li></ul><br />		<ul><li><a class="wiki_link" href="/Magic">Magic</a> (5, fine)</li><li><a class="wiki_link" href="/Orwell">Orwell</a> (7, extra fine)</li><li><a class="wiki_link" href="/Myna">Myna</a> (10, extra fine)</li><li><a class="wiki_link" href="/Hanson">Hanson</a>/<a class="wiki_link" href="/catakleismic">catakleismic</a> (6, extra fine)</li><li><a class="wiki_link" href="/Miracle">Miracle</a> (6, super duper fine)</li><li><a class="wiki_link" href="/Valentine">Valentine</a> (9, super duper fine)</li></ul><br />
	Finally, a temperament in which 3 has two different mappings:<br />		Finally, a temperament in which 3 has two different mappings:<br />
	<ul><li><a class="wiki_link" href="/Pseudo-~~semiphore~~">Pseudo-~~semiphore~~</a> (medium)</li></ul><br />		<ul><li><a class="wiki_link" href="/Pseudo-semaphore">Pseudo-semaphore</a> (medium)</li></ul><br />
	All-around runners up:<br />		All-around runners up:<br />
	<ul><li><a class="wiki_link" href="/Negri">Negri</a> (4)</li><li><a class="wiki_link" href="/Sensi">Sensi</a> (7)</li></ul><br />		<ul><li><a class="wiki_link" href="/Negri">Negri</a> (4)</li><li><a class="wiki_link" href="/Sensi">Sensi</a> (7)</li></ul><br />
	The following temperaments were not included in the list, because they don't stand out as good independent temperaments:<br />		The following temperaments were not included in the list, because they don't stand out as good independent temperaments:<br />
	<ul><li>Augmented (indistinguishable in practice from 12-EDO subsets)</li><li>Roulette (index-2 subtemperament of meantone)</li><li>Injera (pajara is simply better; injera just appears as a little shoulder on its side in a plot of average HE)</li></ul></body></html></pre></div>		<ul><li>Augmented (indistinguishable in practice from 12-EDO subsets)</li><li>Roulette (index-2 subtemperament of meantone)</li><li>Injera (pajara is simply better; injera just appears as a little shoulder on its side in a plot of average HE)</li></ul></body></html></pre></div>