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Talk:Levels of tempering

Hotcrystal0 — Mon, 31 Aug 2026 12:02:41 GMT

Created page with "Has anybody ever actually used MT for meantone? ~~~~"

New page

Has anybody ever actually used MT for meantone? <span style="display: inline-block;transform: rotate(15deg);background:#E1EBF2;font-family:Verdana;text-shadow: 3px 3px 4px #0008;">[[User:Hotcrystal0|hotcrysta]][[User talk: Hotcrystal0|l0]]</span> 12:02, 31 August 2026 (UTC)

Levels of tempering

Inthar — Mon, 31 Aug 2026 05:10:41 GMT

← Older revision		Revision as of 05:10, 31 August 2026
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	The proposal by [[User:Ground]] for some abbreviations, avoiding those commonly seen in a xenharmonic context like MT for meantone and ET for equal temperament:		The proposal by [[User:Ground]] for some abbreviations, avoiding those commonly seen in a xenharmonic context like MT for meantone and ET for equal temperament:
	# '''D: Direct'''. All ratios are exactly as described within practical levels of precision. This category contains both those who use JI tunings and those who may ignore JI and interpret every interval as only its exact definition.		# '''D: Direct'''. All ratios are exactly as described within practical levels of precision. This category contains both those who use JI tunings and those who may ignore JI and interpret every interval as only its exact definition.
	# '''UT: Microtemp''' (based on the SI prefix μ often written as u). Levels of precision are similar to Direct, but allow for the introduction of tempering. People in this category may use large high-accuracy edos to approximate JI, and refer to degrees of the edo based on which JI ratios they map to.		# '''MCT: Microtemp''' (based on the SI prefix μ often written as mc in pharmaceutical contexts). Levels of precision are similar to Direct, but allow for the introduction of tempering. People in this category may use large high-accuracy edos to approximate JI, and refer to degrees of the edo based on which JI ratios they map to.
	# '''ST: Mesotemp''' (also evokes "standard tempered"). This is the most common category in RTT and interpretations of the most common edos. Intervals may be equated even if they are audibly different as long as they share a concordance well or position in the lattice. This is the point where [[Straddle primes\|straddling]] becomes practical.		# '''ST: Mesotemp''' (also evokes "standard tempered"). This is the most common category in RTT and interpretations of the most common edos. Intervals may be equated even if they are audibly different as long as they share a concordance well or position in the lattice. This is the point where [[Straddle primes\|straddling]] becomes practical.
	# '''XT: Exotemp'''. Intervals don't need to sound similar to be equated.		# '''XT: Exotemp'''. Intervals don't need to sound similar to be equated.

Levels of tempering

Ground — Mon, 31 Aug 2026 02:26:33 GMT

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	{{proposed}}		{{proposed}}

	'''Levels of tempering''' is a system of categorizing musicians based on ~~how they interpret different intervals~~. It is intended ~~to allow people~~ to avoid repetitive arguments such as "is 400¢ a 5/4?" by identifying ~~the base assumptions that allow~~ answers to be different.		'''Levels of tempering''' is a system of categorizing how different musicians equate intervals, which appears to be based on unchangeable assumptions. This is intended to avoid repetitive arguments such as "is 400¢ a 5/4?" by identifying what allows answers to be so different.

	The proposal by [[User:Ground]] for some abbreviations, avoiding those commonly seen in a xenharmonic context like MT for meantone and ET for equal temperament:		The proposal by [[User:Ground]] for some abbreviations, avoiding those commonly seen in a xenharmonic context like MT for meantone and ET for equal temperament:
	# '''D: Direct'''. All ratios are exactly as described within practical levels of precision. This category contains both those who use JI tunings and those who may ignore JI and interpret every interval as only its exact definition.		# '''D: Direct'''. All ratios are exactly as described within practical levels of precision. This category contains both those who use JI tunings and those who may ignore JI and interpret every interval as only its exact definition.
	# '''UT: Microtemp''' (based on the SI prefix μ often written as u). Levels of precision are similar to Direct, but allow for the introduction of tempering. People in this category may use large high-accuracy edos to approximate JI, and refer to degrees of the edo based on which JI ratios they map to.		# '''UT: Microtemp''' (based on the SI prefix μ often written as u). Levels of precision are similar to Direct, but allow for the introduction of tempering. People in this category may use large high-accuracy edos to approximate JI, and refer to degrees of the edo based on which JI ratios they map to.
	# '''ST: Mesotemp''' (also evokes "standard tempered"). This is the most common category in RTT and interpretations of the most common edos. Intervals may be equated even if they are audibly different as long as they share a concordance well or position in the lattice.		# '''ST: Mesotemp''' (also evokes "standard tempered"). This is the most common category in RTT and interpretations of the most common edos. Intervals may be equated even if they are audibly different as long as they share a concordance well or position in the lattice. This is the point where [[Straddle primes\|straddling]] becomes practical.
	# '''XT: Exotemp'''. Intervals don't need to sound similar to be equated.		# '''XT: Exotemp'''. Intervals don't need to sound similar to be equated.

Levels of tempering

Ground — Mon, 31 Aug 2026 01:36:04 GMT

Created page with "{{proposed}} '''Levels of tempering''' is a system of categorizing musicians based on how they interpret different intervals. It is intended to allow people to avoid repetitive arguments such as "is 400¢ a 5/4?" by identifying the base assumptions that allow answers to be different. The proposal by User:Ground for some abbreviations, avoiding those commonly seen in a xenharmonic context like MT for meantone and ET for equal temperament: # '''D: Direct'''. All rati..."

New page

{{proposed}}

'''Levels of tempering''' is a system of categorizing musicians based on how they interpret different intervals. It is intended to allow people to avoid repetitive arguments such as "is 400¢ a 5/4?" by identifying the base assumptions that allow answers to be different.

The proposal by [[User:Ground]] for some abbreviations, avoiding those commonly seen in a xenharmonic context like MT for meantone and ET for equal temperament:
# '''D: Direct'''. All ratios are exactly as described within practical levels of precision. This category contains both those who use JI tunings and those who may ignore JI and interpret every interval as only its exact definition.
# '''UT: Microtemp''' (based on the SI prefix μ often written as u). Levels of precision are similar to Direct, but allow for the introduction of tempering. People in this category may use large high-accuracy edos to approximate JI, and refer to degrees of the edo based on which JI ratios they map to.
# '''ST: Mesotemp''' (also evokes "standard tempered"). This is the most common category in RTT and interpretations of the most common edos. Intervals may be equated even if they are audibly different as long as they share a concordance well or position in the lattice.
# '''XT: Exotemp'''. Intervals don't need to sound similar to be equated.

Ground's composition theory

Ground — Mon, 31 Aug 2026 01:18:40 GMT

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	{{UTBox\|g_\|Ground\|7766ff}}		{{UTBox\|g_\|Ground\|7766ff}}

	This article is ~~going to be very long~~. I have more content to add and ~~a plan to~~ revise ~~the existing content eventually~~. It has an emphasis on my other theories, [[Aberrisma\|aberrismic]] and [[Straddle_primes\|straddle-prime]], which may be useful to know.		This article is about composition and the practical considerations of tuning. I always have more content to add and revise. It has an emphasis on my other theories, [[Aberrisma\|aberrismic]] and [[Straddle_primes\|straddle-prime]], which may be useful to know.

	My approach in writing chords is loosely based on non-functional jazz harmony. They ~~generally~~ outline the melody and have few defined cadences involving more than two chords; the most common is probably iv → i.		It should be kept in mind that my style is generally unrestrained and chaotic, and I am in the [[Levels of tempering\|mesotempered category]]. My approach to writing chords is loosely based on non-functional jazz harmony. They usually outline and support the melody and have few defined cadences involving more than two chords; the most common is probably iv → i. As a result, I don't have much to say about harmonic functions, something which is also in dire need of thorough documentation.

	== Interval logic deviation theory ==		== Interval logic deviation theory ==
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	\|Doubling ed4s is very annoying, but it's the only way for me to practically reach large prime edos. 67 is the only one significant enough to be worth the trouble.		\|Doubling ed4s is very annoying, but it's the only way for me to practically reach large prime edos. 67 is the only one significant enough to be worth the trouble.
	\|}		\|}

			== Edo multiplication ==

			I use midi channels interpreted by custom tuning scripts to alter notes by fractions of a step, allowing much larger edos than is otherwise practical in a piano roll without sacrificing the freedom and simplicity of the '''base edo''' to be multiplied. This creates an interesting problem for tuning selection because both the base and any multiples are necessary to consider. I prefer x6 multiplication because it's the superset of x2 and x3, after which it becomes very complicated to juggle all the different channels.

			Example tier list of 37edo multiples:
			* Base ([[37edo\|37]]): S (One of a kind.)
			* x2 ([[74edo\|74]]): B (Nothing special besides the two meantone diasems and various Diashismic straddle temperaments. A pretty straightforward extension of 37's interval set.)
			* x3 ([[111edo\|111]]): A (Very good at approximating the 17-limit and containing a few temperaments I like. More of a self-contained microtemperament, very different from 74.)
			* x6 ([[222edo\|222]]): D (Has extra approximations and generators, but too cumbersome to do much.)
			This is a top-tier edo multiplication setup when you weight these based on their expected frequency, which is estimated based on existing projects:
			* Base: remainder
			* x2: 12%
			* x3: 8%
			* x6: 4%

			When using edo multiples, it's important to keep better track of the intervals in use and minimize the effect of the added complexity to compose efficiently. I do this by switching between discrete scales in the multiples (rather than flowing freely between structures and selecting all notes by ear as in the base edo) and by primarily using notes of the base edo as the tonic to minimize the versions of each scale to keep track of when I modulate. No matter what, the things I write will always be filtered through the base edo. It's natural that the extra precision and features of multiples won't be necessary most of the time, and the freedom of a reasonably sized base edo will outweigh them.

	== Anchor concordances ==		== Anchor concordances ==
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	For example, the patent 3/2 in 25 and 28edo is far off, but their representations of 4:5:6 mask the beating with an accurate 5/4. I prefer to have anchor concordances be 3/2, thirds, or sometimes sixths because they show up the most often in tertian chords, and major thirds in particular. My theory is that this is because the major third range has stronger concordance wells than minor thirds, which have something close to a plateau on the harmonic entropy curve. My favorites to have are 5/4, 9/7, and 13/10.		For example, the patent 3/2 in 25 and 28edo is far off, but their representations of 4:5:6 mask the beating with an accurate 5/4. I prefer to have anchor concordances be 3/2, thirds, or sometimes sixths because they show up the most often in tertian chords, and major thirds in particular. My theory is that this is because the major third range has stronger concordance wells than minor thirds, which have something close to a plateau on the harmonic entropy curve. My favorites to have are 5/4, 9/7, and 13/10.

			I consider 2/1, 3/2, and 5/4 to be the most important anchor concordances. If any two of them are approximated within a few cents, the tuning system will be capable of comfortably stable harmony. This is why I believe 12edo works so well, and why 25, 28, and 37edo are more capable than people expect. The original purpose of straddle primes was to account for the extreme variability of the reasonable tunings of 3/2 in 37edo and stacking both to approximate the pythagorean chain.

			In 37edo, both 4:5:>6 and 4:5:<<6 (a very close approximation of 1/(40:32:27)) serve as reasonable consonances. I've never been able to observe different functions between them. The same applies to the different tunings of 6:8:9, besides <<6:8:<<<<9 = 6:>>8:<<9 which just doesn't sound good. This is most obviously useful when an accurate 3 isn't available, but it's also beneficial for altering intervals to follow a melody with optimal step sizes for my compositional goal:
			* Using the very flat <<3 results in >>16/15, quite audibly wider while keeping some semblance of 4:5:6.
			* <<6/5 is found in the [[Pentagoth]] triad, which uses the exaggerated quality of the minor third and the stability of the major third.

	Compare the different optimizations of 2.3.5 Meantone which I have noticed sound better than those immediately around them:		Compare the different optimizations of 2.3.5 Meantone which I have noticed sound better than those immediately around them:

Diaschismic straddle temperaments

Ground — Sun, 30 Aug 2026 14:59:38 GMT

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	[[Diaschismic]] tempers out 2048/2025 = 2/((9/8)(5/4))^2. The subgroup is usually interpreted as 2.3.5, but 2.9.5 is also possible. This provides a great opportunity for [[Straddle primes\|straddling]] 3, where flat 3 (3/a in SRTT) and sharp 3 (3a) cancel out to make 9.		[[Diaschismic]] tempers out 2048/2025 = 2/((9/8)(5/4))^2. The subgroup is usually interpreted as 2.3.5, but 2.9.5 is also possible. This provides a great opportunity for [[Straddle primes\|straddling]] 3, where flat 3 (3/a in SRTT) and sharp 3 (3a) cancel out to make 9.

	Pajara, Hedgehog, and Echidna are related temperaments, but they temper out 50/49, which damages 5 and 7. Straddling 3 allows both 2048/2025 and 64/63 a^-1 to be tempered out without doing this.		Pajara, Hedgehog, and Echidna are related temperaments, but they temper out 50/49, which damages 5 and 7. Straddling 3 allows both 2048/2025 and 64/63 a^-1 to be tempered out without doing this.

	~~== Straddle Diaschismic ==~~

	'''Straddle Diaschismic''' is the simplest temperament in this category. It is rank-3 and generated by the flat and sharp versions of 3. Of the Diaschismic extensions to the 7-limit:		'''Straddle Diaschismic''' is the simplest temperament in this category. It is rank-3 and generated by the flat and sharp versions of 3. Of the Diaschismic extensions to the 7-limit:
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	Much of what is useful here can be found in a rank-2 restriction, 2.3a.7.17a (the alteration ''a'' is redundant because sharp 3 and 17 are used with no straddling). 2.3.7.17 10&22 tempers out 64/63 and 289/288 and adds 17/14 to Archy, creating a useful symmetry between subminor/supermajor and supraminor/submajor chords.		Much of what is useful here can be found in a rank-2 restriction, 2.3a.7.17a (the alteration ''a'' is redundant because sharp 3 and 17 are used with no straddling). 2.3.7.17 10&22 tempers out 64/63 and 289/288 and adds 17/14 to Archy, creating a useful symmetry between subminor/supermajor and supraminor/submajor chords.

	== Diaschismic Porcupine ==		== Rank-2 Temperaments ==

			[todo: table]

			=== Diaschismic Porcupine ===

	'''Diaschismic Porcupine''' tempers out 2048/2025 and 250/243 a^-5, the Porcupine comma with 3 replaced with 3a, as 3 is usually tuned quite sharp. Without straddling 3, only 22edo tempers out 2048/2025 and 250/243 in 2.3.5.		'''Diaschismic Porcupine''' tempers out 2048/2025 and 250/243 a^-5, the Porcupine comma with 3 replaced with 3a, as 3 is usually tuned quite sharp. Without straddling 3, only 22edo tempers out 2048/2025 and 250/243 in 2.3.5.
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	Mappings of 5.7.11 are the same as Porcupine, and the Diaschismic mapping adds prime 17. 13 may be added using the Oceanfront mapping equating 81/64 a^4 to 13/10.		Mappings of 5.7.11 are the same as Porcupine, and the Diaschismic mapping adds prime 17. 13 may be added using the Oceanfront mapping equating 81/64 a^4 to 13/10.

	17-limit GTO string: 600 \| 2.<3.>3.5.7.11.13.<17.>17 \|\| 0 8 -3 -5 6 -4 -17 8 -3		17-limit GTO string: 600 \| 2.<3.>3.5.7.11.13.<17.>17 \|\| 0 8 -3 -5 6 -4 -17 8 -3; 162.287¢

	One possible erac group resembling the 74edo tuning is 600 \| 2.<3.>>3.>5.>7.11.>13.<17.>>17 \|\| 0 8 -3 -5 6 -4 -17 8 -3. This focuses on approximating 9/5, 9/7, 17/5, 17/7, and 11/9 about equally well, with >17/14 equated to <11/9.		One possible erac group resembling the 74edo tuning is 600 \| 2.<3.>>3.>5.>7.11.>13.<17.>>17 \|\| 0 8 -3 -5 6 -4 -17 8 -3. This focuses on approximating 9/5, 9/7, 17/5, 17/7, and 11/9 about equally well, with >17/14 equated to <11/9.

	== Diaschismic Semioceanfront ==		=== Diaschismic Semioceanfront ===

	'''Diaschismic Semioceanfront''' has a generator of half of an Oceanfront fourth rather than a third. The generator is also equivalent to 13/8 due to the half-octave period.		'''Diaschismic Semioceanfront''' has a generator of half of an Oceanfront fourth rather than a third. The generator is also equivalent to 13/8 due to the half-octave period.

	17-limit GTO string: 600 \| 2.<3.>3.5.7.11.13.<17.>17 \|\| 0 -2 -7 9 4 -15 1 -2 -7		17-limit GTO string: 600 \| 2.<3.>3.5.7.11.13.<17.>17 \|\| 0 -2 -7 9 4 -15 1 -2 -7; 243.180¢

	== Diaschismic Hemiseven ==		=== Diaschismic Hemiseven ===

	[[Hemiseven]] may already be interpreted as straddle-3, but being a 5edo cluster temperament, all versions have a complex 5. This is fixed using Diaschismic with the 9 made by stacking the sharp 3 generator and the flatter 3.		[[Hemiseven]] may already be interpreted as straddle-3, but being a 5edo cluster temperament, all versions have a complex 5. This is fixed using Diaschismic with the 9 made by stacking the sharp 3 generator and the flatter 3.

	Subgroup GTO string: 600 \| 2.<3.>3.5.7.<13.>13.<17.>17 \|\| 0 6 1 -7 -2 2 -3 6 1		Subgroup GTO string: 600 \| 2.<3.>3.5.7.<13.>13.<17.>17 \|\| 0 6 1 -7 -2 2 -3 6 1; 115.976 ¢

	== Diaschismic whole-tone ==		=== Diaschismic whole-tone ===

	'''Diaschismic whole-tone''' is a significant whole-tone scale containing the otonal triads 12:15:17 and 8:10:13. It can be written without straddling as 2.9.5.13.17/3 6&28, but the 3 can be separated in the same way as the other temperaments in this article.		'''Diaschismic whole-tone''' is a significant whole-tone scale containing the otonal triads 12:15:17 and 8:10:13. It can be written without straddling as 2.9.5.13.17/3 6&28, but the 3 can be separated in the same way as the other temperaments in this article.

	Subgroup GTO string: 600 \| 2.<3.>3.5.13.<17.>17.23 \|\| 0 9 -8 -1 4 9 -8 3		Subgroup GTO string: 600 \| 2.<3.>3.5.13.<17.>17.23 \|\| 0 9 -8 -1 4 9 -8 3; 210.145¢

	It is related to Antikythera, but that temperament is 2.9.5.7 and once again tempers out 50/49 as it is specifically a subset of Pajara.		It is related to Antikythera, but that temperament is 2.9.5.7 and once again tempers out 50/49 as it is specifically a subset of Pajara.

			=== Diaschismic oneirotonic ===

			Some tunings of oneirotonic generate a slightly sharp 9/8 that is good for Diaschismic, providing an opportunity to tune 5 well besides A-team. In this case, neither of the straddled versions of 3 are diatonic. This allows the generator to correspond to a simple ratio, 13/7. The non-straddle version of this temperament is 2.9.5.7.11.13 28&46, which may be called '''Diaschismic Sensibeta''' after its 2.9.5.7 commas.

			17-limit GTO string: 600 \| 2.<3.>3.5.7.11.13.<17.>17 \|\| 0 -4 1 3 -11 -5 -12 -4 1; 129.835 ¢

User:Ground

Ground — Fri, 28 Aug 2026 18:15:03 GMT

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	*** [[86edo]] (TODO: make this 43edo)		*** [[86edo]] (TODO: make this 43edo)
	*** Many others without articles or as sections of other articles		*** Many others without articles or as sections of other articles
			** Future video scripts:
			*** [[Ground's intro to tuning diversity]]
			*** Music theory from the perspective of tuning theory
	* I'm a social construct.		* I'm a social construct.
	** There is nothing useful to know about me.		** There is nothing useful to know about me.

Diaschismic straddle temperaments

Ground — Fri, 28 Aug 2026 18:08:14 GMT

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	'''Diaschismic whole-tone''' is a significant whole-tone scale containing the otonal triads 12:15:17 and 8:10:13. It can be written without straddling as 2.9.5.13.17/3 6&28, but the 3 can be separated in the same way as the other temperaments in this article.		'''Diaschismic whole-tone''' is a significant whole-tone scale containing the otonal triads 12:15:17 and 8:10:13. It can be written without straddling as 2.9.5.13.17/3 6&28, but the 3 can be separated in the same way as the other temperaments in this article.

	Subgroup GTO string: 600 \| 2.<3.>3.5.13.<17.>17 \|\| 0 9 -8 -1 4 9 -8		Subgroup GTO string: 600 \| 2.<3.>3.5.13.<17.>17.23 \|\| 0 9 -8 -1 4 9 -8 3

	It is related to Antikythera, but that temperament is 2.9.5.7 and once again tempers out 50/49 as it is specifically a subset of Pajara.		It is related to Antikythera, but that temperament is 2.9.5.7 and once again tempers out 50/49 as it is specifically a subset of Pajara.

User:Ground

Ground — Fri, 28 Aug 2026 17:55:38 GMT

← Older revision		Revision as of 17:55, 28 August 2026
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	** [[List of xenharmonic musicians]]		** [[List of xenharmonic musicians]]
	* I compose and theorize.		* I compose and theorize.
			** A good portion of my theory work has been with [[User:Inthar]] and he's done a great job at documentation.
	** Bad theories:		** Bad theories:
	*** [[Aberrisma]] ~~(shared with [[User:Inthar]])~~		*** [[Aberrisma]]
	*** [[Straddle primes]] ~~(most of it, including nomenclature, eracs, and SRTT)~~		*** [[Straddle primes]]
	*** [[Ground's composition theory]]		*** [[Ground's composition theory]]
	** Even worse temperaments:		** Even worse temperaments:
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	*** [[Monarch\|Monarch and Interarch]]		*** [[Monarch\|Monarch and Interarch]]
	*** [[Straddle cluster temperaments]]		*** [[Straddle cluster temperaments]]
	*** [[86edo]]		*** [[Diaschismic straddle temperaments]]
	*** Many others without articles		*** [[86edo]] (TODO: make this 43edo)
			*** Many others without articles or as sections of other articles
	* I'm a social construct.		* I'm a social construct.
	** There is nothing useful to know about me.		** There is nothing useful to know about me.

Diaschismic straddle temperaments

Ground — Fri, 28 Aug 2026 14:27:45 GMT

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	{{proposed}}		{{proposed}}
	~~{{stub}}~~

	[[Diaschismic]] tempers out 2048/2025 = 2/((9/8)(5/4))^2. ~~Normally the~~ subgroup is interpreted as 2.3.5, but 2.9.5 is also possible. This provides a great opportunity for [[Straddle primes\|straddling]] 3, where flat 3 (3/a in SRTT) and sharp 3 (3a) cancel out to make 9.		[[Diaschismic]] tempers out 2048/2025 = 2/((9/8)(5/4))^2. The subgroup is usually interpreted as 2.3.5, but 2.9.5 is also possible. This provides a great opportunity for [[Straddle primes\|straddling]] 3, where flat 3 (3/a in SRTT) and sharp 3 (3a) cancel out to make 9.

	Of the Diaschismic extensions to the 7-limit, Septimal Diaschismic tempers out 126/125, ~~another~~ 2.9.5.7 comma~~, and~~ Pajara uses the sharp 3, tempering out 64/(3a*7) = 64/63 a^-1. These are ~~equivalent~~.		Pajara, Hedgehog, and Echidna are related temperaments, but they temper out 50/49, which damages 5 and 7. Straddling 3 allows both 2048/2025 and 64/63 a^-1 to be tempered out without doing this.

			== Straddle Diaschismic ==

			'''Straddle Diaschismic''' is the simplest temperament in this category. It is rank-3 and generated by the flat and sharp versions of 3. Of the Diaschismic extensions to the 7-limit:
			* Septimal Diaschismic tempers out 126/125, a 2.9.5.7 comma requiring no straddling because it has 9 rather than 3.
			* Pajara uses the sharp 3, tempering out 64/(3a*7) = 64/63 a^-1.
			These both result in the same extension. And as in Diaschismic, the half-octave can be interpreted as 17/12, implying that 17 is straddled like 3. The final resulting SRTT subgroup is 2.3a.5.7.17a.a^2, where 3a/(a^2) = 3/a. This can be converted to standard RTT to avoid straddling, resulting in 2.9.5.7.11.289 22&30.

			Much of what is useful here can be found in a rank-2 restriction, 2.3a.7.17a (the alteration ''a'' is redundant because sharp 3 and 17 are used with no straddling). 2.3.7.17 10&22 tempers out 64/63 and 289/288 and adds 17/14 to Archy, creating a useful symmetry between subminor/supermajor and supraminor/submajor chords.

	== Diaschismic Porcupine ==		== Diaschismic Porcupine ==
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	'''Diaschismic Porcupine''' tempers out 2048/2025 and 250/243 a^-5, the Porcupine comma with 3 replaced with 3a, as 3 is usually tuned quite sharp. Without straddling 3, only 22edo tempers out 2048/2025 and 250/243 in 2.3.5.		'''Diaschismic Porcupine''' tempers out 2048/2025 and 250/243 a^-5, the Porcupine comma with 3 replaced with 3a, as 3 is usually tuned quite sharp. Without straddling 3, only 22edo tempers out 2048/2025 and 250/243 in 2.3.5.

	~~Porcupine mappings extend to~~ the ~~11-limit~~ and ~~the~~ Diaschismic ~~mapping adds prime 17, implying that 17~~ is ~~straddled like 3~~. ~~13 may be added using the Oceanfront mapping of 81~~/~~64 a~~^~~4 = 13~~/10.		The simplest interpretation of the generator is 9/7, which also generates Hedgehog and Stearn. Straddle Diaschismic is equivalent to Stearn in the 2.9.7 subgroup, tempering out 118098/117649 = ((9/7)/(7/6)*(9/7))^2/2. This is an extremely efficient temperament.

	One possible ~~GTO string~~ resembling the 74edo tuning is 600 \| 2.<3.>>3.>5.>7.11.<17.>>17 \|\| 0 8 -3 -5 6 -4 8 -3. This focuses on approximating 9/5, 9/7, 17/5, 17/7, and 11/9 about equally well, with >17/14 equated to <11/9.		Mappings of 5.7.11 are the same as Porcupine, and the Diaschismic mapping adds prime 17. 13 may be added using the Oceanfront mapping equating 81/64 a^4 to 13/10.

			17-limit GTO string: 600 \| 2.<3.>3.5.7.11.13.<17.>17 \|\| 0 8 -3 -5 6 -4 -17 8 -3

			One possible erac group resembling the 74edo tuning is 600 \| 2.<3.>>3.>5.>7.11.>13.<17.>>17 \|\| 0 8 -3 -5 6 -4 -17 8 -3. This focuses on approximating 9/5, 9/7, 17/5, 17/7, and 11/9 about equally well, with >17/14 equated to <11/9.

	== Diaschismic Semioceanfront ==		== Diaschismic Semioceanfront ==

	'''Diaschismic Semioceanfront''' has a generator of half of an Oceanfront fourth rather than a third. The generator is also equivalent to 13/8 due to the half-octave period.		'''Diaschismic Semioceanfront''' has a generator of half of an Oceanfront fourth rather than a third. The generator is also equivalent to 13/8 due to the half-octave period.

			17-limit GTO string: 600 \| 2.<3.>3.5.7.11.13.<17.>17 \|\| 0 -2 -7 9 4 -15 1 -2 -7

			== Diaschismic Hemiseven ==

			[[Hemiseven]] may already be interpreted as straddle-3, but being a 5edo cluster temperament, all versions have a complex 5. This is fixed using Diaschismic with the 9 made by stacking the sharp 3 generator and the flatter 3.

			Subgroup GTO string: 600 \| 2.<3.>3.5.7.<13.>13.<17.>17 \|\| 0 6 1 -7 -2 2 -3 6 1

			== Diaschismic whole-tone ==

			'''Diaschismic whole-tone''' is a significant whole-tone scale containing the otonal triads 12:15:17 and 8:10:13. It can be written without straddling as 2.9.5.13.17/3 6&28, but the 3 can be separated in the same way as the other temperaments in this article.

			Subgroup GTO string: 600 \| 2.<3.>3.5.13.<17.>17 \|\| 0 9 -8 -1 4 9 -8

			It is related to Antikythera, but that temperament is 2.9.5.7 and once again tempers out 50/49 as it is specifically a subset of Pajara.

User talk:Ground

Inthar — Fri, 28 Aug 2026 02:38:33 GMT

Inthar deleted page User talk:Ground content was: "== 27-08-2027 incident == I apologize for the thing in #blabber, it was just a bunch of my issues or competing interests flaring up. I'm okay now, I have no thoughts of harming myself or others. Inthar (talk) 02:34, 28 August 2026 (UTC)", and the only contributor was "Inthar" (talk)

Template:Navbox EDO

Ground — Fri, 28 Aug 2026 02:34:26 GMT

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	\| Data 7 = [[60edo\|60]] • [[63edo\|63]] • [[64edo\|64]] • [[65edo\|65]] • [[67edo\|67]] • [[68edo\|68]] • [[70edo\|70]]		\| Data 7 = [[60edo\|60]] • [[63edo\|63]] • [[64edo\|64]] • [[65edo\|65]] • [[67edo\|67]] • [[68edo\|68]] • [[70edo\|70]]
	\| Header 8 = 72-83		\| Header 8 = 72-83
	\| Data 8 = [[72edo\|72]] • [[77edo\|77]] • [[80edo\|80]] • [[81edo\|81]] • [[82edo\|82]]		\| Data 8 = [[72edo\|72]] • [[74edo\|74]] • [[77edo\|77]] • [[80edo\|80]] • [[81edo\|81]] • [[82edo\|82]]
	\| Header 9 = 84-95		\| Header 9 = 84-95
	\| Data 9 = [[84edo\|84]] • [[87edo\|87]] • [[89edo\|89]] • [[90edo\|90]] • [[93edo\|93]] • [[94edo\|94]]		\| Data 9 = [[84edo\|84]] • [[87edo\|87]] • [[89edo\|89]] • [[90edo\|90]] • [[93edo\|93]] • [[94edo\|94]]

74edo

Ground — Fri, 28 Aug 2026 02:33:26 GMT

Redirected page to 37edo#74edo

New page

#redirect [[37edo#74edo]]

37edo

Ground — Fri, 28 Aug 2026 02:32:35 GMT

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	== Multiples ==		== Multiples ==
	=== 74edo ===		=== 74edo ===
	74edo is best known as a nearly optimal tuning of Grosstone, an extension of Septimal Meantone. Its flat 3 and 37edo's sharp 3 stack to make a sharp 9, which fills some gaps in 37edo's sharp-tending 13-limit. It ~~adds~~ a half-octave period which turns Porcupine into Stearn, a very efficient 2.3.7 restriction of Hedgehog. It is a decent [[Monarch]] tuning.		74edo is best known as a nearly optimal tuning of Grosstone, an extension of Septimal Meantone. Its flat 3 and 37edo's sharp 3 stack to make a sharp 9, which fills some gaps in 37edo's sharp-tending 13-limit. It is an excellent tuning of [[Diaschismic straddle temperaments\|Diaschismic straddle temperaments]], adding a half-octave period which turns Porcupine into Stearn, a very efficient 2.3.7 restriction of Hedgehog. It is a decent [[Monarch]] tuning.
	{{Harmonics in ED\|74\|37\|0}}		{{Harmonics in ED\|74\|37\|0}}
	=== 111edo ===		=== 111edo ===
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	{{UserTag\|g_\|Ground\|7766ff\|With no regard for structure, 37edo is an extremely special set of notes, which transcends structure. If you assume that approximating successive primes is good, this is a good tuning. If you think straddling 3 causes things to bend in a desirable way, this is the perfect tuning. After years of experience, I think 37edo is best described as 2.<<3.>3.5.7.11.13.17.19.		{{UserTag\|g_\|Ground\|7766ff\|With no regard for structure, 37edo is an extremely special set of notes, which transcends structure. If you assume that approximating successive primes is good, this is a good tuning. If you think straddling 3 causes things to bend in a desirable way, this is the perfect tuning. After years of experience, I think 37edo is best described as 2.<<3.>3.5.7.11.13.17.19.

	74 doesn't change much over 37, just adding a flat 3 and a reasonable 23, but the sharp 9 makes a big difference in available intervals, especially 9/7, which I think is the most obvious gap in 37edo. A more granular erac subgroup is 2.<3.>>>3.<<<5.>5.<<<7.>7.11.<<<13.>13.<<17.>>17.<19.>>>19.<<<23.>23 (this distribution is about optimal for overall accuracy of erac representations, but an interval directly between <<<<3 and >>3 as in 37edo would normally be <3).		74 doesn't change much over 37, just adding a flat 3 and a reasonable 23, but the sharp 9 makes a big difference in available intervals, especially 9/7, which I think is the most obvious gap in 37edo. I think that Diaschismic Porcupine/Semioceanfront are by far the most important new structures, not just to 74 as a multiple of 37, but 74 in general.

			A more granular erac subgroup in 74edo is 2.<3.>>>3.<<<5.>5.<<<7.>7.11.<<<13.>13.<<17.>>17.<19.>>>19.<<<23.>23 (this distribution is about optimal for overall accuracy of erac representations, but an interval directly between <<<<3 and >>3 as in 37edo would normally be <3).

	I use 37edo in the piano roll with six channels to reach 222, the superset of 74 and 111, which are a weird combination. 74 is a similar extension in spirit of 37, filling in missing intervals and continuing to straddle 3. On the other hand, 111 is a microtempered Buzzard edo (53+58) where the 17-limit JI lattice is more useful than straddling in most cases. 222 just adds extra intervals including a better flat 5 and 7 which I don't expect to use often.		I use 37edo in the piano roll with six channels to reach 222, the superset of 74 and 111, which are a weird combination. 74 is a similar extension in spirit of 37, filling in missing intervals and continuing to straddle 3. On the other hand, 111 is a microtempered Buzzard edo (53+58) where the 17-limit JI lattice is more useful than straddling in most cases. 222 just adds extra intervals including a better flat 5 and 7 which I don't expect to use often.

Diaschismic straddle temperaments

Ground — Fri, 28 Aug 2026 02:17:48 GMT

Created page with "{{proposed}} {{stub}} Diaschismic tempers out 2048/2025 = 2/((9/8)(5/4))^2. Normally the subgroup is interpreted as 2.3.5, but 2.9.5 is also possible. This provides a great opportunity for straddling 3, where flat 3 (3/a in SRTT) and sharp 3 (3a) cancel out to make 9. Of the Diaschismic extensions to the 7-limit, Septimal Diaschismic tempers out 126/125, another 2.9.5.7 comma, and Pajara uses the sharp 3, tempering out 64/(3a*7) = 64/63 a^-1. Th..."

New page

{{proposed}}
{{stub}}

[[Diaschismic]] tempers out 2048/2025 = 2/((9/8)(5/4))^2. Normally the subgroup is interpreted as 2.3.5, but 2.9.5 is also possible. This provides a great opportunity for [[Straddle primes|straddling]] 3, where flat 3 (3/a in SRTT) and sharp 3 (3a) cancel out to make 9.

Of the Diaschismic extensions to the 7-limit, Septimal Diaschismic tempers out 126/125, another 2.9.5.7 comma, and Pajara uses the sharp 3, tempering out 64/(3a*7) = 64/63 a^-1. These are equivalent.

== Diaschismic Porcupine ==

'''Diaschismic Porcupine''' tempers out 2048/2025 and 250/243 a^-5, the Porcupine comma with 3 replaced with 3a, as 3 is usually tuned quite sharp. Without straddling 3, only 22edo tempers out 2048/2025 and 250/243 in 2.3.5.

Porcupine mappings extend to the 11-limit and the Diaschismic mapping adds prime 17, implying that 17 is straddled like 3. 13 may be added using the Oceanfront mapping of 81/64 a^4 = 13/10.

One possible GTO string resembling the 74edo tuning is 600 | 2.<3.>>3.>5.>7.11.<17.>>17 || 0 8 -3 -5 6 -4 8 -3. This focuses on approximating 9/5, 9/7, 17/5, 17/7, and 11/9 about equally well, with >17/14 equated to <11/9.

== Diaschismic Semioceanfront ==

'''Diaschismic Semioceanfront''' has a generator of half of an Oceanfront fourth rather than a third. The generator is also equivalent to 13/8 due to the half-octave period.

53edo

Hotcrystal0 — Thu, 27 Aug 2026 17:13:03 GMT

standardize 1 decimal place cent values

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	\|-		\|-
	!Cents		!Cents
	\|249		\|249.1
	\|~~272~~		\|271.7
	\|'''294'''		\|'''294.3'''
	\|317		\|317
	\|~~340~~		\|339.6
	\|362		\|362.3
	\|~~385~~		\|384.9
	\|'''~~408~~'''		\|'''407.5'''
	\|430		\|430.2
	\|~~453~~		\|452.8
	\|-		\|-
	!Just interpretation		!Just interpretation

Aberschismic

Hotcrystal0 — Thu, 27 Aug 2026 17:06:35 GMT

The current name is aberschismic so that doens’t apply anymore

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	# equates the Pythagorean chroma with 15/14		# equates the Pythagorean chroma with 15/14
	# equates the Pythagorean augmented fourth with 10/7		# equates the Pythagorean augmented fourth with 10/7
	# splits 36/35 into two equal parts each representing the "common comma" 81/80 ~ 64/63 ~~(hence 36/35 might humorously be referred to as the "famity")~~		# splits 36/35 into two equal parts each representing the "common comma" 81/80 ~ 64/63
	# splits 7/5 into two equal parts each representing 32/27		# splits 7/5 into two equal parts each representing 32/27
	# makes 28/27 - 21/20 - 16/15 - 27/25 equidistant and 35/32 - 10/9 - 9/8 - 8/7 equidistant		# makes 28/27 - 21/20 - 16/15 - 27/25 equidistant and 35/32 - 10/9 - 9/8 - 8/7 equidistant

37edo

Ground — Thu, 27 Aug 2026 13:01:12 GMT

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	</math>}}		</math>}}

	{{Adv\|I don’t know if this is audibly significant, but at least it’s a cool justification for specifically 37edo as a neogothic blackdye tuning.}}		{{Adv\|I don’t know if this is audibly significant, but at least it’s a cool justification for specifically 37edo as a neogothic blackdye tuning. It is also a good tuning of "[[Ground's composition theory#Quasi-diatonic scales\|hiberidian]]"}}

			74edo scales:
			* 12:5:2 Diasem (eridian meantone)
			* 12:4:3 Diasem (Animist detemper of semiquartal, supermajor third is ~9/7 and subfourth is ~13/10)
			* Half-octave Porcupine (9/7)
			* Half-octave Oceanfront (17/14)

			111edo scales:
			* 5L2m3s4t
			* Vengeance

	== Multiples ==		== Multiples ==
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	{{UserTag\|g_\|Ground\|7766ff\|With no regard for structure, 37edo is an extremely special set of notes, which transcends structure. If you assume that approximating successive primes is good, this is a good tuning. If you think straddling 3 causes things to bend in a desirable way, this is the perfect tuning. After years of experience, I think 37edo is best described as 2.<<3.>3.5.7.11.13.17.19.		{{UserTag\|g_\|Ground\|7766ff\|With no regard for structure, 37edo is an extremely special set of notes, which transcends structure. If you assume that approximating successive primes is good, this is a good tuning. If you think straddling 3 causes things to bend in a desirable way, this is the perfect tuning. After years of experience, I think 37edo is best described as 2.<<3.>3.5.7.11.13.17.19.

	74 doesn't change much over 37, just adding a flat 3 and a reasonable 23, but the sharp 9 makes a big difference in available intervals, especially 9/7, which I think is the most obvious gap in 37edo. A more granular erac subgroup is 2.<3.>>>3.<<<5.>5.<<<7.>7.11.<<<13.>13.<<17.>>17.<19.>>>19.<<<23.>23 (this distribution is about optimal for overall accuracy).		74 doesn't change much over 37, just adding a flat 3 and a reasonable 23, but the sharp 9 makes a big difference in available intervals, especially 9/7, which I think is the most obvious gap in 37edo. A more granular erac subgroup is 2.<3.>>>3.<<<5.>5.<<<7.>7.11.<<<13.>13.<<17.>>17.<19.>>>19.<<<23.>23 (this distribution is about optimal for overall accuracy of erac representations, but an interval directly between <<<<3 and >>3 as in 37edo would normally be <3).

	I use 37edo in the piano roll with six channels to reach 222, the superset of 74 and 111, which are weird ~~together~~. 74 is a ~~pretty~~ similar extension of 37, ~~whereas~~ 111 is a microtempered Buzzard edo, 53+58.		I use 37edo in the piano roll with six channels to reach 222, the superset of 74 and 111, which are a weird combination. 74 is a similar extension in spirit of 37, filling in missing intervals and continuing to straddle 3. On the other hand, 111 is a microtempered Buzzard edo (53+58) where the 17-limit JI lattice is more useful than straddling in most cases. 222 just adds extra intervals including a better flat 5 and 7 which I don't expect to use often.
	}}		}}

	{{Navbox EDO}}		{{Navbox EDO}}
	{{Cat\|edos}}		{{Cat\|edos}}

Main Page

Inthar — Thu, 27 Aug 2026 06:45:01 GMT

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	Welcome to the '''Xenharmonic Reference (XR)''', a place for documentation and sharing of xenharmonic music and theory, dedicated to ''~~the~~ development of xenharmonic practice''. As a result of being focused on compositional theory, it aims to be accessible to the interested beginner while also containing [[compositional theory\|niche compositional theories]] that may become a new established compositional approach.		Welcome to the '''Xenharmonic Reference (XR)''', a place for documentation and sharing of xenharmonic music and theory, dedicated to the ''development of xenharmonic practice''. As a result of being focused on compositional theory, it aims to be accessible to the interested beginner while also containing [[compositional theory\|niche compositional theories]] that may become a new established compositional approach.

	==== If you can explain how a concept was derived and why it's useful, it belongs here. ====		==== If you can explain how a concept was derived and why it's useful, it belongs here. ====

37edo

Ground — Thu, 27 Aug 2026 05:10:08 GMT

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	== Multiples ==		== Multiples ==
			=== 74edo ===
			74edo is best known as a nearly optimal tuning of Grosstone, an extension of Septimal Meantone. Its flat 3 and 37edo's sharp 3 stack to make a sharp 9, which fills some gaps in 37edo's sharp-tending 13-limit. It adds a half-octave period which turns Porcupine into Stearn, a very efficient 2.3.7 restriction of Hedgehog. It is a decent [[Monarch]] tuning.
			{{Harmonics in ED\|74\|37\|0}}
	=== 111edo ===		=== 111edo ===
	111edo is a ~~significant~~ multiple of 37edo that adds an accurate prime 3. It is notable for being an [[Aberschismic]] system (tempering out 5120/5103, equating 81/80 and 64/63)~~; it is also~~ a [[Buzzard]] tuning with exact [[interordinal]] intervals. It is consistent in the 21-odd-limit ~~though~~ sharp-tending.		111edo is a multiple of 37edo that adds an accurate prime 3 to 37edo's 13-limit, making it significant to RTT. It is notable for being an [[Aberschismic]] system (tempering out 5120/5103, equating 81/80 and 64/63) and a [[Buzzard]] tuning with exact [[interordinal]] intervals. It is highly accurate in the 17-limit and consistent in the 21-odd-limit, although notably sharp-tending. It contains good tunings of [[Pentagoth\|Cohemimabila]] and [[Hibernal]].
	{{Harmonics in ED\|111\|37\|0}}		{{Harmonics in ED\|111\|37\|0}}

			{{UserTag\|g_\|Ground\|7766ff\|With no regard for structure, 37edo is an extremely special set of notes, which transcends structure. If you assume that approximating successive primes is good, this is a good tuning. If you think straddling 3 causes things to bend in a desirable way, this is the perfect tuning. After years of experience, I think 37edo is best described as 2.<<3.>3.5.7.11.13.17.19.

			74 doesn't change much over 37, just adding a flat 3 and a reasonable 23, but the sharp 9 makes a big difference in available intervals, especially 9/7, which I think is the most obvious gap in 37edo. A more granular erac subgroup is 2.<3.>>>3.<<<5.>5.<<<7.>7.11.<<<13.>13.<<17.>>17.<19.>>>19.<<<23.>23 (this distribution is about optimal for overall accuracy).

			I use 37edo in the piano roll with six channels to reach 222, the superset of 74 and 111, which are weird together. 74 is a pretty similar extension of 37, whereas 111 is a microtempered Buzzard edo, 53+58.
			}}

	{{Navbox EDO}}		{{Navbox EDO}}
	{{Cat\|edos}}		{{Cat\|edos}}

Generator sequence

Inthar — Wed, 26 Aug 2026 06:51:01 GMT

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	{{Expert}}		{{Expert}}

	A '''generator sequence''' (GS) is a cyclically repeating sequence of stacked intervals called ''generators''. A GS can be denoted:		A '''generator sequence''' ('''GS''') is a cyclically repeating sequence of stacked intervals called ''generators''. A GS can be denoted:

	GS(interval1, interval2, interval3, ..., intervaln),		GS(interval1, interval2, interval3, ..., intervaln),

GS

Inthar — Wed, 26 Aug 2026 06:50:38 GMT

Redirected page to Generator sequence

New page

#redirect [[Generator sequence]]

Aberrisma

Inthar — Wed, 26 Aug 2026 06:50:10 GMT

Extension to higher ranks

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	== Extension to higher ranks ==		== Extension to higher ranks ==
	Aberrismic theory is typically concerned with ternary scales, especially those with two chains of generators of the same length or differing by one, meaning that they have ~~an AGS~~ with only two intervals. This is because they are not much harder to navigate than an MOS while offering significantly more flexibility.		Aberrismic theory is typically concerned with ternary scales, especially those with two chains of generators of the same length or differing by one, meaning that they have a [[GS]] with only two intervals. This is because they are not much harder to navigate than an MOS while offering significantly more flexibility.

	{{UserTag\|g_\|Ground\|7766ff\|I propose that we call this type of scale '''bicatenary'''.}}		{{UserTag\|g_\|Ground\|7766ff\|I propose that we call this type of scale '''bicatenary'''.}}

Aberrisma

Ground — Wed, 26 Aug 2026 03:50:17 GMT

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	* '''Solitone''': The melodic function of the L step in quasi-diatonic aberrismic scales.		* '''Solitone''': The melodic function of the L step in quasi-diatonic aberrismic scales.
	* '''Subaberrisma''': A step so small (smaller than the typical limit for an aberrisma) that its status as a melodic step is unclear.		* '''Subaberrisma''': A step so small (smaller than the typical limit for an aberrisma) that its status as a melodic step is unclear.

			== Extension to higher ranks ==
			Aberrismic theory is typically concerned with ternary scales, especially those with two chains of generators of the same length or differing by one, meaning that they have an AGS with only two intervals. This is because they are not much harder to navigate than an MOS while offering significantly more flexibility.

			{{UserTag\|g_\|Ground\|7766ff\|I propose that we call this type of scale '''bicatenary'''.}}

			However, this is often not enough to accurately represent a larger JI subgroup. Portent penslen (5L5m6s) is a somewhat unique counterexample, being bicatenary yet microtempered in the 11-limit. The solution is to add more step sizes while maintaining as much lower-rank structure as possible. For example, while penslen is found in edos like 41 and 46, the equivalent in the similarly LCJI-accurate 53 and 58 is 5L2m3s4t, a quaternary subset of Buzzard[28] with a JI tuning of 35/32 : 21/20 : 36/35 : 64/63. [[Ground's composition theory#Interval logic deviation theory\|ILD]] is a simple way to access notes outside of a scale with a higher rank.

			A common tempering of blackdye involves a sharper 3 to significantly inflate 81/80, both for melodic utility and to equate 27/20 to a superfourth like 15/11 or 11/8. It also becomes natural to temper out 64/63. Possibly the most common ratio representation of an aberrisma is 36/35, but few ternary scales use it without tempering out 81/80 or 64/63.

			(81/80)^2 = 43¢ is notably in the middle of the usual aberrisma size range as well. This does not exist in blackdye, but becomes accessible in certain variants of 5L2m8s as well as numerous quaternary scales with even more options for available intervals.

	== Compositional examples ==		== Compositional examples ==

Ground's composition theory

Ground — Mon, 24 Aug 2026 04:16:56 GMT

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	== Anchor concordances ==		== Anchor concordances ==

	~~Hiiiiiiiiiiiiiiii another incomplete section :3~~		We conventionally make the distinction between consonance, which is dependent on musical context, and concordance, which is not. This gives the false impression that concordance does not vary significantly between people, and based on register and timbre. Dyadic harmonic entropy is the most well-known measure of concordance, and it works pretty well for me, but not everyone. Triadic harmonic entropy fits less well because I've personally noticed that having at least one interval in a chord tuned close to LCJI sounds more concordant than balancing the errors of multiple LCJI intervals. I refer to this special interval as an '''anchor concordance''' because it stabilizes the whole chord. This way of optimization is opposed to delta-rational, which I also find useful.

	~~I have found that tempered chords can be substantially stabilized by having a somewhat accurate LCJI ratio.~~ For example, the patent 3/2 in 25 and 28edo is far off, but their representations of 4:5:6 mask the beating with an accurate 5/4. I prefer to have ~~these intervals~~ be thirds or sometimes sixths because they show up the most often chords, ~~especially~~ major thirds. My theory is that this is because the major third range has stronger concordance wells than minor thirds, which have something close to a plateau on the harmonic entropy curve. My favorites to have are 5/4, 9/7, and 13/10.		For example, the patent 3/2 in 25 and 28edo is far off, but their representations of 4:5:6 mask the beating with an accurate 5/4. I prefer to have anchor concordances be 3/2, thirds, or sometimes sixths because they show up the most often in tertian chords, and major thirds in particular. My theory is that this is because the major third range has stronger concordance wells than minor thirds, which have something close to a plateau on the harmonic entropy curve. My favorites to have are 5/4, 9/7, and 13/10.

			Compare the different optimizations of 2.3.5 Meantone which I have noticed sound better than those immediately around them:
			* ~69edo: DR 4:5:6, 10:12:15
			* ~31edo: 5/4 anchor concordance
			* ~74edo: DR 3:4:5
			* ~67edo: 1/6-comma (equal error on 9 and 5, around where 3 becomes truly stackable by my standards)

	== Quartertone composition ==		== Quartertone composition ==

19edo

Hotcrystal0 — Sun, 23 Aug 2026 21:17:48 GMT

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	=== 57edo ===		=== 57edo ===
	57edo is a good no-threes no-fives 19-limit or 2.5/3.7.11.13.19 subgroup system.		57edo is a good no-threes no-fives 19-limit or 2.5/3.7.11.13.17.19 subgroup system.
	{{Harmonics in ED\|57}}		{{Harmonics in ED\|57}}

	=== 171edo ===		=== 171edo ===
	Combining 19edo's accurate 28/27 and 6/5 with [[9edo]]'s accurate 7/6 results in 171edo which is a very accurate model of 7-limit JI, even more accurate than [[99edo]]. ~~Since its 14/13 is~~ inherited from ~~19edo~~, ~~its 13th harmonic~~ is acceptable too. 171edo supports [[Schismic]] and [[Ennealimmal]].		Combining 19edo's accurate 28/27 and 6/5 with [[9edo]]'s accurate 7/6 results in 171edo which is a very accurate model of 7-limit JI, even more accurate than [[99edo]]. Its 13th harmonic, inherited from 57edo, is acceptable too. 171edo supports [[Schismic]] and [[Ennealimmal]].
	{{Harmonics in ED\|171}}		{{Harmonics in ED\|171}}

Hemiseven

Ground — Sun, 23 Aug 2026 17:38:58 GMT

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	Vengeance is easy to add by making this temperament half-octave. Perhaps it could be called Bisectanseventh as a pun. It can be shown that the generator is the same by approximating the subset temperament generators in a subgroup: 2.7.15/13.11/5.25/17 52 & 82. 52 & 66 provides a similarly useful neutral third generator but is unrelated.		Vengeance is easy to add by making this temperament half-octave. Perhaps it could be called Bisectanseventh as a pun. It can be shown that the generator is the same by approximating the subset temperament generators in a subgroup: 2.7.15/13.11/5.25/17 52 & 82. 52 & 66 provides a similarly useful neutral third generator but is unrelated.

			In short:
			* 52+5n is Hemiseven (Island Slendric, the mapping of 5 varies)
			* 52+15n adds Porcupine by using the very sharp 3 as the Hemiseven generator
			* 52+30n adds Vengeance generated by Porcupine's extremely inflated 81/80 plus 1\2
	}}		}}

	{{Navbox regtemp}}		{{Navbox regtemp}}
	{{Cat\|temperaments}}		{{Cat\|temperaments}}

Ground's composition theory

Ground — Fri, 21 Aug 2026 21:46:25 GMT

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User:Unque

Unque — Fri, 21 Aug 2026 12:55:54 GMT

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	Additionally, it can be noted that if one makes an [[Delta-rational chord\|isodifferential]] triad with an interval of 400c (the familiar major third from 12edo) between the bottom two pitches, this chord will have a "fifth" which very closely resembles the "fifth" of 5edo. Thus, we can assume that a tuning which contains 3edo and 5edo as subsets has a close approximation of this chord. This can be seen as an alternative way to "fix" the lack of harmonic effect in the 12edo major triad, which detunes the fifth rather than the third.		Additionally, it can be noted that if one makes an [[Delta-rational chord\|isodifferential]] triad with an interval of 400c (the familiar major third from 12edo) between the bottom two pitches, this chord will have a "fifth" which very closely resembles the "fifth" of 5edo. Thus, we can assume that a tuning which contains 3edo and 5edo as subsets has a close approximation of this chord. This can be seen as an alternative way to "fix" the lack of harmonic effect in the 12edo major triad, which detunes the fifth rather than the third.

			=== [[17edo]] ===
			While it's a common choice, it is so for good reason. It's the second edo to support diatonic, the first to distinguish enharmonic notes, and the first to contain nondiatonic intervals alongside diatonic ones, in the form of neutral intervals. It additionally has multiple 7-form MOS scales, including diatonic, mosh, and antidiatonic, allowing for modal interchange between these scales to create more complex chromatic harmonies.

	=== [[30edt]] ===		=== [[30edt]] ===
	There isn't too much to say about 30edt. Its most obvious application is as a stretched version of 19edo, which is desirable due to 19edo's consistent flat temperament of prime harmonics.		There isn't too much to say about 30edt. Its most obvious application is as a stretched version of 19edo, which is desirable due to 19edo's consistent flat temperament of prime harmonics.

	However, it can also be used on its own as a tritave-equivalent system. Because the tritave is significantly wider than the octave, using the semi-tritave as a period can be a good way to make scales more melodically coherent. The semi-tritave interval, unlike the semi-octave, is relatively consonant, and a very intuitive size for a period due to being precisely ~~half way~~ between the perfect fifth and the octave (the two most common spans for a scale).		However, it can also be used on its own as a tritave-equivalent system. Because the tritave is significantly wider than the octave, using the semi-tritave as a period can be a good way to make scales more melodically coherent. The semi-tritave interval, unlike the semi-octave, is relatively consonant, and a very intuitive size for a period due to being precisely halfway between the perfect fifth and the octave (the two most common spans for a scale).

	=== [[29edo]] ===		=== [[29edo]] ===

Trivial EDO

Hotcrystal0 — Thu, 20 Aug 2026 20:29:38 GMT

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	== 3edo ==		== 3edo ==
	The step of 3edo serves as 12edo's major third and also as 15edo's. When interpreted as 5/4, it supports ~~augmented~~ temperament, as 3edo is equivalent to an augmented triad in 12edo. It also serves as the period of ~~misty~~ temperament, which splits the deficit between three 5/4s and the octave into three. In general, 3edo provides a basic outline for triadic harmony. 3edo may be considered a 2.5.11/7.19/3 temperament, being the first edo to approximate a low prime other than 2.		The step of 3edo serves as 12edo's major third and also as 15edo's. When interpreted as 5/4, it supports [[Augmented (temperament)\|Augmented]] temperament, as 3edo is equivalent to an augmented triad in 12edo. It also serves as the period of [[Misty]] temperament, which splits the deficit between three 5/4s and the octave into three. In general, 3edo provides a basic outline for triadic harmony. 3edo may be considered a 2.5.11/7.19/3 temperament, being the first edo to approximate a low prime other than 2.

	3edo also simplifies 3:4:5-based harmony, reflecting the inaccurate temperament Father which equates 5/4 and 4/3.		3edo also simplifies 3:4:5-based harmony, reflecting the inaccurate temperament Father which equates 5/4 and 4/3.

User:Hotcrystal0/Sandbox

Hotcrystal0 — Thu, 20 Aug 2026 20:27:03 GMT

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	This page is for hotcrystal0's random tests.		This page is for hotcrystal0's random tests.

			{{Harmonics in ED\|270\|41\|0}}

	{{Harmonics in ED\|311\|41\|0}}		{{Harmonics in ED\|311\|41\|0}}

	~~{{UserTag\|hc0\|Hotcrystal0\|e1ebf2\|User tag test}}~~

	==Intervals==		==Intervals==

User:Hotcrystal0

Hotcrystal0 — Thu, 20 Aug 2026 20:22:23 GMT

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	[[User:Hotcrystal0/Proposals for xenreference\|Proposals for xenreference]]		[[User:Hotcrystal0/Proposals for xenreference\|Proposals for xenreference]]

	~~[[User:Hotcrystal0/Aleph briefer\|Aleph briefer]]~~

	[[User:Hotcrystal0/Earth26\|Earth#26]]		[[User:Hotcrystal0/Earth26\|Earth#26]]

	[[User:Hotcrystal0/27edo\|27edo]]		[[User:Hotcrystal0/27edo\|27edo]]

User:Hotcrystal0/Aleph briefer

Hotcrystal0 — Thu, 20 Aug 2026 20:22:12 GMT

Replaced content with "Unnecessary page. I might repurpose this for something else or ask for it to be deleted."

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	~~'''This~~ page ~~is intended to be TEMPORARILY hosted here until I can find a better wiki to host it~~.~~'''~~		Unnecessary page. I might repurpose this for something else or ask for it to be deleted.

	~~Aleph is a character created by a Discord user who~~ I~~'ll call "xp2" here. This is her explanation on the character:~~
	~~<blockquote>~~
	~~Aleph (she/her):~~
	~~Winged. Transfem. Autistic, ADHDer, and aroace. Colon three.~~
	~~Extremely averse to social interaction with most people she doesn't know, but extremely affectionate towards those she does.~~
	~~</blockquote>~~

	~~[insert explanation of Aleph's backstory here]~~

	~~Past this, it splits, with multiple "iterations" depending on where xp2 has inserted Aleph into. The iterations will be explained separately. The "point of divergence" is when~~ this ~~iteration's timeline diverges from that of other iterations.~~

	~~==Iterations==~~

	~~===1st iteration: Old Te[52]===~~
	~~Point of divergence: N/A (incorporated as fragments into the main Aleph backstory)~~
	~~<br />The first iteration of Aleph was created by xp2~~ for ~~a roleplay from 2020.~~

	~~===2nd iteration: Trans-wwei===~~
	~~Point of divergence: N/A (incorporated into the main Aleph backstory)~~
	~~<br />This is the second iteration, created~~ for ~~a thread on the CGoL forums in February 2025. Although the character was actually created by me,~~ it ~~was mainly developed by xp2~~.

	~~===3rd iteration===~~

	~~===4th iteration===~~

	~~===5th iteration===~~

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	[[User:Hotcrystal0/Proposals for xenreference\|Proposals for xenreference]]		[[User:Hotcrystal0/Proposals for xenreference\|Proposals for xenreference]]

	~~[[User:Hotcrystal0/Aleph briefer\|Aleph briefer]]~~

	[[User:Hotcrystal0/Earth26\|Earth#26]]		[[User:Hotcrystal0/Earth26\|Earth#26]]

	[[User:Hotcrystal0/27edo\|27edo]]		[[User:Hotcrystal0/27edo\|27edo]]