User:Unque - Revision history

Unque at 18:06, 2 March 2026

2026-03-02T18:06:18Z

Unque: 43edo

2026-01-29T22:50:26Z

43edo

← Older revision		Revision as of 22:50, 29 January 2026
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	Additionally, and perhaps more convincingly for practical composers, 41edo maps the perfect fifth to 24 steps. Just like with the perfect fourth of 29edo, this highly divisible interval allows for many useful melodic structures, including (but not limited to) [[Rastmic clan\|Neutral]], [[Slendric]], [[Tetracot]], and [[Miracle]].		Additionally, and perhaps more convincingly for practical composers, 41edo maps the perfect fifth to 24 steps. Just like with the perfect fourth of 29edo, this highly divisible interval allows for many useful melodic structures, including (but not limited to) [[Rastmic clan\|Neutral]], [[Slendric]], [[Tetracot]], and [[Miracle]].

			=== [[43edo]] ===
			By the size of 43edo, even rank-2 thinking is rather difficult for conceptualizing how the system fits together. However, 43edo can very intuitively be taken as a Septimal Meantone system with 45/44~56/55~100/99 as a secondary chroma to provide access to 11-limit intervals. Because the chromatic semitone of the diatonic scale is three of these undecimal chromata, any interval in the system can be notated with no more than one accidental of each kind.

			This threefold division of the chroma additionally allows the wholetone to be altered into a down-wholetone such that three of them make a perfect fourth instead of an augmented one; the structure begat by these down-wholetones resembles [[Porcupine]] in its form, but does not contain the classical minor third.

			Finally, 43edo offers a rather accurate tuning of [[Bleu]], the temperament which cleaves the perfect fifth into five parts; two of these parts reaches 7/6, three reach 14/11, and four reach 11/8.

	== Music ==		== Music ==

Unque: Replaced some wrong info on 29edo temperaments

2026-01-27T00:28:17Z

Replaced some wrong info on 29edo temperaments

← Older revision		Revision as of 00:28, 27 January 2026
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	I strongly believe that 29edo should be alongside systems such as 19edo and 31edo as introductions to xenharmonic tunings for beginners. Not only does it represent the third harmonic within less than two cents of error (making the diatonic scale roughly equivalent to its familiar representations in western music), but it additionally contains distinct, unambiguous representations for [[interordinal]] intervals. The ability to place these unfamiliar intervals onto the familiar circle of fifths is extremely beneficial, as it allows beginners to more clearly get a feel for how these intervals can be used to create sounds unavailable in 12edo. This provides a benefit over systems such as Meantone, in that the circle of fifths requires less relearning to account for the difference in intonation compared to 12edo, and in that the interordinals represented are distinct and unambiguous (compare to 31edo, where there is no clear representation for, say, the semifourth or the semisixth).		I strongly believe that 29edo should be alongside systems such as 19edo and 31edo as introductions to xenharmonic tunings for beginners. Not only does it represent the third harmonic within less than two cents of error (making the diatonic scale roughly equivalent to its familiar representations in western music), but it additionally contains distinct, unambiguous representations for [[interordinal]] intervals. The ability to place these unfamiliar intervals onto the familiar circle of fifths is extremely beneficial, as it allows beginners to more clearly get a feel for how these intervals can be used to create sounds unavailable in 12edo. This provides a benefit over systems such as Meantone, in that the circle of fifths requires less relearning to account for the difference in intonation compared to 12edo, and in that the interordinals represented are distinct and unambiguous (compare to 31edo, where there is no clear representation for, say, the semifourth or the semisixth).

	Additionally, 29edo finds the perfect fourth at 12 steps, a highly divisible size, supporting [[Porcupine]], [[28812/28561#Tesseract\|Tesseract]], ~~and~~ [[~~Unicorn~~]]. This helps provide an introduction into systems that divide simple intervals into a certain number of steps, and how those divisions can apply to writing melodies and chord progressions.		Additionally, 29edo finds the perfect fourth at 12 steps, a highly divisible size, supporting [[Porcupine]], [[28812/28561#Tesseract\|Tesseract]], [[Negri]], [[Semaphore and godzilla\|Semaphore]], and other similar structures. This helps provide an introduction into systems that divide simple intervals into a certain number of steps, and how those divisions can apply to writing melodies and chord progressions; see my treatise [[User:Unque/On Voice Leading\|on voice leading]] for a more detailed explanation of why this is important.

	Finally, for those who like microtemperaments, simple harmonics such as 5 and 7 are very easy to find in supersets such as [[58edo]] and [[87edo]], since these harmonics have a relative error very close to simple fractions. The perfect fifth of 29edo is optimal for [[parapyth]] tuning, which makes supersets of 29edo extremely desirable if one seeks an extremely high accuracy equal temperament sequence.		Finally, for those who like microtemperaments, simple harmonics such as 5 and 7 are very easy to find in supersets such as [[58edo]] and [[87edo]], since these harmonics have a relative error very close to simple fractions. The perfect fifth of 29edo is optimal for [[parapyth]] tuning, which makes supersets of 29edo extremely desirable if one seeks an extremely high accuracy equal temperament sequence.

Unque at 16:13, 15 January 2026

2026-01-15T16:13:41Z

← Older revision		Revision as of 16:13, 15 January 2026
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	* Methane Lamentation (31-EDO): https://youtu.be/CBmYRoej2yQ		* Methane Lamentation (31-EDO): https://youtu.be/CBmYRoej2yQ
	* Autumn (27-EDO): https://youtu.be/dcQe6ebpGFU		* Autumn (27-EDO): https://youtu.be/dcQe6ebpGFU
			* North Star (15-EDT): [https://youtu.be/ftfnW9DsFVE?si=ho0-eiFfeOYSAo96&t=795 https://youtu.be/ftfnW9DsFVE&t=795]
	* Winter (37-EDO): [https://youtu.be/rE9L56yZ1Kw?si=K9LGwj_VsbbAJn3H https://youtu.be/rE9L56yZ1Kw]		* Winter (37-EDO): [https://youtu.be/rE9L56yZ1Kw?si=K9LGwj_VsbbAJn3H https://youtu.be/rE9L56yZ1Kw]
			* September Sunset (18-EDO): https://youtu.be/HWUsCJUXOqg

	== Main Space Contributions ==		== Main Space Contributions ==
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	* The [[User:Unque/Dietic Minor\|Dietic Minor]] scale		* The [[User:Unque/Dietic Minor\|Dietic Minor]] scale
	* A [[User:Unque/29edo Counterpoint Treatise\|treatise]] on 29edo counterpoint		* A [[User:Unque/29edo Counterpoint Treatise\|treatise]] on 29edo counterpoint
			* A discussion [[User:Unque/On Voice Leading\|on voice leading]]
	I confine many of my pages to my user space to contain idiosyncrasies, jokes, niche topics, subpar writing quality, and other stuff that I don't believe deserves to be put on the main wiki space. Some of these pages may be moved over to main space pages if they are deemed to be well-written pages about legitimately applicable ideas, but for now they remain here.		I confine many of my pages to my user space to contain idiosyncrasies, jokes, niche topics, subpar writing quality, and other stuff that I don't believe deserves to be put on the main wiki space. Some of these pages may be moved over to main space pages if they are deemed to be well-written pages about legitimately applicable ideas, but for now they remain here.

Unque at 16:32, 5 December 2025

2025-12-05T16:32:50Z

← Older revision		Revision as of 16:32, 5 December 2025
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	[[File:Unque EDO tier list.png\|thumb\|An updated version of the tier list in question]]		[[File:Unque EDO tier list.png\|thumb\|An updated version of the tier list in question]]
	I am Uncreative Name, or Unque! I'm a music theorist, composer, performer, mathematician, worldbuilder, and linguist. Members of the community may know me as a vocal advocate for [[15edo\|tunings of suboptimal popularity]]; an [[User:Unque/Dhembrwood\|exotempering troll]]; a spewer of [[Regular temperament theory\|recreational mathematics]]; ~~or even~~ the author of [[:File:EDOs V2.png\|a tier list]] which was used in [https://www.tumblr.com/orteil42/779067449007079424/learning-music-theory-taking-me-places-i-dont a Tumblr post] teasing the xen wiki.		I am Uncreative Name, or Unque! I'm a music theorist, composer, performer, mathematician, worldbuilder, and linguist. Members of the community may know me as a vocal advocate for [[15edo\|tunings of suboptimal popularity]]; an [[User:Unque/Dhembrwood\|exotempering troll]]; a spewer of [[Regular temperament theory\|recreational mathematics]]; the author of [[:File:EDOs V2.png\|a tier list]] which was used in [https://www.tumblr.com/orteil42/779067449007079424/learning-music-theory-taking-me-places-i-dont a Tumblr post] teasing the xen wiki; and a [[User:Lériendil\|co-conspirator]] in the [[:Category:Worldbuilding\|plot]] that supposedly presents a vague and unclear threat to the [[History and philosophy of xenharmonic music\|real-life xenharmonic culture]]<sup>[citation needed]</sup>.

	I learned of the Arabic Maqamat several years ago when I was studying non-Western music traditions, and I was especially intrigued by the presence of the neutral intervals in Jins Rast and other scales. This kickstarted a weird rabbit hole where I learned about the harmonic series, Just Intonation, and various Equal Divisions of the [your name here].		I learned of the Arabic Maqamat several years ago when I was studying non-Western music traditions, and I was especially intrigued by the presence of the neutral intervals in Jins Rast and other scales. This kickstarted a weird rabbit hole where I learned about the harmonic series, Just Intonation, and various Equal Divisions of the [your name here].

Unque at 17:59, 17 November 2025

2025-11-17T17:59:11Z

← Older revision		Revision as of 17:59, 17 November 2025
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	** Tempering out the Octaphore provides a highly accurate temperament that cleaves 4/3 into eight equal parts.		** Tempering out the Octaphore provides a highly accurate temperament that cleaves 4/3 into eight equal parts.
	* The [[131072/130321]] comma		* The [[131072/130321]] comma
	** Tempering out this comma provides ~~an interpretation of 4~~-~~EDO,~~ much more accurate than [[648/625\|Diminished]].		** Tempering out this comma provides a quarter-octave temperament that is much more accurate than [[648/625\|Diminished]].
	* The [[28812/28561\|Tesseract comma]] and its related temperaments		* The [[28812/28561\|Tesseract comma]] and its related temperaments
	** Tempering out this comma provides cleaves 4/3 into four equal parts; it's not quite as accurate as the Octaphore, but it's supported by more ET sequences at lower complexity, and quite frankly has a much better name.		** Tempering out this comma provides cleaves 4/3 into four equal parts; it's not quite as accurate as the Octaphore, but it's supported by more ET sequences at lower complexity, and quite frankly has a much better name.

Unque at 15:46, 2 September 2025

2025-09-02T15:46:05Z

← Older revision		Revision as of 15:46, 2 September 2025
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	While 15edo does not provide accurate representations of the harmonic series, it does provide extremely useful melodic frameworks. The [[5L 5s\|Blackwood Decatonic scale]], for instance, contains several copies of [[Nicetone]] over each degree, allowing for diatonic-like chord progressions to move smoothly between keys that may seem unrelated in systems with a more accurate chain of fifths.		While 15edo does not provide accurate representations of the harmonic series, it does provide extremely useful melodic frameworks. The [[5L 5s\|Blackwood Decatonic scale]], for instance, contains several copies of [[Nicetone]] over each degree, allowing for diatonic-like chord progressions to move smoothly between keys that may seem unrelated in systems with a more accurate chain of fifths.

	Additionally, it can be noted that if one makes an [[Delta-rational chord\|Isodifferential ~~chord~~]] 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.

	=== [[30edt]] ===		=== [[30edt]] ===

Unque at 17:11, 31 August 2025

2025-08-31T17:11:45Z

← Older revision		Revision as of 17:11, 31 August 2025
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	* A [[User:Unque/Barbershop Tuning Theory\|study on tuning theory as applied to Barbershop music]]		* A [[User:Unque/Barbershop Tuning Theory\|study on tuning theory as applied to Barbershop music]]
	* The [[User:Unque/Dietic Minor\|Dietic Minor]] scale		* The [[User:Unque/Dietic Minor\|Dietic Minor]] scale
			* A [[User:Unque/29edo Counterpoint Treatise\|treatise]] on 29edo counterpoint
	I confine many of my pages to my user space to contain idiosyncrasies, jokes, niche topics, subpar writing quality, and other stuff that I don't believe deserves to be put on the main wiki space. Some of these pages may be moved over to main space pages if they are deemed to be well-written pages about legitimately applicable ideas, but for now they remain here.		I confine many of my pages to my user space to contain idiosyncrasies, jokes, niche topics, subpar writing quality, and other stuff that I don't believe deserves to be put on the main wiki space. Some of these pages may be moved over to main space pages if they are deemed to be well-written pages about legitimately applicable ideas, but for now they remain here.

Unque at 02:52, 25 August 2025

2025-08-25T02:52:07Z

← Older revision		Revision as of 02:52, 25 August 2025
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	[[File:Unque EDO tier list.png\|thumb\|An updated version of the tier list in question]]		[[File:Unque EDO tier list.png\|thumb\|An updated version of the tier list in question]]
	I am Uncreative Name, or Unque! I'm a music theorist, composer, performer, mathematician, worldbuilder, and linguist. Members of the community may know me as a vocal advocate for tunings of suboptimal popularity; an [[User:Unque/Dhembrwood\|exotempering troll]]; a spewer of [[Regular temperament theory\|recreational mathematics]]; or even the author of [[:File:EDOs V2.png\|a tier list]] which was used in [https://www.tumblr.com/orteil42/779067449007079424/learning-music-theory-taking-me-places-i-dont a Tumblr post] teasing the xen wiki.		I am Uncreative Name, or Unque! I'm a music theorist, composer, performer, mathematician, worldbuilder, and linguist. Members of the community may know me as a vocal advocate for [[15edo\|tunings of suboptimal popularity]]; an [[User:Unque/Dhembrwood\|exotempering troll]]; a spewer of [[Regular temperament theory\|recreational mathematics]]; or even the author of [[:File:EDOs V2.png\|a tier list]] which was used in [https://www.tumblr.com/orteil42/779067449007079424/learning-music-theory-taking-me-places-i-dont a Tumblr post] teasing the xen wiki.

	I learned of the Arabic Maqamat several years ago when I was studying non-Western music traditions, and I was especially intrigued by the presence of the neutral intervals in Jins Rast and other scales. This kickstarted a weird rabbit hole where I learned about the harmonic series, Just Intonation, and various Equal Divisions of the [your name here].		I learned of the Arabic Maqamat several years ago when I was studying non-Western music traditions, and I was especially intrigued by the presence of the neutral intervals in Jins Rast and other scales. This kickstarted a weird rabbit hole where I learned about the harmonic series, Just Intonation, and various Equal Divisions of the [your name here].

Unque at 15:35, 22 August 2025

2025-08-22T15:35:09Z

← Older revision		Revision as of 15:35, 22 August 2025
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	I strongly believe that 29edo should be alongside systems such as 19edo and 31edo as introductions to xenharmonic tunings for beginners. Not only does it represent the third harmonic within less than two cents of error (making the diatonic scale roughly equivalent to its familiar representations in western music), but it additionally contains distinct, unambiguous representations for [[interordinal]] intervals. The ability to place these unfamiliar intervals onto the familiar circle of fifths is extremely beneficial, as it allows beginners to more clearly get a feel for how these intervals can be used to create sounds unavailable in 12edo. This provides a benefit over systems such as Meantone, in that the circle of fifths requires less relearning to account for the difference in intonation compared to 12edo, and in that the interordinals represented are distinct and unambiguous (compare to 31edo, where there is no clear representation for, say, the semifourth or the semisixth).		I strongly believe that 29edo should be alongside systems such as 19edo and 31edo as introductions to xenharmonic tunings for beginners. Not only does it represent the third harmonic within less than two cents of error (making the diatonic scale roughly equivalent to its familiar representations in western music), but it additionally contains distinct, unambiguous representations for [[interordinal]] intervals. The ability to place these unfamiliar intervals onto the familiar circle of fifths is extremely beneficial, as it allows beginners to more clearly get a feel for how these intervals can be used to create sounds unavailable in 12edo. This provides a benefit over systems such as Meantone, in that the circle of fifths requires less relearning to account for the difference in intonation compared to 12edo, and in that the interordinals represented are distinct and unambiguous (compare to 31edo, where there is no clear representation for, say, the semifourth or the semisixth).

	Additionally, 29edo finds the perfect fourth at 12 steps, a highly divisible size. This helps provide an introduction into systems that divide simple intervals into a certain number of steps, and how those divisions can apply to writing melodies and chord progressions.		Additionally, 29edo finds the perfect fourth at 12 steps, a highly divisible size, supporting [[Porcupine]], [[28812/28561#Tesseract\|Tesseract]], and [[Unicorn]]. This helps provide an introduction into systems that divide simple intervals into a certain number of steps, and how those divisions can apply to writing melodies and chord progressions.

	Finally, for those who like microtemperaments, simple harmonics such as 5 and 7 are very easy to find in supersets such as [[58edo]] and [[87edo]], since these harmonics have a relative error very close to simple fractions. The perfect fifth of 29edo is optimal for [[parapyth]] tuning, which makes supersets of 29edo extremely desirable if one seeks an extremely high accuracy equal temperament sequence.		Finally, for those who like microtemperaments, simple harmonics such as 5 and 7 are very easy to find in supersets such as [[58edo]] and [[87edo]], since these harmonics have a relative error very close to simple fractions. The perfect fifth of 29edo is optimal for [[parapyth]] tuning, which makes supersets of 29edo extremely desirable if one seeks an extremely high accuracy equal temperament sequence.
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	=== [[36edo]] ===		=== [[36edo]] ===
	I don't have too much to say regarding 36edo. It is a superset of 12edo, which provides a very accurate representation of prime 7. It is roughly the optimal tuning for [[Slendric]] temperament, as well as a good EDO representation for 2.3.7 JI scales such as [[Nicetone\|Septimal Zarlino]] and [[Diasem]].		I don't have too much to say regarding 36edo. It is a superset of 12edo, which provides a very accurate representation of prime 7. It is roughly the optimal tuning for [[Slendric]] temperament, as well as a good EDO representation for 2.3.7 JI scales such as [[Nicetone\|Septimal Zarlino]] and [[Diasem]].

			=== [[41edo]] ===
			While a bit larger than the typical optimal size for practicality, physical instruments such as the [[Kite Guitar]] have made 41edo significantly more accessible than it may seem at first.

			In terms of tone organization, 41edo is extremely accurate and efficient. The first fifteen harmonics are practically indistinguishable from JI (potentially excluding 13), and the edostep acts as an all-purpose formal comma, representing [[100/99\|S10]]~[[81/80\|S9]]~[[78/77]]~[[66/65]]~[[64/63\|S8]]~[[49/48\|S7]]~[[45/44]]. Additionally, 41edo is the unique intersection of [[Magic]], [[Sensamagic]], and [[Pentacircle]], all extremely intuitive relationships that make it a perfect choice for composers who want to access strong low-complexity JI-like sound while retaining all the benefits of an equal temperament sequence.

			Additionally, and perhaps more convincingly for practical composers, 41edo maps the perfect fifth to 24 steps. Just like with the perfect fourth of 29edo, this highly divisible interval allows for many useful melodic structures, including (but not limited to) [[Rastmic clan\|Neutral]], [[Slendric]], [[Tetracot]], and [[Miracle]].

	== Music ==		== Music ==
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	* Methane Lamentation (31-EDO): https://youtu.be/CBmYRoej2yQ		* Methane Lamentation (31-EDO): https://youtu.be/CBmYRoej2yQ
	* Autumn (27-EDO): https://youtu.be/dcQe6ebpGFU		* Autumn (27-EDO): https://youtu.be/dcQe6ebpGFU
	* Winter (37-EDO): https://~~www~~.~~youtube.com~~/~~watch~~?v=rE9L56yZ1Kw		* Winter (37-EDO): [https://youtu.be/rE9L56yZ1Kw?si=K9LGwj_VsbbAJn3H https://youtu.be/rE9L56yZ1Kw]

	== Main Space Contributions ==		== Main Space Contributions ==

← Older revision		Revision as of 18:06, 2 March 2026
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	While 15edo does not provide accurate representations of the harmonic series, it does provide extremely useful melodic frameworks. The [[5L 5s\|Blackwood Decatonic scale]], for instance, contains several copies of [[Nicetone]] over each degree, allowing for diatonic-like chord progressions to move smoothly between keys that may seem unrelated in systems with a more accurate chain of fifths.		While 15edo does not provide accurate representations of the harmonic series, it does provide extremely useful melodic frameworks. The [[5L 5s\|Blackwood Decatonic scale]], for instance, contains several copies of [[Nicetone]] over each degree, allowing for diatonic-like chord progressions to move smoothly between keys that may seem unrelated in systems with a more accurate chain of fifths.

	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.

	=== [[30edt]] ===		=== [[30edt]] ===
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	I strongly believe that 29edo should be alongside systems such as 19edo and 31edo as introductions to xenharmonic tunings for beginners. Not only does it represent the third harmonic within less than two cents of error (making the diatonic scale roughly equivalent to its familiar representations in western music), but it additionally contains distinct, unambiguous representations for [[interordinal]] intervals. The ability to place these unfamiliar intervals onto the familiar circle of fifths is extremely beneficial, as it allows beginners to more clearly get a feel for how these intervals can be used to create sounds unavailable in 12edo. This provides a benefit over systems such as Meantone, in that the circle of fifths requires less relearning to account for the difference in intonation compared to 12edo, and in that the interordinals represented are distinct and unambiguous (compare to 31edo, where there is no clear representation for, say, the semifourth or the semisixth).		I strongly believe that 29edo should be alongside systems such as 19edo and 31edo as introductions to xenharmonic tunings for beginners. Not only does it represent the third harmonic within less than two cents of error (making the diatonic scale roughly equivalent to its familiar representations in western music), but it additionally contains distinct, unambiguous representations for [[interordinal]] intervals. The ability to place these unfamiliar intervals onto the familiar circle of fifths is extremely beneficial, as it allows beginners to more clearly get a feel for how these intervals can be used to create sounds unavailable in 12edo. This provides a benefit over systems such as Meantone, in that the circle of fifths requires less relearning to account for the difference in intonation compared to 12edo, and in that the interordinals represented are distinct and unambiguous (compare to 31edo, where there is no clear representation for, say, the semifourth or the semisixth).

	Additionally, 29edo finds the perfect fourth at 12 steps, a highly divisible size, supporting [[Porcupine]], [[28812/28561#Tesseract\|Tesseract]], [[Negri]], [[Semaphore and godzilla\|Semaphore]], and other similar structures. This helps provide an introduction into systems that divide simple intervals into a certain number of steps, and how those divisions can apply to writing melodies and chord progressions; see my treatise [[User:Unque/On Voice Leading\|on voice leading]] for a more detailed explanation of why this is important.		Additionally, 29edo finds the perfect fourth at 12 steps, a highly divisible size, supporting [[Porcupine]], [[28812/28561#Tesseract\|Tesseract]], [[Negri]], [[Semaphore and godzilla\|Semaphore]], and other similar structures. This helps provide an introduction into systems that divide simple intervals into a certain number of steps, and how those divisions can apply to writing melodies and chord progressions; see my treatise [[User:Unque/On Voice Leading\|on voice leading]] for a more detailed explanation of why I find this important.

	Finally, for those who like microtemperaments, simple harmonics such as 5 and 7 are very easy to find in supersets such as ~~[[58edo]] and~~ [[87edo]], since these harmonics have a relative error very close to simple fractions. The perfect fifth of 29edo is optimal for [[~~parapyth~~]] tuning, which makes supersets of 29edo extremely desirable if one seeks an extremely high accuracy equal temperament sequence.		Finally, for those who like microtemperaments, simple harmonics such as 5 and 7 are very easy to find in supersets such as [[87edo]], since these harmonics have a relative error very close to simple fractions. The perfect fifth of 29edo is optimal for [[Parapyth]] tuning, which makes supersets of 29edo extremely desirable if one seeks an extremely high accuracy equal temperament sequence; additionally, 87edo supports [[Rodan]] temperament, an extremely efficient system which extends the harmonies of the chain of fifths by adding a formal chroma [[81/80\|S9]]~[[64/63\|S8]]~[[49/48\|S7]] at (8/7) ^ 5. Rodan also contains [[Slendric]] and [[Hemifamity]] as subset parts of its structure

	=== [[36edo]] ===		=== [[36edo]] ===
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	=== [[43edo]] ===		=== [[43edo]] ===
	By the size of 43edo, even rank-2 thinking is rather difficult for conceptualizing how the system fits together. However, 43edo can very intuitively be taken as a Septimal Meantone system with 45/44~56/55~100/99 as a secondary chroma to provide access to 11-limit intervals. Because the chromatic semitone of the diatonic scale is three of these undecimal chromata, any interval in the system can be notated with no more than one accidental of each kind.		By the size of 43edo, even rank-2 thinking is rather difficult for conceptualizing how the system fits together. However, 43edo can very intuitively be taken as a Septimal Meantone system with 45/44~56/55~100/99 as a secondary chroma to provide access to 11-limit intervals. Because the chromatic semitone of the diatonic scale is three of these undecimal chromata, any interval size in the system can be notated with no more than one accidental of each kind.

	This threefold division of the chroma additionally allows the wholetone to be altered into a down-wholetone such that three of them make a perfect fourth instead of an augmented one; the structure ~~begat~~ by these down-wholetones resembles [[Porcupine]] in its form, but does not contain the classical minor third.		This threefold division of the chroma additionally allows the wholetone to be altered into a down-wholetone such that three of them make a perfect fourth instead of an augmented one; the structure begotten by these down-wholetones resembles [[Porcupine]] in its form, but does not contain the classical minor third.

	Finally, 43edo offers a rather accurate tuning of [[Bleu]], the temperament which cleaves the perfect fifth into five parts; two of these parts reaches 7/6, three reach 14/11, and four reach 11/8.		Finally, 43edo offers a rather accurate tuning of [[Bleu]], the temperament which cleaves the perfect fifth into five parts; two of these parts reaches 7/6, three reach 14/11, and four reach 11/8.
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	Here are some other contributions I've made to theory on my personal user space:		Here are some other contributions I've made to theory on my personal user space:

	* A ~~convention~~ for describing [[User:Unque/15edo-Chords\|15-EDO Chords]]		* A system for describing [[User:Unque/15edo-Chords\|15-EDO Chords]]
	*[[User:Unque/Dhembrwood\|Dhembrwood Temperament]]		*[[User:Unque/Dhembrwood\|Dhembrwood Temperament]]
	*[[User:Unque/Redeye scale\|Redeye Scale]]		*[[User:Unque/Redeye scale\|Redeye Scale]]
	*[[User:Unque/On Imaginary Harmonics\|A study on Imaginary Harmonics]]		*[[User:Unque/On Imaginary Harmonics\|A study on Imaginary Harmonics]]
	*[[User:Unque/Chord interlacing (scale building method)\|Chord Interlace Scales]]		*[[User:Unque/Chord interlacing (scale building method)\|Chord Interlace Scales]]
	* A [[User:Unque/TERNAMS\|naming convention]] for MV3 ternary scales (which ~~is currently undergoing serious reconstruction~~)		* A [[User:Unque/TERNAMS\|naming convention]] for MV3 ternary scales (a largely forgotten project which probably remain perpetually unfinished)
	* A [[User:Unque/19-function System\|system of nineteen functions]] to describe xenmelody
	* A [[User:Unque/Barbershop Tuning Theory\|study on tuning theory as applied to Barbershop music]]		* A [[User:Unque/Barbershop Tuning Theory\|study on tuning theory as applied to Barbershop music]]
	* The [[User:Unque/Dietic Minor\|Dietic Minor]] scale		* The [[User:Unque/Dietic Minor\|Dietic Minor]] scale