Functional harmony in rank-2 temperaments - Revision history

FloraC: /* Role swapping vs breaking out of mos */ expand, addressing the nonswapping approach in each example

2025-02-07T10:44:23Z

Role swapping vs breaking out of mos: expand, addressing the nonswapping approach in each example

← Older revision		Revision as of 10:44, 7 February 2025
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	== Role swapping vs breaking out of mos ==		== Role swapping vs breaking out of mos ==
	~~{{Todo\|inline=1\|expand\|comment=Discuss the non-swapping technique in each example. }}~~

	A greater problem arises when the temperament in question is not generated by a [[3/2\|perfect fifth (3/2)]]. In septimal meantone, the most [[consonance and dissonance\|consonant]] [[interval]] besides the [[octave]], the perfect fifth, is a [[generator]], so that the most consonant interval is also the most abundant interval in any [[mos scale]]. This is not true for any temperament that is not generated by a perfect fifth. In these temperaments, we have to choose what our dominant and subdominant functions are based on.		A greater problem arises when the temperament in question is not generated by a [[3/2\|perfect fifth (3/2)]]. In septimal meantone, the most [[consonance and dissonance\|consonant]] [[interval]] besides the [[octave]], the perfect fifth, is a [[generator]], so that the most consonant interval is also the most abundant interval in any [[mos scale]]. This is not true for any temperament that is not generated by a perfect fifth. In these temperaments, we have to choose what our dominant and subdominant functions are based on.

	Here we discuss the technique where "dominant" and "subdominant" are no longer related to tonic by the fifth, but by the generator. This can be called ''role swapping'' as we are essentially swapping the functional roles of harmonic 3 and a higher harmonic according to their temperamental complexities (number of generator steps). Since mos scales are created from stacking generators, this leads to an approach closely adherent to mos scales, and depending on the specific scale pattern, it can sound very xenharmonic.		Here we discuss the technique where "dominant" and "subdominant" are no longer related to tonic by the fifth, but by the generator. This can be called ''role swapping'' as we are essentially swapping the functional roles of harmonic 3 and a higher harmonic according to their temperamental complexities (number of generator steps). Since mos scales are created from stacking generators, this leads to an approach closely adherent to mos scales, and depending on the specific scale pattern, it can sound very xenharmonic.

	In contrast to role swapping, ~~there is a~~ one-for-all solution to harmony in temperaments regardless of how far the primes are on the generator chain~~. It~~ entails using the same old chords as the basis of harmony, and keeping the roles of each prime as in meantone. This means dominant is always a perfect fifth over tonic, for example. A consequence is we must step out of the logic of mos scales, as they are often too restrictive without the many fifths to stack. Try starting unlimited and thinking directly about [[ratio]]s and [[comma pump]]s. You might end up with a diatonic-like scale with comma-level inflections here and there, but it is also possible to slap whatever scale you like, using inflections to reach the ratios.		In contrast to role swapping, the one-for-all solution to harmony in temperaments regardless of how far the primes are on the generator chain entails using the same old chords as the basis of harmony, and keeping the roles of each prime as in meantone. This means dominant is always a perfect fifth over tonic, for example. A consequence is we must step out of the logic of mos scales, as they are often too restrictive without the many fifths to stack. Try starting unlimited and thinking directly about [[ratio]]s and [[comma pump]]s. You might end up with a diatonic-like scale with comma-level inflections here and there, but it is also possible to slap whatever scale you like, using inflections to reach the ratios.

	=== Example: magic ===		=== Example: magic ===
	In terms of generator steps of [[magic]], the just major triad (1–5/4–3/2) and just minor triad (1–6/5–3/2) are 0–1–5 and 0–4–5; this is similar, but also in clear contrast to the 0–4–1 and 0–(−3)–1 of meantone. These chords can be used as the basis of harmony, with the roles of 3 and 5 swapped according to their temperamental complexities. Thus a "dominant" chord is 5/4 over tonic; a "subdominant" chord is 5/4 under tonic. The 7-tone mos contains a tonic and a "dominant" chord. The 10-tone mos is good for encapsulating tonic, "pre-dominant", and "dominant" functions.		In terms of generator steps of [[magic]], the just major triad (1–5/4–3/2) and just minor triad (1–6/5–3/2) are 0–1–5 and 0–4–5; this is similar, but also in clear contrast to the 0–4–1 and 0–(−3)–1 of meantone. These chords can be used as the basis of harmony, with the roles of 3 and 5 swapped according to their temperamental complexities. Thus a "dominant" chord is 5/4 over tonic; a "subdominant" chord is 5/4 under tonic. The 7-tone mos contains a tonic and a "dominant" chord. The 10-tone mos is good for encapsulating tonic, "pre-dominant", and "dominant" functions. The suspended chord of meantone is made of two generators stacked, and doing the same in magic, we also have the augmented triad (1–5/4–14/9) as the magic analog of the suspended chord of meantone.

	~~The suspended chord of meantone is made~~ of ~~two generators stacked. Doing the same in magic~~, we ~~also have~~ the ~~augmented~~ triad (~~1–5/4–14/9~~) as the ~~magic analog~~ of the ~~suspended chord of meantone~~.		Instead of thinking about mos scales, we can keep the tonic, dominant, and subdominant functions related by fifths. The classical major triad on tonic is 0–1–5; on dominant, 5–6–10; and on subdominant, (-5)–(-4)–0. This is essentially working in a rank-3 space (or 2-dimensional [[lattice]] up to octave equivalence) but the edges are wrapped around due to the equivalence from the commas tempered out. The only thing that distinguishes it from JI is that you should in some way use the commas tempered out to prove the worth of the intonational compromises; otherwise you could simply choose JI.

	=== Example: hanson ===		=== Example: hanson ===
	In terms of generator steps of [[hanson]], the just major triad (1–5/4–3/2) and just minor triad (1–6/5–3/2) are 0–5–6 and 0–1–6; like in magic, these chords can be used as the basis of harmony, but the intervals whose roles are being swapped here are 3 and 3/5. A "dominant" chord is 6/5 over tonic; a "subdominant" chord is 6/5 under tonic. The 11-tone mos might be a good place to start since the 7-tone mos barely contains anything more a tonic chord.		In terms of generator steps of [[hanson]], the just major triad (1–5/4–3/2) and just minor triad (1–6/5–3/2) are 0–5–6 and 0–1–6; like in magic, these chords can be used as the basis of harmony, but the intervals whose roles are being swapped here are 3 and 3/5. A "dominant" chord is 6/5 over tonic; a "subdominant" chord is 6/5 under tonic. The 11-tone mos might be a good place to start since the 7-tone mos barely contains anything more a tonic chord.

			If instead we keep the tonic, dominant, and subdominant functions related by fifths, then the classical major triad on tonic is 0–5–6; on dominant, 6–11–12; and on subdominant, (-6)–(-1)–0.

	=== Example: orwell ===		=== Example: orwell ===
	[[Orwell]] is generated by ~7/6, so ~~the most basic form of harmony~~ could be 1–7/6–3/2 (0–1–7) and 1–9/7–3/2 (0–6–7), or 1–7/4–3 (0–8–7) and 1–12/7–3 (0–(-1)–7), or tetrads such as 1–7/6–3/2–7/4 (0–1–7–8). The intervals whose roles are being swapped here are 3 and 5 and then 5 and 7. A "dominant" chord is either 7/6 or 12/7 over tonic; a "subdominant" chord is either 7/6 or 12/7 under tonic. The 9-tone mos contains a tonic and a "dominant" triad. The 13-tone mos is good for encapsulating tonic, "pre-dominant", and "dominant" functions, triads and ~~tetrads alike~~.		[[Orwell]] is generated by ~7/6, so chords could be built around it as 1–7/6–3/2 (0–1–7) and 1–9/7–3/2 (0–6–7), or 1–7/4–3 (0–8–7) and 1–12/7–3 (0–(-1)–7), or tetrads such as 1–7/6–3/2–7/4 (0–1–7–8). The intervals whose roles are being swapped here are 3 and 5 and then 5 and 7. Since two generator steps give ~11/8, 1–11/8–3/2–7/4 (0–2–7–8) is also a characteristic otonal tetrad of orwell. To 1–7/6–3/2–7/4 we may add 11/8, or to 1–11/8–3/2–7/4 we may add 7/6, to form an essentially tempered pentad, 1–7/6–11/8–3/2–7/4 (0–1–2–7–8). Its inverse is 1–12/11–9/7–3/2–12/7 (0–5–6–7–(-1)), which can serve as a minor counterpart. A "dominant" chord is either 7/6 or 12/7 over tonic; a "subdominant" chord is either 7/6 or 12/7 under tonic. The 9-tone mos contains a tonic and a "dominant" triad. The 13-tone mos is good for encapsulating tonic, "pre-dominant", and "dominant" functions, triads to pentads alike.

			If instead we keep the tonic, dominant, and subdominant functions related by fifths, then the above pentad on tonic is 0–1–2–7–8; on dominant, 7–8–9–14–15; and on subdominant, (-7)–(-6)–(-5)–0–1.

	=== Example: sensi ===		=== Example: sensi ===
	[[Sensi]] is generated by ~9/7, so ~~the most basic form of harmony~~ could be 1–9/7–3/2 (0–1–7) and 1–7/6–3/2 (0–6–7). The intervals whose roles are being swapped here are 3 and 5 and then 5 and 9/7. A "dominant" chord is 9/7 over tonic; a "subdominant" chord is 9/7 under tonic. The 11-tone mos might be a good place to start since the 8-tone mos barely contains anything more than a tonic chord.		[[Sensi]] is generated by ~9/7, so chords could be built around it as 1–9/7–3/2 (0–1–7) and 1–7/6–3/2 (0–6–7). The intervals whose roles are being swapped here are 3 and 5 and then 5 and 9/7. Since two generator steps give ~5/3, 1–7/6–3/2–5/3 (0–6–7–2) is also a characteristic otonal tetrad of sensi; its inverse is 1–9/7–3/2–9/5 (0–1–7–5). In addition, 1–9/7–3/2–5/3 (0–1–7–2) is a very characteristic essentially tempered tetrad; its inverse is 1–7/6–3/2–9/5 (0–6–7–5). A "dominant" chord is 9/7 over tonic; a "subdominant" chord is 9/7 under tonic. The 11-tone mos might be a good place to start since the 8-tone mos barely contains anything more than a tonic chord.

			If instead we keep the tonic, dominant, and subdominant functions related by fifths, then the above tetrads on tonic are 0–6–7–2 and 0–1–7–2; on dominant, 7–13–14–9 and 7–8–14–9; and on subdominant, (-7)–(-1)–0–(-5) and (–7)–(-6)–0–(-5).

	=== Example: würschmidt ===		=== Example: würschmidt ===
	[[Würschmidt]] is just about on the complexity level where role swapping stops to work. The 10-tone mos contains a tonic and a "dominant" chord. For full analog of traditional harmony you need an unwieldy 13-tone mos, with more notes than the traditional chromatic scale and nearly twice the number of notes as the diatonic scale.		[[Würschmidt]] is just about on the complexity level where role swapping stops to work. The 10-tone mos contains a tonic and a "dominant" chord. For full analog of traditional harmony you need an unwieldy 13-tone mos, with more notes than the traditional chromatic scale and nearly twice the number of notes as the diatonic scale.

	~~Instead of thinking about~~ the ~~mos scheme~~, ~~we can pick our desired intervals however we like. If you want~~ the classical major triad on tonic~~, use~~ 0–1–8; on dominant, ~~use~~ 8–9–16; and on subdominant, ~~use~~ (-8)–(-7)–0. The possibilities are theoretically infinite. This is essentially working in a rank-3 space (or 2-dimensional [[lattice]] up to octave equivalence) but the edges are wrapped around due to the equivalence from the commas tempered out. The only thing that distinguishes it from JI is that you should in some way use the commas tempered out to prove the worth of the intonational compromises; otherwise you could simply choose JI.		If instead we keep the tonic, dominant, and subdominant functions related by fifths, then the classical major triad on tonic is 0–1–8; on dominant, 8–9–16; and on subdominant, (-8)–(-7)–0.

	== Generalization ==		== Generalization ==

FloraC: Rework to discuss the problem and solutions in one place and facilitate adding non-swapping technique to specific temps.

2025-02-03T14:04:14Z

Rework to discuss the problem and solutions in one place and facilitate adding non-swapping technique to specific temps.

← Older revision		Revision as of 14:04, 3 February 2025
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	== Functions based on septimal harmony ==		== Functions based on septimal harmony ==
	In septimal meantone, the interval class of 5 is reached by a relatively low number of generator steps whereas the interval class of 7 is way more complex, so classical intervals are very accessible in the [[5L 2s\|diatonic scale]] whereas septimal intervals are only available in the [[7L 5s\|chromatic]] or even [[12L 7s\|enharmonic scale]] (except for [[~]][[7/5]], which is mapped to the [[tritone]]). This greatly shapes the way we use harmony in it: the classical chords are the basis of harmony, and septimal chords are only used sparingly for additional spices. For other temperaments generated by a fifth, the condition tends to differ. In [[superpyth]], the interval class of 7 is only -2 generator steps, while the interval class of 5 is +9 steps away, so the temperament inherently favors prime 7 over 5. If we continue to use it the old way, the harmonic resource we get from a finite scale will be sparse. To address that, ~~we can swap the functional roles of primes 5 and 7 according to their temperamental complexities (number of generator steps), and this means using~~ septimal chords as the basic form of harmony, ~~only inserting~~ classical chords occasionally for additional spices.		In septimal meantone, the interval class of 5 is reached by a relatively low number of generator steps whereas the interval class of 7 is way more complex, so classical intervals are very accessible in the [[5L 2s\|diatonic scale]] whereas septimal intervals are only available in the [[7L 5s\|chromatic]] or even [[12L 7s\|enharmonic scale]] (except for [[~]][[7/5]], which is mapped to the [[tritone]]). This greatly shapes the way we use harmony in it: the classical chords are the basis of harmony, and septimal chords are only used sparingly for additional spices. For other temperaments generated by a fifth, the condition tends to differ. In [[superpyth]], the interval class of 7 is only -2 generator steps, while the interval class of 5 is +9 steps away, so the temperament inherently favors prime 7 over 5. If we continue to use it the old way, the harmonic resource we get from a finite scale will be sparse. To address that, septimal chords can be used as the basic form of harmony, with classical chords occasionally inserted for additional spices.

	=== Example: superpyth ===		=== Example: superpyth ===
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	Superpyth[7], the diatonic scale of superpyth, contains 3 copies of the tetrads and 4 copies of the rooted/uprooted triads, and with Superpyth[12], the chromatic scale, classical chords are available here and there.		Superpyth[7], the diatonic scale of superpyth, contains 3 copies of the tetrads and 4 copies of the rooted/uprooted triads, and with Superpyth[12], the chromatic scale, classical chords are available here and there.

	== ~~Functions based on generators~~ ==		== Role swapping vs breaking out of mos ==
	~~In septimal meantone, the most [[consonance and dissonance~~\|~~consonant]] [[interval]] besides the [[octave]], the [[3/2~~\|~~perfect fifth (3/2)]], is a [[generator]], so that~~ the ~~most consonant interval is also the most abundant interval~~ in ~~any [[mos scale]]. This is not true for any temperament that is not generated by a perfect fifth. In these temperaments, we have to choose what our dominant and subdominant functions are based on~~.		{{Todo\|inline=1\|expand\|comment=Discuss the non-swapping technique in each example. }}

	Here we discuss the technique where "dominant" and "subdominant" are no longer related to tonic by the fifth, but by the generator. Since mos scales are created from stacking generators, this leads to an approach closely adherent to mos scales, and depending on the specific scale pattern, it can sound very xenharmonic. ~~For~~ the ~~other approach see [[#Breaking~~ out of mos]].		A greater problem arises when the temperament in question is not generated by a [[3/2\|perfect fifth (3/2)]]. In septimal meantone, the most [[consonance and dissonance\|consonant]] [[interval]] besides the [[octave]], the perfect fifth, is a [[generator]], so that the most consonant interval is also the most abundant interval in any [[mos scale]]. This is not true for any temperament that is not generated by a perfect fifth. In these temperaments, we have to choose what our dominant and subdominant functions are based on.

			Here we discuss the technique where "dominant" and "subdominant" are no longer related to tonic by the fifth, but by the generator. This can be called ''role swapping'' as we are essentially swapping the functional roles of harmonic 3 and a higher harmonic according to their temperamental complexities (number of generator steps). Since mos scales are created from stacking generators, this leads to an approach closely adherent to mos scales, and depending on the specific scale pattern, it can sound very xenharmonic.

			In contrast to role swapping, there is a one-for-all solution to harmony in temperaments regardless of how far the primes are on the generator chain. It entails using the same old chords as the basis of harmony, and keeping the roles of each prime as in meantone. This means dominant is always a perfect fifth over tonic, for example. A consequence is we must step out of the logic of mos scales, as they are often too restrictive without the many fifths to stack. Try starting unlimited and thinking directly about [[ratio]]s and [[comma pump]]s. You might end up with a diatonic-like scale with comma-level inflections here and there, but it is also possible to slap whatever scale you like, using inflections to reach the ratios.

	=== Example: magic ===		=== Example: magic ===
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	=== Example: hanson ===		=== Example: hanson ===
	In terms of generator steps of [[hanson]], the just major triad (1–5/4–3/2) and just minor triad (1–6/5–3/2) are 0–5–6 and 0–1–6; like in magic, these chords can be used as the basis of harmony, but the intervals whose roles are being swapped here are 3 and 3/5. A "dominant" chord is 6/5 over tonic; a "subdominant" chord is 6/5 under tonic. The 11-tone mos might be a good place to start since the 7-tone mos barely contains anything more a tonic chord.		In terms of generator steps of [[hanson]], the just major triad (1–5/4–3/2) and just minor triad (1–6/5–3/2) are 0–5–6 and 0–1–6; like in magic, these chords can be used as the basis of harmony, but the intervals whose roles are being swapped here are 3 and 3/5. A "dominant" chord is 6/5 over tonic; a "subdominant" chord is 6/5 under tonic. The 11-tone mos might be a good place to start since the 7-tone mos barely contains anything more a tonic chord.

	~~== Combination of techniques ==~~
	~~We can combine the two techniques discussed above, which means we can:~~
	* Treat septimal chords as the basic form of harmony, and
	* Define the functions on the generator.

	=== Example: orwell ===		=== Example: orwell ===
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	[[Sensi]] is generated by ~9/7, so the most basic form of harmony could be 1–9/7–3/2 (0–1–7) and 1–7/6–3/2 (0–6–7). The intervals whose roles are being swapped here are 3 and 5 and then 5 and 9/7. A "dominant" chord is 9/7 over tonic; a "subdominant" chord is 9/7 under tonic. The 11-tone mos might be a good place to start since the 8-tone mos barely contains anything more than a tonic chord.		[[Sensi]] is generated by ~9/7, so the most basic form of harmony could be 1–9/7–3/2 (0–1–7) and 1–7/6–3/2 (0–6–7). The intervals whose roles are being swapped here are 3 and 5 and then 5 and 9/7. A "dominant" chord is 9/7 over tonic; a "subdominant" chord is 9/7 under tonic. The 11-tone mos might be a good place to start since the 8-tone mos barely contains anything more than a tonic chord.

	== ~~Other role-swapping techniques~~ ==		=== Example: würschmidt ===
	~~All~~ the ~~techniques discussed above may be collectively called ''~~role swapping''. As the ~~name suggests, they all involve swapping~~ the ~~functional roles~~ of ~~certain primes according to their temperamental complexities~~. ~~To better illustrate this~~, we abstract the following hierarchy of functional roles for intervals of 3, 5, and 7 from meantone:		[[Würschmidt]] is just about on the complexity level where role swapping stops to work. The 10-tone mos contains a tonic and a "dominant" chord. For full analog of traditional harmony you need an unwieldy 13-tone mos, with more notes than the traditional chromatic scale and nearly twice the number of notes as the diatonic scale.

			Instead of thinking about the mos scheme, we can pick our desired intervals however we like. If you want the classical major triad on tonic, use 0–1–8; on dominant, use 8–9–16; and on subdominant, use (-8)–(-7)–0. The possibilities are theoretically infinite. This is essentially working in a rank-3 space (or 2-dimensional [[lattice]] up to octave equivalence) but the edges are wrapped around due to the equivalence from the commas tempered out. The only thing that distinguishes it from JI is that you should in some way use the commas tempered out to prove the worth of the intonational compromises; otherwise you could simply choose JI.

			== Generalization ==
			We abstract the following hierarchy of functional roles for intervals of 3, 5, and 7 from meantone:

	; Primary role		; Primary role
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	{\| class="wikitable center-all"		{\| class="wikitable center-all"
	\|+ style="font-size: 105%;" \| Functional roles in meantone and ~~Orwell~~		\|+ style="font-size: 105%;" \| Functional roles in meantone and orwell
	\|-		\|-
	! Temperament !! Intervals of 3 !! Intervals of 5 !! Intervals of 7		! Temperament !! Intervals of 3 !! Intervals of 5 !! Intervals of 7
Line 79:		Line 83:

	Adding that to the table, it might give us a clue about using temperaments in [[non-octave]] settings.		Adding that to the table, it might give us a clue about using temperaments in [[non-octave]] settings.

	~~== Breaking out of mos ==~~
	~~In contrast to role swapping, there is a one-for-all solution to harmony in temperaments regardless of how far the primes are on the generator chain.~~

	This approach entails using the same old chords as the basis of harmony, and keeping the roles of each prime as in meantone. This means dominant is always ~3/2 over tonic, for example. A consequence is we must step out of the logic of mos scales, as they are often too restrictive without the many fifths to stack. Try starting unlimited and thinking directly about [[ratio]]s and [[comma pump]]s. You might end up with a diatonic-like scale with comma-level inflections here and there, but it is also possible to slap whatever scale you like, using inflections to reach the ratios.

	~~=== Example: würschmidt ===~~
	[[Würschmidt]] is just about on the complexity level where role swapping stops to work. The 10-tone mos contains a tonic and a "dominant" chord. For full analog of traditional harmony you need an unwieldy 13-tone mos, with more notes than the traditional chromatic scale and nearly twice the number of notes as the diatonic scale.

	Instead of thinking about the mos scheme, we can pick our desired intervals however we like. If you want the classical major triad on tonic, use 0–1–8; on dominant, use 8–9–16; and on subdominant, use (−8)–(−7)–0. The possibilities are theoretically infinite. This is essentially working in a rank-3 space (or 2-dimensional [[lattice]] up to octave equivalence) but the edges are wrapped around due to the equivalence from the commas tempered out. The only thing that distinguishes it from JI is that you should in some way use the commas tempered out to prove the worth of the intonational compromises; otherwise you could simply choose JI.

	[[Category:Regular temperament theory]]		[[Category:Regular temperament theory]]
	[[Category:Harmony]]		[[Category:Harmony]]
	[[Category:Guides]]		[[Category:Guides]]

ArrowHead294 at 13:31, 3 February 2025

2025-02-03T13:31:28Z

← Older revision		Revision as of 13:31, 3 February 2025
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	{\| class="wikitable center-all"		{\| class="wikitable center-all"
	\|+Functional ~~Roles~~ in ~~Meantone~~ and Orwell		\|+ style="font-size: 105%;" \| Functional roles in meantone and Orwell
	\|-		\|-
	! Temperament !! Intervals of 3 !! Intervals of 5 !! Intervals of 7		! Temperament !! Intervals of 3 !! Intervals of 5 !! Intervals of 7
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	[[Würschmidt]] is just about on the complexity level where role swapping stops to work. The 10-tone mos contains a tonic and a "dominant" chord. For full analog of traditional harmony you need an unwieldy 13-tone mos, with more notes than the traditional chromatic scale and nearly twice the number of notes as the diatonic scale.		[[Würschmidt]] is just about on the complexity level where role swapping stops to work. The 10-tone mos contains a tonic and a "dominant" chord. For full analog of traditional harmony you need an unwieldy 13-tone mos, with more notes than the traditional chromatic scale and nearly twice the number of notes as the diatonic scale.

	Instead of thinking about the mos scheme, we can pick our desired intervals however we like. If you want the classical major triad on tonic, use 0–1–8; on dominant, use 8–9–16; and on subdominant, use (-8)–(-7)–0. The possibilities are theoretically infinite. This is essentially working in a rank-3 space (or 2-dimensional [[lattice]] up to octave equivalence) but the edges are wrapped around due to the equivalence from the commas tempered out. The only thing that distinguishes it from JI is that you should in some way use the commas tempered out to prove the worth of the intonational compromises; otherwise you could simply choose JI.		Instead of thinking about the mos scheme, we can pick our desired intervals however we like. If you want the classical major triad on tonic, use 0–1–8; on dominant, use 8–9–16; and on subdominant, use (−8)–(−7)–0. The possibilities are theoretically infinite. This is essentially working in a rank-3 space (or 2-dimensional [[lattice]] up to octave equivalence) but the edges are wrapped around due to the equivalence from the commas tempered out. The only thing that distinguishes it from JI is that you should in some way use the commas tempered out to prove the worth of the intonational compromises; otherwise you could simply choose JI.

	[[Category:Regular temperament theory]]		[[Category:Regular temperament theory]]
	[[Category:Harmony]]		[[Category:Harmony]]
	[[Category:Guides]]		[[Category:Guides]]

FloraC: /* Functions based on generators */ clarify based on community feedback

2025-02-03T12:34:21Z

Functions based on generators: clarify based on community feedback

← Older revision		Revision as of 12:34, 3 February 2025
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	== Functions based on generators ==		== Functions based on generators ==
	In septimal meantone, the most [[consonance and dissonance\|consonant]] [[interval]] besides the [[octave]], the [[3/2\|perfect fifth (3/2)]], is a [[generator]], ~~meaning it~~ is the most abundant interval in any [[mos scale]]. This is ~~often~~ not true for ~~other~~ temperaments. Here we discuss the technique where "dominant" and "subdominant" are no longer related to tonic by the fifth, but by the generator. ~~This~~ leads to an approach closely adherent to mos scales, and depending on the specific scale pattern, it can sound very xenharmonic.		In septimal meantone, the most [[consonance and dissonance\|consonant]] [[interval]] besides the [[octave]], the [[3/2\|perfect fifth (3/2)]], is a [[generator]], so that the most consonant interval is also the most abundant interval in any [[mos scale]]. This is not true for any temperament that is not generated by a perfect fifth. In these temperaments, we have to choose what our dominant and subdominant functions are based on.

			Here we discuss the technique where "dominant" and "subdominant" are no longer related to tonic by the fifth, but by the generator. Since mos scales are created from stacking generators, this leads to an approach closely adherent to mos scales, and depending on the specific scale pattern, it can sound very xenharmonic. For the other approach see [[#Breaking out of mos]].

	=== Example: magic ===		=== Example: magic ===
	In terms of generator steps of [[magic]], the just major triad (1–5/4–3/2) and just minor triad (1–6/5–3/2) are 0–1–5 and 0–4–5; this is similar, but also in clear contrast to the 0–4–1 and 0–(−3)–1 of meantone. These chords can be used as the basis of harmony, with the roles of 3 and 5 swapped according to their temperamental complexities. Thus a "dominant" chord is 5/4 over tonic; a "subdominant" chord is 5/4 under tonic. The 7-tone mos contains a tonic and a "dominant" chord. The 10-tone mos is good for encapsulating tonic, "pre-dominant", and "dominant" functions. We also have the augmented triad (1–5/4–14/9) as the magic analog of the suspended chord of meantone.		In terms of generator steps of [[magic]], the just major triad (1–5/4–3/2) and just minor triad (1–6/5–3/2) are 0–1–5 and 0–4–5; this is similar, but also in clear contrast to the 0–4–1 and 0–(−3)–1 of meantone. These chords can be used as the basis of harmony, with the roles of 3 and 5 swapped according to their temperamental complexities. Thus a "dominant" chord is 5/4 over tonic; a "subdominant" chord is 5/4 under tonic. The 7-tone mos contains a tonic and a "dominant" chord. The 10-tone mos is good for encapsulating tonic, "pre-dominant", and "dominant" functions.

			The suspended chord of meantone is made of two generators stacked. Doing the same in magic, we also have the augmented triad (1–5/4–14/9) as the magic analog of the suspended chord of meantone.

	=== Example: hanson ===		=== Example: hanson ===
	In terms of generator steps of [[hanson]], the just major triad (1–5/4–3/2) and just minor triad (1–6/5–3/2) are 0–5–6 and 0–1–6; like in magic, these chords can be used as the basis of harmony, but the intervals whose roles are being swapped here are 3 and 3/5. A "dominant" chord is 6/5 over tonic; a "subdominant" chord is 6/5 under tonic. The 11-tone mos might be a good place to start since the 7-tone mos barely contains anything more a tonic chord.		In terms of generator steps of [[hanson]], the just major triad (1–5/4–3/2) and just minor triad (1–6/5–3/2) are 0–5–6 and 0–1–6; like in magic, these chords can be used as the basis of harmony, but the intervals whose roles are being swapped here are 3 and 3/5. A "dominant" chord is 6/5 over tonic; a "subdominant" chord is 6/5 under tonic. The 11-tone mos might be a good place to start since the 7-tone mos barely contains anything more a tonic chord.

	== Combination of techniques ==		== Combination of techniques ==

FloraC: Orwell could use 1-7/4-3 or 1-7/4-7/6-3

2024-10-18T10:46:37Z

Orwell could use 1-7/4-3 or 1-7/4-7/6-3

← Older revision		Revision as of 10:46, 18 October 2024
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	=== Example: hanson ===		=== Example: hanson ===
	In terms of generator steps of [[hanson]], the just major triad (1–5/4–3/2) and just minor triad (1–6/5–3/2) are 0–5–6 and 0–1–6; like in magic, these chords can be used as the basis of harmony, but the intervals whose roles are being swapped here are 3 and 3/5. A "dominant" chord is 6/5 over tonic; a "subdominant" chord is 6/5 under tonic. The 11-tone mos might be a good start since the 7-tone mos barely contains anything more a tonic chord.		In terms of generator steps of [[hanson]], the just major triad (1–5/4–3/2) and just minor triad (1–6/5–3/2) are 0–5–6 and 0–1–6; like in magic, these chords can be used as the basis of harmony, but the intervals whose roles are being swapped here are 3 and 3/5. A "dominant" chord is 6/5 over tonic; a "subdominant" chord is 6/5 under tonic. The 11-tone mos might be a good place to start since the 7-tone mos barely contains anything more a tonic chord.

	== Combination of techniques ==		== Combination of techniques ==
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	=== Example: orwell ===		=== Example: orwell ===
	[[Orwell]] is generated by ~7/6, so the most basic form of harmony could be 1–7/6–3/2 (0–1–7) and 1–9/7–3/2 (0–6–7). The intervals whose roles are being swapped here are 3 and 5 and then 5 and 7/3. A "dominant" chord is 7/6 over tonic; a "subdominant" chord is 7/6 under tonic. The 9-tone mos contains a tonic and a "dominant" ~~chord~~. The 13-tone mos is good for encapsulating tonic, "pre-dominant", and "dominant" functions.		[[Orwell]] is generated by ~7/6, so the most basic form of harmony could be 1–7/6–3/2 (0–1–7) and 1–9/7–3/2 (0–6–7), or 1–7/4–3 (0–8–7) and 1–12/7–3 (0–(-1)–7), or tetrads such as 1–7/6–3/2–7/4 (0–1–7–8). The intervals whose roles are being swapped here are 3 and 5 and then 5 and 7. A "dominant" chord is either 7/6 or 12/7 over tonic; a "subdominant" chord is either 7/6 or 12/7 under tonic. The 9-tone mos contains a tonic and a "dominant" triad. The 13-tone mos is good for encapsulating tonic, "pre-dominant", and "dominant" functions, triads and tetrads alike.

	=== Example: sensi ===		=== Example: sensi ===
	[[Sensi]] is generated by ~9/7, so the most basic form of harmony could be 1–9/7–3/2 (0–1–7) and 1–7/6–3/2 (0–6–7). The intervals whose roles are being swapped here are 3 and 5 and then 5 and 9/7. A "dominant" chord is 9/7 over tonic; a "subdominant" chord is 9/7 under tonic. The 11-tone mos might be a good start since the 8-tone mos barely contains anything more than a tonic chord.		[[Sensi]] is generated by ~9/7, so the most basic form of harmony could be 1–9/7–3/2 (0–1–7) and 1–7/6–3/2 (0–6–7). The intervals whose roles are being swapped here are 3 and 5 and then 5 and 9/7. A "dominant" chord is 9/7 over tonic; a "subdominant" chord is 9/7 under tonic. The 11-tone mos might be a good place to start since the 8-tone mos barely contains anything more than a tonic chord.

	== Other role-swapping techniques ==		== Other role-swapping techniques ==

FloraC: As I promised

2024-10-08T09:29:13Z

As I promised

New page

Using '''functonal harmony in rank-2 temperaments''' can be challenging. [[Diatonic functional harmony]] is virtually based on one [[regular temperament|temperament]] over the centuries: [[meantone]], but what works in meantone may or may not work in other [[rank-2 temperament]]s. This guide discusses some techniques that may be helpful for transferring functional harmony to those temperaments.

== Functions based on septimal harmony ==
In septimal meantone, the interval class of 5 is reached by a relatively low number of generator steps whereas the interval class of 7 is way more complex, so classical intervals are very accessible in the [[5L 2s|diatonic scale]] whereas septimal intervals are only available in the [[7L 5s|chromatic]] or even [[12L 7s|enharmonic scale]] (except for [[~]][[7/5]], which is mapped to the [[tritone]]). This greatly shapes the way we use harmony in it: the classical chords are the basis of harmony, and septimal chords are only used sparingly for additional spices. For other temperaments generated by a fifth, the condition tends to differ. In [[superpyth]], the interval class of 7 is only -2 generator steps, while the interval class of 5 is +9 steps away, so the temperament inherently favors prime 7 over 5. If we continue to use it the old way, the harmonic resource we get from a finite scale will be sparse. To address that, we can swap the functional roles of primes 5 and 7 according to their temperamental complexities (number of generator steps), and this means using septimal chords as the basic form of harmony, only inserting classical chords occasionally for additional spices.

=== Example: superpyth ===
Intuitively, one can think of the basic form of septimal harmony as follows:
* 1–7/6–3/2 (0–(−3)–1)
* 1–9/7–3/2 (0–4–1)

As [[Flora Canou]]'s essay [[User:FloraC/Analysis on the 13-limit just intonation space: episode i|''Analysis on the 13-Limit Just Intonation Space: Episode I'']] has identified, the classical triads 1–5/4–3/2 and 1–6/5–3/2 are [[rooted]] and [[uprooted]], respectively, but these septimal triads are not. Another possible complaint is that the chroma between the "major" and "minor" triads is too large. One solution is to insert another tone to the chords, resulting in
* 1–7/6–21/16–3/2 (0–(−3)–(−1)–1)
* 1–8/7–9/7–3/2 (0–2–4–1)

Particular to superpyth, note that 1–7/6–21/16–3/2 can also be expressed as 1–7/6–4/3–3/2 and that 1–8/7–9/7–3/2 can also be expressed as 1–9/8–9/7–3/2.

Another solution, proposed in the essay, is to adopt ''semiquartal harmony'', a chord construction scheme analogous to {{w|tertian harmony}} but for septimal chords:
* 1–8/7–4/3 (0–2–(−1))
* 1–7/6–4/3 (0–(−3)–(−1)))

Flora specifically proposes this voicing:
* 1–7/4–3 (0–(−2)–1)
* 1–12/7–3 (0–3–1))

which is suppoed to sound better with very wide voicing variations.

Superpyth[7], the diatonic scale of superpyth, contains 3 copies of the tetrads and 4 copies of the rooted/uprooted triads, and with Superpyth[12], the chromatic scale, classical chords are available here and there.

== Functions based on generators ==
In septimal meantone, the most [[consonance and dissonance|consonant]] [[interval]] besides the [[octave]], the [[3/2|perfect fifth (3/2)]], is a [[generator]], meaning it is the most abundant interval in any [[mos scale]]. This is often not true for other temperaments. Here we discuss the technique where "dominant" and "subdominant" are no longer related to tonic by the fifth, but by the generator. This leads to an approach closely adherent to mos scales, and depending on the specific scale pattern, it can sound very xenharmonic.

=== Example: magic ===
In terms of generator steps of [[magic]], the just major triad (1–5/4–3/2) and just minor triad (1–6/5–3/2) are 0–1–5 and 0–4–5; this is similar, but also in clear contrast to the 0–4–1 and 0–(−3)–1 of meantone. These chords can be used as the basis of harmony, with the roles of 3 and 5 swapped according to their temperamental complexities. Thus a "dominant" chord is 5/4 over tonic; a "subdominant" chord is 5/4 under tonic. The 7-tone mos contains a tonic and a "dominant" chord. The 10-tone mos is good for encapsulating tonic, "pre-dominant", and "dominant" functions. We also have the augmented triad (1–5/4–14/9) as the magic analog of the suspended chord of meantone.

=== Example: hanson ===
In terms of generator steps of [[hanson]], the just major triad (1–5/4–3/2) and just minor triad (1–6/5–3/2) are 0–5–6 and 0–1–6; like in magic, these chords can be used as the basis of harmony, but the intervals whose roles are being swapped here are 3 and 3/5. A "dominant" chord is 6/5 over tonic; a "subdominant" chord is 6/5 under tonic. The 11-tone mos might be a good start since the 7-tone mos barely contains anything more a tonic chord.

== Combination of techniques ==
We can combine the two techniques discussed above, which means we can:
* Treat septimal chords as the basic form of harmony, and
* Define the functions on the generator.

=== Example: orwell ===
[[Orwell]] is generated by ~7/6, so the most basic form of harmony could be 1–7/6–3/2 (0–1–7) and 1–9/7–3/2 (0–6–7). The intervals whose roles are being swapped here are 3 and 5 and then 5 and 7/3. A "dominant" chord is 7/6 over tonic; a "subdominant" chord is 7/6 under tonic. The 9-tone mos contains a tonic and a "dominant" chord. The 13-tone mos is good for encapsulating tonic, "pre-dominant", and "dominant" functions.

=== Example: sensi ===
[[Sensi]] is generated by ~9/7, so the most basic form of harmony could be 1–9/7–3/2 (0–1–7) and 1–7/6–3/2 (0–6–7). The intervals whose roles are being swapped here are 3 and 5 and then 5 and 9/7. A "dominant" chord is 9/7 over tonic; a "subdominant" chord is 9/7 under tonic. The 11-tone mos might be a good start since the 8-tone mos barely contains anything more than a tonic chord.

== Other role-swapping techniques ==
All the techniques discussed above may be collectively called ''role swapping''. As the name suggests, they all involve swapping the functional roles of certain primes according to their temperamental complexities. To better illustrate this, we abstract the following hierarchy of functional roles for intervals of 3, 5, and 7 from meantone:

; Primary role
: The spine on which the functions are defined.
; Secondary role
: The main flavors for harmony.
; Tertiary role
: The additional spices for harmony.

Note that "intervals of 5" may be 5, 5/3, sometimes 9/5 or even 15, all depending on the temperament; same with intervals of 7. Other primes may be swapped in too. For instance, intervals of 13 have a great tertiary role in hanson. The following table shows a more typical role swapping result in orwell.

{| class="wikitable center-all"
|+Functional Roles in Meantone and Orwell
|-
! Temperament !! Intervals of 3 !! Intervals of 5 !! Intervals of 7
|-
| Meantone || Primary || Secondary || Tertiary
|-
| Orwell || Secondary || Tertiary || Primary
|}

We can, as a bonus, identify the "nullary" role for prime 2:

; Nullary role
: The [[interval of equivalence]].

Adding that to the table, it might give us a clue about using temperaments in [[non-octave]] settings.

== Breaking out of mos ==
In contrast to role swapping, there is a one-for-all solution to harmony in temperaments regardless of how far the primes are on the generator chain.

This approach entails using the same old chords as the basis of harmony, and keeping the roles of each prime as in meantone. This means dominant is always ~3/2 over tonic, for example. A consequence is we must step out of the logic of mos scales, as they are often too restrictive without the many fifths to stack. Try starting unlimited and thinking directly about [[ratio]]s and [[comma pump]]s. You might end up with a diatonic-like scale with comma-level inflections here and there, but it is also possible to slap whatever scale you like, using inflections to reach the ratios.

=== Example: würschmidt ===
[[Würschmidt]] is just about on the complexity level where role swapping stops to work. The 10-tone mos contains a tonic and a "dominant" chord. For full analog of traditional harmony you need an unwieldy 13-tone mos, with more notes than the traditional chromatic scale and nearly twice the number of notes as the diatonic scale.

Instead of thinking about the mos scheme, we can pick our desired intervals however we like. If you want the classical major triad on tonic, use 0–1–8; on dominant, use 8–9–16; and on subdominant, use (-8)–(-7)–0. The possibilities are theoretically infinite. This is essentially working in a rank-3 space (or 2-dimensional [[lattice]] up to octave equivalence) but the edges are wrapped around due to the equivalence from the commas tempered out. The only thing that distinguishes it from JI is that you should in some way use the commas tempered out to prove the worth of the intonational compromises; otherwise you could simply choose JI.

[[Category:Regular temperament theory]]
[[Category:Harmony]]
[[Category:Guides]]