Superpyth - Revision history

Zhenlige: /* Tuning considerations and optima */simplify explanation

2026-04-24T18:07:27Z

Tuning considerations and optima: simplify explanation

← Older revision		Revision as of 18:07, 24 April 2026
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	A case can also be made for tuning archy even sharper than 27edo, which involves the notion of splitting the error of 4/3 into that of 8/7 and 7/6. This is a similar logic to Zarlino's preference for [[2/7-comma meantone]], treating [[~]][[6:7:8]] as the fundamental chord of the 2.3.7 subgroup, and in this case would imply 2/5-comma archy, where [[49/48]] is tuned justly, and 8/7 and 7/6 are both 1/5 a septimal comma off, and which is closely approximated by [[32edo]]. Unlike in the case of meantone, [[CEE]] optimization agrees with the notion of such a sharp tuning, where 3 is twice as sharp as 7. In this range, the best extension to prime 5 is ultrapyth.		A case can also be made for tuning archy even sharper than 27edo, which involves the notion of splitting the error of 4/3 into that of 8/7 and 7/6. This is a similar logic to Zarlino's preference for [[2/7-comma meantone]], treating [[~]][[6:7:8]] as the fundamental chord of the 2.3.7 subgroup, and in this case would imply 2/5-comma archy, where [[49/48]] is tuned justly, and 8/7 and 7/6 are both 1/5 a septimal comma off, and which is closely approximated by [[32edo]]. Unlike in the case of meantone, [[CEE]] optimization agrees with the notion of such a sharp tuning, where 3 is twice as sharp as 7. In this range, the best extension to prime 5 is ultrapyth.

	Finally, it may be noted that the {{w\|plastic ratio}} has a value of ~486.822 cents, which, taken as a generator (~4/3) and assuming a pure-octave period, constitutes an extremely sharp variety of archy. ~~This can be explained since archy equates [[21/16]] and [[4/3]]~~, ~~making~~ the ~9:12:~~16:21 chord evenly spaced by ~4/3, and when keeping {{nowrap\| ~9~~ + ~~~12 {{=}} ~21 }} the generator becomes the plastic number~~.		Finally, it may be noted that the {{w\|plastic ratio}} has a value of ~486.822 cents, which, taken as a generator (~4/3) and assuming a pure-octave period, constitutes an extremely sharp variety of archy. In fact, it is the tuning that makes ~6:7:8 become +1+1 [[delta-rational]].

	=== Norm-based tunings ===		=== Norm-based tunings ===

Squib: /* Tuning considerations and optima */ point to intended article instead of disambiguation page

2026-04-09T19:40:49Z

Tuning considerations and optima: point to intended article instead of disambiguation page

← Older revision		Revision as of 19:40, 9 April 2026
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	A case can also be made for tuning archy even sharper than 27edo, which involves the notion of splitting the error of 4/3 into that of 8/7 and 7/6. This is a similar logic to Zarlino's preference for [[2/7-comma meantone]], treating [[~]][[6:7:8]] as the fundamental chord of the 2.3.7 subgroup, and in this case would imply 2/5-comma archy, where [[49/48]] is tuned justly, and 8/7 and 7/6 are both 1/5 a septimal comma off, and which is closely approximated by [[32edo]]. Unlike in the case of meantone, [[CEE]] optimization agrees with the notion of such a sharp tuning, where 3 is twice as sharp as 7. In this range, the best extension to prime 5 is ultrapyth.		A case can also be made for tuning archy even sharper than 27edo, which involves the notion of splitting the error of 4/3 into that of 8/7 and 7/6. This is a similar logic to Zarlino's preference for [[2/7-comma meantone]], treating [[~]][[6:7:8]] as the fundamental chord of the 2.3.7 subgroup, and in this case would imply 2/5-comma archy, where [[49/48]] is tuned justly, and 8/7 and 7/6 are both 1/5 a septimal comma off, and which is closely approximated by [[32edo]]. Unlike in the case of meantone, [[CEE]] optimization agrees with the notion of such a sharp tuning, where 3 is twice as sharp as 7. In this range, the best extension to prime 5 is ultrapyth.

	Finally, it may be noted that the {{w\|plastic ~~number~~}} has a value of ~486.822 cents, which, taken as a generator (~4/3) and assuming a pure-octave period, constitutes an extremely sharp variety of archy. This can be explained since archy equates [[21/16]] and [[4/3]], making the ~9:12:16:21 chord evenly spaced by ~4/3, and when keeping {{nowrap\| ~9 + ~12 {{=}} ~21 }} the generator becomes the plastic number.		Finally, it may be noted that the {{w\|plastic ratio}} has a value of ~486.822 cents, which, taken as a generator (~4/3) and assuming a pure-octave period, constitutes an extremely sharp variety of archy. This can be explained since archy equates [[21/16]] and [[4/3]], making the ~9:12:16:21 chord evenly spaced by ~4/3, and when keeping {{nowrap\| ~9 + ~12 {{=}} ~21 }} the generator becomes the plastic number.

	=== Norm-based tunings ===		=== Norm-based tunings ===

Overthink: /* Scales */ expand sentence

2026-04-07T18:06:47Z

Scales: expand sentence

← Older revision		Revision as of 18:06, 7 April 2026
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	; 5-note mos ([[2L 3s]], proper)		; 5-note mos ([[2L 3s]], proper)

	In contrast to the meantone pentic scale, the superpyth pentic is much softer and mellow in quality, which is related to the fact that the intervals of the 2.3.7 subgroup cluster around [[5edo]]. As such, this system may be preferred over diatonic for interval classification, with 7/6 becoming a major interval and 8/7~9/8 becoming a minor one, and 49/48~[[28/27]] becoming a chroma. See [[Pentatonic Functional Just System]] for further ~~reading~~.		In contrast to the meantone pentic scale, the superpyth pentic is much softer and mellow in quality, which is related to the fact that the intervals of the 2.3.7 subgroup cluster around [[5edo]]. As such, this system may be preferred over diatonic for interval classification, with 7/6 becoming a major interval and 8/7~9/8 becoming a minor one, and 49/48~[[28/27]] becoming a chroma. See [[Pentatonic Functional Just System]] for a further explanation of what such a system would look like.

	; 7-note mos ([[5L 2s]], improper)		; 7-note mos ([[5L 2s]], improper)

Overthink: /* Chords and harmony */ note 28:36:42:49

2026-03-31T22:10:40Z

Chords and harmony: note 28:36:42:49

← Older revision		Revision as of 22:10, 31 March 2026
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	Therefore, it may be helpful to also consider the [[9-odd-limit]] [[anomalous saturated suspension\|saturated suspensions]], 1–7/6–3/2–7/4 ([[12:14:18:21]]) and 1–9/7–3/2–12/7 ([[14:18:21:24]]), which extend the chords above and are good for creating tensions and resolutions: 1–9/7–3/2–12/7 on the fifth degree creates a leading tone that wants to go to the tonic; 1–7/6–3/2–7/4 on the fourth degree creates a flat sixth that wants to go to the fifth.		Therefore, it may be helpful to also consider the [[9-odd-limit]] [[anomalous saturated suspension\|saturated suspensions]], 1–7/6–3/2–7/4 ([[12:14:18:21]]) and 1–9/7–3/2–12/7 ([[14:18:21:24]]), which extend the chords above and are good for creating tensions and resolutions: 1–9/7–3/2–12/7 on the fifth degree creates a leading tone that wants to go to the tonic; 1–7/6–3/2–7/4 on the fourth degree creates a flat sixth that wants to go to the fifth.

			In meantone, the dominant seventh chord, a tempering of [[20:25:30:36\|1–5/4–3/2–9/5]], is often used on the dominant to resolve to the tonic. A similar chord in superpyth is [[28:36:42:49\|1–9/7–3/2–7/4]], with a leading tone at 9/7 above the perfect fifth, or [[27/14]]. This chord contains a [[49/36]] [[tritone]] between the 9/7 and 7/4, which creates tension in the chord. (However, 11-limit superpyth maps it to [[15/11]], making it a [[15-odd-limit]] [[swetismic chords\|swetismic essentially tempered chord]].)

	== Scales ==		== Scales ==

Overthink: /* Chords and harmony */ fix errors

2026-03-17T18:45:40Z

Chords and harmony: fix errors

← Older revision		Revision as of 18:45, 17 March 2026
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	Another approach takes account of the fact that, in the 5-limit, the major triad can be constructed by octave-reducing odd harmonics 1, 3, and 5, giving us 4:5:6, with the minor triad being its utonal inversion. A similar construction of septimal chords gives us 1–7/6–4/3 ([[6:7:8]]) and its inversion 1–8/7–4/3 ([[21:24:28]]). These intervals contrast by [[49/48]], similarly to how 5-limit thirds contrast by [[25/24]]. There are some issues, however. For example, the 6:7:8 chord has the root on the top rather than the bottom, and the notes may clash from being too close to each other. However, the wide voicing of these chords, those being 1–7/4–3 (4:7:12) and 1–12/7–3 (7:12:21), solve both of these issues. These triads span a twelfth. In terms of the [[chain of fifths]], these chords are simpler in superpyth than the 5-limit triads in meantone.		Another approach takes account of the fact that, in the 5-limit, the major triad can be constructed by octave-reducing odd harmonics 1, 3, and 5, giving us 4:5:6, with the minor triad being its utonal inversion. A similar construction of septimal chords gives us 1–7/6–4/3 ([[6:7:8]]) and its inversion 1–8/7–4/3 ([[21:24:28]]). These intervals contrast by [[49/48]], similarly to how 5-limit thirds contrast by [[25/24]]. There are some issues, however. For example, the 6:7:8 chord has the root on the top rather than the bottom, and the notes may clash from being too close to each other. However, the wide voicing of these chords, those being 1–7/4–3 (4:7:12) and 1–12/7–3 (7:12:21), solve both of these issues. These triads span a twelfth. In terms of the [[chain of fifths]], these chords are simpler in superpyth than the 5-limit triads in meantone.

	Therefore, it may be helpful to also consider the [[9-odd-limit]] [[anomalous saturated suspension\|saturated suspensions]], 1–7/6–3/~~2–12~~/7 ([[12:14:18:21]]) and 1–9/7–3/2–12/7 ([[14:18:21:24]]), which extend the chords above and are good for creating tensions and resolutions: 1–9/7–3/2–12/7 on the fifth degree creates a leading tone that wants to go to the tonic; 1–7/6–3/~~2–12~~/7 on the fourth degree creates a flat sixth that wants to go to the fifth.		Therefore, it may be helpful to also consider the [[9-odd-limit]] [[anomalous saturated suspension\|saturated suspensions]], 1–7/6–3/2–7/4 ([[12:14:18:21]]) and 1–9/7–3/2–12/7 ([[14:18:21:24]]), which extend the chords above and are good for creating tensions and resolutions: 1–9/7–3/2–12/7 on the fifth degree creates a leading tone that wants to go to the tonic; 1–7/6–3/2–7/4 on the fourth degree creates a flat sixth that wants to go to the fifth.

	== Scales ==		== Scales ==

FloraC: /* Chords and harmony */ + the final kinds of chords

2026-03-17T11:47:45Z

Chords and harmony: + the final kinds of chords

← Older revision		Revision as of 11:47, 17 March 2026
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	Perhaps a more interesting approach is for the tonic chords of superpyth to be considered the tetrad 1–7/6–4/3–3/2 ([[6:7:8:9]]) and its utonal inverse 1–9/8–9/7–3/2 (representing [[14:16:18:21]] as [[64/63]] is tempered out), the former of which is a subminor chord with added fourth, and the latter a supermajor chord with added second (resembling the {{w\|mu chord}} of {{w\|Steely Dan}} fame). Both of these have distinct moods, and are stable and consonant, if somewhat more sophisticated than their classic 5-limit counterparts. To this group we could also add 1–9/8–4/3–3/2 (a sus2-4 chord). These three chords comprise the three ways to divide the superpyth perfect fifth into two whole tones and one septimal minor third. In the diatonic major scale, the 1–7/6–4/3–3/2 chord occurs on II, III, and VI, while its inverse occurs on I, IV, and V. Compared to meantone, major and minor swap places in a sense, though in a different way from in [[mavila]]. [[Chromatic]] or [[enharmonic]] alterations of them also exist, for example, the 1–9/8–9/7–3/2 chord may be altered to 1–9/8–11/8–3/2 (8:9:11:12), which is impressive-sounding, resembling a sus4 but with even more tension; it resolves quite nicely to 1–9/8–9/7–3/2.		Perhaps a more interesting approach is for the tonic chords of superpyth to be considered the tetrad 1–7/6–4/3–3/2 ([[6:7:8:9]]) and its utonal inverse 1–9/8–9/7–3/2 (representing [[14:16:18:21]] as [[64/63]] is tempered out), the former of which is a subminor chord with added fourth, and the latter a supermajor chord with added second (resembling the {{w\|mu chord}} of {{w\|Steely Dan}} fame). Both of these have distinct moods, and are stable and consonant, if somewhat more sophisticated than their classic 5-limit counterparts. To this group we could also add 1–9/8–4/3–3/2 (a sus2-4 chord). These three chords comprise the three ways to divide the superpyth perfect fifth into two whole tones and one septimal minor third. In the diatonic major scale, the 1–7/6–4/3–3/2 chord occurs on II, III, and VI, while its inverse occurs on I, IV, and V. Compared to meantone, major and minor swap places in a sense, though in a different way from in [[mavila]]. [[Chromatic]] or [[enharmonic]] alterations of them also exist, for example, the 1–9/8–9/7–3/2 chord may be altered to 1–9/8–11/8–3/2 (8:9:11:12), which is impressive-sounding, resembling a sus4 but with even more tension; it resolves quite nicely to 1–9/8–9/7–3/2.

	Another approach takes account of the fact that, in the 5-limit, the major triad can be constructed by octave-reducing odd harmonics 1, 3, and 5, giving us 4:5:6, with the minor triad being its utonal inversion. A similar construction of septimal chords gives us 1–7/6–4/3 ([[6:7:8]]) and its inversion 1–8/7–4/3 ([[21:24:28]]). These intervals contrast by [[49/48]], similarly to how 5-limit thirds contrast by [[25/24]]. There are some issues, however. For example, the 6:7:8 chord has the root on the top rather than the bottom, and the notes may clash from being too close to each other. However, the wide voicing of these chords, those being 1–7/4–3 (4:7:12) and 1–12/7–3 (7:12:21), solve both of these issues. These triads span a twelfth. In terms of the [[chain of fifths]], these chords are simpler in superpyth than the 5-limit triads in meantone.		Another approach takes account of the fact that, in the 5-limit, the major triad can be constructed by octave-reducing odd harmonics 1, 3, and 5, giving us 4:5:6, with the minor triad being its utonal inversion. A similar construction of septimal chords gives us 1–7/6–4/3 ([[6:7:8]]) and its inversion 1–8/7–4/3 ([[21:24:28]]). These intervals contrast by [[49/48]], similarly to how 5-limit thirds contrast by [[25/24]]. There are some issues, however. For example, the 6:7:8 chord has the root on the top rather than the bottom, and the notes may clash from being too close to each other. However, the wide voicing of these chords, those being 1–7/4–3 (4:7:12) and 1–12/7–3 (7:12:21), solve both of these issues. These triads span a twelfth. In terms of the [[chain of fifths]], these chords are simpler in superpyth than the 5-limit triads in meantone.

			Therefore, it may be helpful to also consider the [[9-odd-limit]] [[anomalous saturated suspension\|saturated suspensions]], 1–7/6–3/2–12/7 ([[12:14:18:21]]) and 1–9/7–3/2–12/7 ([[14:18:21:24]]), which extend the chords above and are good for creating tensions and resolutions: 1–9/7–3/2–12/7 on the fifth degree creates a leading tone that wants to go to the tonic; 1–7/6–3/2–12/7 on the fourth degree creates a flat sixth that wants to go to the fifth.

	== Scales ==		== Scales ==

Overthink: - unneeded link

2026-03-06T23:37:17Z

- unneeded link

← Older revision		Revision as of 23:37, 6 March 2026
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	Since the generator is a perfect fifth, superpyth can be notated using the same standard [[chain-of-fifths notation]] that is also used for [[meantone]], with the understanding that sharps are sharper than flats (for example, A♯ is sharper than B♭) just like in [[Pythagorean tuning]], in contrast to meantone where sharps are flatter than or equal to the corresponding flats. [[22edo\|13\22]] (~1/4 septimal comma) and [[27edo\|16\27]] (~1/3 septimal comma) are the most common tunings of the generator.		Since the generator is a perfect fifth, superpyth can be notated using the same standard [[chain-of-fifths notation]] that is also used for [[meantone]], with the understanding that sharps are sharper than flats (for example, A♯ is sharper than B♭) just like in [[Pythagorean tuning]], in contrast to meantone where sharps are flatter than or equal to the corresponding flats. [[22edo\|13\22]] (~1/4 septimal comma) and [[27edo\|16\27]] (~1/3 septimal comma) are the most common tunings of the generator.

	If intervals of [[5/1\|5]] are desired, the 5th harmonic is canonically mapped to +9 generators through tempering out [[245/243]], so [[5/4]] is an augmented second (e.g. C–D♯). This mapping equates the pythagorean limma, [[256/243]], to the syntonic [[81/80]], tempering out [[20480/19683]], so that 5/4 is mapped to a major third minus a limma. Therefore superpyth is the "opposite" of meantone in several different ways: most notably, meantone (including [[12edo]]) has the fifth tuned flat so that intervals of [[harmonic]] 5 are simple while intervals of [[7/1\|7]] are complex, while superpyth has the fifth tuned sharp so that intervals of 7 are simple while intervals of 5 are complex.		If intervals of [[5/1\|5]] are desired, the 5th harmonic is canonically mapped to +9 generators through tempering out [[245/243]], so [[5/4]] is an augmented second (e.g. C–D♯). This mapping equates the pythagorean limma, [[256/243]], to the syntonic [[81/80]], tempering out [[20480/19683]], so that 5/4 is mapped to a major third minus a limma. Therefore superpyth is the "opposite" of meantone in several different ways: most notably, meantone (including [[12edo]]) has the fifth tuned flat so that intervals of harmonic 5 are simple while intervals of [[7/1\|7]] are complex, while superpyth has the fifth tuned sharp so that intervals of 7 are simple while intervals of 5 are complex.

	If intervals of 11 are desired, the canonical way is to map [[11/8]] to +16 generators, or a doubly augmented second (C–D𝄪), tempering out [[100/99]]. A simpler but less accurate way to map it is to −6 generators, or a diminished fifth (C–G♭), by tempering out [[99/98]]. The latter is called '''suprapyth''', a name coined by [[Mike Battaglia]] in 2011<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_96882.html#96895 Yahoo! Tuning Group \| ''A few full 11-limit 896/891 temperaments'']</ref>. The [[2.3.7.11 subgroup]] restriction of suprapyth, known as [[supra]], is also notable. The two mappings unite on [[22edo]].		If intervals of 11 are desired, the canonical way is to map [[11/8]] to +16 generators, or a doubly augmented second (C–D𝄪), tempering out [[100/99]]. A simpler but less accurate way to map it is to −6 generators, or a diminished fifth (C–G♭), by tempering out [[99/98]]. The latter is called '''suprapyth''', a name coined by [[Mike Battaglia]] in 2011<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_96882.html#96895 Yahoo! Tuning Group \| ''A few full 11-limit 896/891 temperaments'']</ref>. The [[2.3.7.11 subgroup]] restriction of suprapyth, known as [[supra]], is also notable. The two mappings unite on [[22edo]].

FloraC: Cleanup on infobox

2026-02-09T09:46:42Z

Cleanup on infobox

← Older revision		Revision as of 09:46, 9 February 2026
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	\| Mapping = 1; 1 9 -2 16		\| Mapping = 1; 1 9 -2 16
	\| Edo join 1 = 22 \| Edo join 2 = 27e		\| Edo join 1 = 22 \| Edo join 2 = 27e
	\| ~~Generator~~ = 3/2		\| Generators = 3/2
	\| ~~Generator~~ tuning = 710.1		\| Generators tuning = 710.1
	\| Optimization method = CWE		\| Optimization method = CWE
	\| Pergen = (P8, P5)		\| Pergen = (P8, P5)
	\| Color name = Ruti		\| Color name = Ruti
	\| MOS scales = [[2L 3s]], [[5L 2s]], [[5L 7s]], [[5L 12s]], [[5L 17s]]		\| MOS scales = [[2L 3s]], [[5L 2s]], [[5L 7s]], [[5L 12s]], [[5L 17s]]
	\| Odd limit 1 = (2.3.7) 7 \| Mistuning 1 = 9.09 \| Complexity 1 = 5		\| Odd limit 1 = 2.3.7 7 \| Mistuning 1 = 9.09 \| Complexity 1 = 5
	\| Odd limit 2 = 9 \| Mistuning 2 = 15.27 \| Complexity 2 = 12		\| Odd limit 2 = 9 \| Mistuning 2 = 15.27 \| Complexity 2 = 12
	}}		}}

Overthink: /* Scales */ This article assumes full 7- and 11-limit superpyth; the range is more restricted than archy

2026-02-05T17:01:31Z

Scales: This article assumes full 7- and 11-limit superpyth; the range is more restricted than archy

← Older revision		Revision as of 17:01, 5 February 2026
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	; 7-note mos ([[5L 2s]], improper)		; 7-note mos ([[5L 2s]], improper)

	In contrast to the meantone diatonic scale, the superpyth diatonic is improper. Since the fifth is sharp rather than flat in meantone, the large steps (major seconds) are wider, being around ~~212–222~~{{c}} ~~depending on the tuning~~. The small steps (minor seconds) are thus narrower, being around ~~44–71~~{{c}}. This has the effect of large and small steps being more distinct compared to meantone diatonic, as well as stronger leading tones due to narrower small steps, though one may want to bend the leading tone down by a small step to avoid it being too close to the tonic.		In contrast to the meantone diatonic scale, the superpyth diatonic is improper. Since the fifth is sharp rather than flat in meantone, the large steps (major seconds) are wider, being around 220{{c}} in size. The small steps (minor seconds) are thus narrower, being around 50{{c}} (a quartertone) wide. This has the effect of large and small steps being more distinct compared to meantone diatonic, as well as stronger leading tones due to narrower small steps, though one may want to bend the leading tone down by a small step to avoid it being too close to the tonic.

	; 12-note mos ([[5L 7s]], improper)		; 12-note mos ([[5L 7s]], improper)

FloraC: /* Scales */ update

2026-01-28T09:04:55Z

Scales: update

← Older revision		Revision as of 09:04, 28 January 2026
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	=== Scala files ===		=== Scala files ===
	* [[Archy5]] – in ~~472edo~~ tuning		* [[Archy5]] – in 49edo tuning
	* [[Archy7]] – in ~~472edo~~ tuning		* [[Archy7]] – in 49edo tuning
	* [[Archy12]] – in ~~472edo~~ tuning		* [[Archy12]] – in 49edo tuning
	* [[12-22a]] – in 22edo tuning		* [[12-22a]] – in 22edo tuning

← Older revision		Revision as of 23:37, 6 March 2026
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	Since the generator is a perfect fifth, superpyth can be notated using the same standard [[chain-of-fifths notation]] that is also used for [[meantone]], with the understanding that sharps are sharper than flats (for example, A♯ is sharper than B♭) just like in [[Pythagorean tuning]], in contrast to meantone where sharps are flatter than or equal to the corresponding flats. [[22edo\|13\22]] (~1/4 septimal comma) and [[27edo\|16\27]] (~1/3 septimal comma) are the most common tunings of the generator.		Since the generator is a perfect fifth, superpyth can be notated using the same standard [[chain-of-fifths notation]] that is also used for [[meantone]], with the understanding that sharps are sharper than flats (for example, A♯ is sharper than B♭) just like in [[Pythagorean tuning]], in contrast to meantone where sharps are flatter than or equal to the corresponding flats. [[22edo\|13\22]] (~1/4 septimal comma) and [[27edo\|16\27]] (~1/3 septimal comma) are the most common tunings of the generator.

	If intervals of [[5/1\|5]] are desired, the 5th harmonic is canonically mapped to +9 generators through tempering out [[245/243]], so [[5/4]] is an augmented second (e.g. C–D♯). This mapping equates the pythagorean limma, [[256/243]], to the syntonic [[81/80]], tempering out [[20480/19683]], so that 5/4 is mapped to a major third minus a limma. Therefore superpyth is the "opposite" of meantone in several different ways: most notably, meantone (including [[12edo]]) has the fifth tuned flat so that intervals of [[harmonic]] 5 are simple while intervals of [[7/1\|7]] are complex, while superpyth has the fifth tuned sharp so that intervals of 7 are simple while intervals of 5 are complex.		If intervals of [[5/1\|5]] are desired, the 5th harmonic is canonically mapped to +9 generators through tempering out [[245/243]], so [[5/4]] is an augmented second (e.g. C–D♯). This mapping equates the pythagorean limma, [[256/243]], to the syntonic [[81/80]], tempering out [[20480/19683]], so that 5/4 is mapped to a major third minus a limma. Therefore superpyth is the "opposite" of meantone in several different ways: most notably, meantone (including [[12edo]]) has the fifth tuned flat so that intervals of harmonic 5 are simple while intervals of [[7/1\|7]] are complex, while superpyth has the fifth tuned sharp so that intervals of 7 are simple while intervals of 5 are complex.

	If intervals of 11 are desired, the canonical way is to map [[11/8]] to +16 generators, or a doubly augmented second (C–D𝄪), tempering out [[100/99]]. A simpler but less accurate way to map it is to −6 generators, or a diminished fifth (C–G♭), by tempering out [[99/98]]. The latter is called '''suprapyth''', a name coined by [[Mike Battaglia]] in 2011<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_96882.html#96895 Yahoo! Tuning Group \| ''A few full 11-limit 896/891 temperaments'']</ref>. The [[2.3.7.11 subgroup]] restriction of suprapyth, known as [[supra]], is also notable. The two mappings unite on [[22edo]].		If intervals of 11 are desired, the canonical way is to map [[11/8]] to +16 generators, or a doubly augmented second (C–D𝄪), tempering out [[100/99]]. A simpler but less accurate way to map it is to −6 generators, or a diminished fifth (C–G♭), by tempering out [[99/98]]. The latter is called '''suprapyth''', a name coined by [[Mike Battaglia]] in 2011<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_96882.html#96895 Yahoo! Tuning Group \| ''A few full 11-limit 896/891 temperaments'']</ref>. The [[2.3.7.11 subgroup]] restriction of suprapyth, known as [[supra]], is also notable. The two mappings unite on [[22edo]].