3.5.11 subgroup - Revision history

CompactStar: /* Rank-2 temperaments */

2026-11-02T05:27:10Z

Rank-2 temperaments

← Older revision		Revision as of 05:27, 2 November 2026
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	=== Rank-2 temperaments ===		=== Rank-2 temperaments ===
	{{Todo\|inline=1\|text=add more maybe?}}		{{Todo\|inline=1\|text=add more temperaments maybe?}}

	==== Delta centauri ====		==== Delta centauri ====

CompactStar: /* Rank-2 temperaments */

2026-11-02T05:21:17Z

Rank-2 temperaments

← Older revision		Revision as of 05:21, 2 November 2026
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	=== Rank-2 temperaments ===		=== Rank-2 temperaments ===
	{{Todo~~\|cleanup~~\|inline=1\|text=add more maybe?}}		{{Todo\|inline=1\|text=add more maybe?}}

	==== Delta centauri ====		==== Delta centauri ====

CompactStar: /* Rank-2 temperaments */

2026-11-02T05:21:11Z

Rank-2 temperaments

← Older revision		Revision as of 05:21, 2 November 2026
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	=== Rank-2 temperaments ===		=== Rank-2 temperaments ===
			{{Todo\|cleanup\|inline=1\|text=add more maybe?}}

	==== Delta centauri ====		==== Delta centauri ====
	{{Main\|Delta centauri}}		{{Main\|Delta centauri}}

CompactStar: /* Chords and harmony */

2026-11-02T05:15:10Z

Chords and harmony

← Older revision		Revision as of 05:15, 2 November 2026
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	Simple [[isoharmonic]] chords are largely absent from this subgroup, but non-isoharmonic chords can be used instead. An obvious place to start would be with the chord 3:5:11, which tritave-reduces to 9:11:15, analogous to how 2:3:5 octave-reduces to [[4:5:6]]. This chord is actually surprisingly vaguely familiar, at least compared to the Bohlen-Pierce [[3:5:7]] triad, because 9:11:15 (1-[[11/9]]-[[5/3]]) somewhat resembles 12:15:20 (1-[[5/4]]-[[5/3]]), which is an octave-equivalent inversion of the familiar [[10:12:15]] minor triad, although there is still a very large difference between [[11/9]] and [[5/4]] such that 9:11:15 still clearly sounds xenharmonic. 9:11:15 also has a utonal counterpart, 33:45:55 (which resembles 3:4:5), and a simpler, but very wide, tritave-equivalent inversion, 5:9:11. The tetrad 9:11:15:25 adds [[25/9]] or ([[5/3]])<sup>2</sup> above the root. If you have another tritave above the root then this will become 9:11:15:25:27 with a dissonant [[27/25]] semitone between the 25/9 and 27/9 (3/1), which is a similar situation to the [[16/15]] at the top of the 5-limit major seventh chord [[8:10:12:15]](:16). In the 3.5.11.13 subgroup, 9:11:15 can be extended to the isoharmonic chord 9:11:13:15, which is sort of a [[diminished seventh chord]].		Simple [[isoharmonic]] chords are largely absent from this subgroup, but non-isoharmonic chords can be used instead. An obvious place to start would be with the chord 3:5:11, which tritave-reduces to 9:11:15, analogous to how 2:3:5 octave-reduces to [[4:5:6]]. This chord is actually surprisingly vaguely familiar, at least compared to the Bohlen-Pierce [[3:5:7]] triad, because 9:11:15 (1-[[11/9]]-[[5/3]]) somewhat resembles 12:15:20 (1-[[5/4]]-[[5/3]]), which is an octave-equivalent inversion of the familiar [[10:12:15]] minor triad, although there is still a very large difference between [[11/9]] and [[5/4]] such that 9:11:15 still clearly sounds xenharmonic. 9:11:15 also has a utonal counterpart, 33:45:55 (which resembles 3:4:5), and a simpler, but very wide, tritave-equivalent inversion, 5:9:11. The tetrad 9:11:15:25 adds [[25/9]] or ([[5/3]])<sup>2</sup> above the root. If you have another tritave above the root then this will become 9:11:15:25:27 with a dissonant [[27/25]] semitone between the 25/9 and 27/9 (3/1), which is a similar situation to the [[16/15]] at the top of the 5-limit major seventh chord [[8:10:12:15]](:16). In the 3.5.11.13 subgroup, 9:11:15 can be extended to the isoharmonic chord 9:11:13:15, which is sort of a [[diminished seventh chord]].

	It is interesting to note that if you run with the analogy of 9:11:15 as a "minor triad" equivalent and 33:44:55 as a "major triad" equivalent, [[9/7]] acts as a sort of "neutral third" equivalent, being near √[[5/3]] ([[BPS]] temperament realizes this).		It is interesting to note that if you run with the analogy of 9:11:15 as a "minor triad" equivalent and 33:44:55 as a "major triad" equivalent, [[9/7]] acts as a sort of "neutral third" equivalent, being near √([[5/3]]) ([[BPS]] temperament realizes this).

	''One should bear in mind that, in the world of tritave equivalence, ratios of 3 in the denominator are fungible instead of those of 2.'' For example, while the octave-reduced fifth harmonic is 5:4, the tritave-reduced fifth harmonic would be 5:3 instead, which would be a "major sixth" by conventional [[pitch class\|pitch-class]] terminology. The tritave-reduced 11th harmonic is 11:9.		''One should bear in mind that, in the world of tritave equivalence, ratios of 3 in the denominator are fungible instead of those of 2.'' For example, while the octave-reduced fifth harmonic is 5:4, the tritave-reduced fifth harmonic would be 5:3 instead, which would be a "major sixth" by conventional [[pitch class\|pitch-class]] terminology. The tritave-reduced 11th harmonic is 11:9.

CompactStar: /* Chords and harmony */

2026-11-02T05:14:58Z

Chords and harmony

← Older revision		Revision as of 05:14, 2 November 2026
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	== Chords and harmony ==		== Chords and harmony ==
	Simple [[isoharmonic]] chords are largely absent from this subgroup, but non-isoharmonic chords can be used instead. An obvious place to start would be with the chord 3:5:11, which tritave-reduces to 9:11:15, analogous to how 2:3:5 octave-reduces to [[4:5:6]]. This chord is actually surprisingly vaguely familiar, at least compared to the Bohlen-Pierce [[3:5:7]] triad, because 9:11:15 (1-[[11/9]]-[[5/3]]) somewhat resembles 12:15:20 (1-[[5/4]]-[[5/3]]), which is an octave-equivalent inversion of the familiar [[10:12:15]] minor triad, although there is still a very large difference between [[11/9]] and [[5/4]] such that 9:11:15 still clearly sounds xenharmonic. 9:11:15 also has a utonal counterpart, 33:45:55 (which resembles 3:4:5), and a simpler, but very wide, tritave-equivalent inversion, 5:9:11. The tetrad 9:11:15:25 adds [[25/9]] or ([[5/3]])<sup>2</sup> above the root. If you have another tritave above the root then this will become 9:11:15:25:27 with a dissonant [[27/25]] semitone between the 25/9 and 27/9 (3/1), which is a similar situation to the [[16/15]] at the top of the 5-limit major seventh chord [[8:10:12:15]](:16). In the 3.5.11.13 subgroup, 9:11:15 can be extended to the isoharmonic chord 9:11:13:15, which is sort of a [[diminished seventh chord]].		Simple [[isoharmonic]] chords are largely absent from this subgroup, but non-isoharmonic chords can be used instead. An obvious place to start would be with the chord 3:5:11, which tritave-reduces to 9:11:15, analogous to how 2:3:5 octave-reduces to [[4:5:6]]. This chord is actually surprisingly vaguely familiar, at least compared to the Bohlen-Pierce [[3:5:7]] triad, because 9:11:15 (1-[[11/9]]-[[5/3]]) somewhat resembles 12:15:20 (1-[[5/4]]-[[5/3]]), which is an octave-equivalent inversion of the familiar [[10:12:15]] minor triad, although there is still a very large difference between [[11/9]] and [[5/4]] such that 9:11:15 still clearly sounds xenharmonic. 9:11:15 also has a utonal counterpart, 33:45:55 (which resembles 3:4:5), and a simpler, but very wide, tritave-equivalent inversion, 5:9:11. The tetrad 9:11:15:25 adds [[25/9]] or ([[5/3]])<sup>2</sup> above the root. If you have another tritave above the root then this will become 9:11:15:25:27 with a dissonant [[27/25]] semitone between the 25/9 and 27/9 (3/1), which is a similar situation to the [[16/15]] at the top of the 5-limit major seventh chord [[8:10:12:15]](:16). In the 3.5.11.13 subgroup, 9:11:15 can be extended to the isoharmonic chord 9:11:13:15, which is sort of a [[diminished seventh chord]].

			It is interesting to note that if you run with the analogy of 9:11:15 as a "minor triad" equivalent and 33:44:55 as a "major triad" equivalent, [[9/7]] acts as a sort of "neutral third" equivalent, being near √[[5/3]] ([[BPS]] temperament realizes this).

	''One should bear in mind that, in the world of tritave equivalence, ratios of 3 in the denominator are fungible instead of those of 2.'' For example, while the octave-reduced fifth harmonic is 5:4, the tritave-reduced fifth harmonic would be 5:3 instead, which would be a "major sixth" by conventional [[pitch class\|pitch-class]] terminology. The tritave-reduced 11th harmonic is 11:9.		''One should bear in mind that, in the world of tritave equivalence, ratios of 3 in the denominator are fungible instead of those of 2.'' For example, while the octave-reduced fifth harmonic is 5:4, the tritave-reduced fifth harmonic would be 5:3 instead, which would be a "major sixth" by conventional [[pitch class\|pitch-class]] terminology. The tritave-reduced 11th harmonic is 11:9.

CompactStar at 05:07, 2 November 2026

2026-11-02T05:07:32Z

← Older revision		Revision as of 05:07, 2 November 2026
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	=== Rank-2 temperaments ===		=== Rank-2 temperaments ===
	=== Delta centauri ===		==== Delta centauri ====
	{{Main\|Delta centauri}}		{{Main\|Delta centauri}}
	[[Delta centauri]] (b28 & b71) is a [[microtemperament]] that is generated by [[~]][[5/3]], and 9 generators stacked represent [[11/9]]. It tempers out the [[4-cent comma]] (1953125/1948617). The best tuning is to flatten 5/3 by around half a cent, though just 5/3 or slightly sharp 5/3 will work fine too. For a microtemperament, it is quite low in [[complexity]], and can be used with a 13-note {{MOS scalesig\|2L 11s<3/1>\|link=1}} or 15-note {{MOS scalesig\|13L 2s<3/1>\|link=1}} (these are melodically comparable to 8-10 note octave-equivalent scales), making it a very [[efficiency\|efficient]] temperament. Due to the ~3/1 period and ~5/3 generator, there is one-to-one correspondence between the [[3.5 subgroup]] and mapped intervals. It also has the advantage (or disadvantage, depending on your point of view) that 11/9 is simple enough to appear in the 13-note and 15-note MOS scales but the quasi-octave 243/121 is too complex, being at -18 generators.		[[Delta centauri]] (b28 & b71) is a [[microtemperament]] that is generated by [[~]][[5/3]], and 9 generators stacked represent [[11/9]]. It tempers out the [[4-cent comma]] (1953125/1948617). The best tuning is to flatten 5/3 by around half a cent, though just 5/3 or slightly sharp 5/3 will work fine too. For a microtemperament, it is quite low in [[complexity]], and can be used with a 13-note {{MOS scalesig\|2L 11s<3/1>\|link=1}} or 15-note {{MOS scalesig\|13L 2s<3/1>\|link=1}} (these are melodically comparable to 8-10 note octave-equivalent scales), making it a very [[efficiency\|efficient]] temperament. Due to the ~3/1 period and ~5/3 generator, there is one-to-one correspondence between the [[3.5 subgroup]] and mapped intervals. It also has the advantage (or disadvantage, depending on your point of view) that 11/9 is simple enough to appear in the 13-note and 15-note MOS scales but the quasi-octave 243/121 is too complex, being at -18 generators.

	=== Deneb ===		==== Deneb ====
	{{Main\|Deneb}}		{{Main\|Deneb}}
	[[Deneb]] (b11 & b28) tempers out the comma [[6655/6561]] and equates three [[11/9]]'s with [[9/5]]. It is supported by [[11edt]] and the 11-note {{mos scalesig\|6L 5s<3/1>\|link=1}} scale can be used with diatonic-like melodic properties. In some tunings, 5 generators up is an accurate [[8/3]]–if this is included, rendering it a 3.5.8.11 or 8.3.5.11 temperament, then it is actually related to [[hitchcock]]. [[28edt]] is notable for being the unique intersection of deneb and delta centauri. The 3.5.7.11 temperament [[Mintra]], although it's mainly known as a strong extension of [[Mintaka]], is additionally a weak extension of Deneb and a weak extension of [[BPS]], combining all three temperaments into an efficient no-2s 11-limit temperament.		[[Deneb]] (b11 & b28) tempers out the comma [[6655/6561]] and equates three [[11/9]]'s with [[9/5]]. It is supported by [[11edt]] and the 11-note {{mos scalesig\|6L 5s<3/1>\|link=1}} scale can be used with diatonic-like melodic properties. In some tunings, 5 generators up is an accurate [[8/3]]–if this is included, rendering it a 3.5.8.11 or 8.3.5.11 temperament, then it is actually related to [[hitchcock]]. [[28edt]] is notable for being the unique intersection of deneb and delta centauri. The 3.5.7.11 temperament [[Mintra]], although it's mainly known as a strong extension of [[Mintaka]], is additionally a weak extension of Deneb and a weak extension of [[BPS]], combining all three temperaments into an efficient no-2s 11-limit temperament.

	=== Polaris ===		==== Polaris ====
	[[Polaris]] (b17 & b32) constructs an ~11/9 from stacking ~5/3 generators, similar to delta centauri, but here ~5/3 is tuned quite sharp so that -6 5/3s stacked (or 6 [[9/5]]s stacked) tritave-reduced is ~11/9.		[[Polaris]] (b17 & b32) constructs an ~11/9 from stacking ~5/3 generators, similar to delta centauri, but here ~5/3 is tuned quite sharp so that -6 5/3s stacked (or 6 [[9/5]]s stacked) tritave-reduced is ~11/9.

CompactStar: /* Rank-2 temperaments */

2026-11-02T05:07:17Z

Rank-2 temperaments

← Older revision		Revision as of 05:07, 2 November 2026
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	=== Rank-2 temperaments ===		=== Rank-2 temperaments ===
			=== Delta centauri ===
			{{Main\|Delta centauri}}
	[[Delta centauri]] (b28 & b71) is a [[microtemperament]] that is generated by [[~]][[5/3]], and 9 generators stacked represent [[11/9]]. It tempers out the [[4-cent comma]] (1953125/1948617). The best tuning is to flatten 5/3 by around half a cent, though just 5/3 or slightly sharp 5/3 will work fine too. For a microtemperament, it is quite low in [[complexity]], and can be used with a 13-note {{MOS scalesig\|2L 11s<3/1>\|link=1}} or 15-note {{MOS scalesig\|13L 2s<3/1>\|link=1}} (these are melodically comparable to 8-10 note octave-equivalent scales), making it a very [[efficiency\|efficient]] temperament. Due to the ~3/1 period and ~5/3 generator, there is one-to-one correspondence between the [[3.5 subgroup]] and mapped intervals. It also has the advantage (or disadvantage, depending on your point of view) that 11/9 is simple enough to appear in the 13-note and 15-note MOS scales but the quasi-octave 243/121 is too complex, being at -18 generators.		[[Delta centauri]] (b28 & b71) is a [[microtemperament]] that is generated by [[~]][[5/3]], and 9 generators stacked represent [[11/9]]. It tempers out the [[4-cent comma]] (1953125/1948617). The best tuning is to flatten 5/3 by around half a cent, though just 5/3 or slightly sharp 5/3 will work fine too. For a microtemperament, it is quite low in [[complexity]], and can be used with a 13-note {{MOS scalesig\|2L 11s<3/1>\|link=1}} or 15-note {{MOS scalesig\|13L 2s<3/1>\|link=1}} (these are melodically comparable to 8-10 note octave-equivalent scales), making it a very [[efficiency\|efficient]] temperament. Due to the ~3/1 period and ~5/3 generator, there is one-to-one correspondence between the [[3.5 subgroup]] and mapped intervals. It also has the advantage (or disadvantage, depending on your point of view) that 11/9 is simple enough to appear in the 13-note and 15-note MOS scales but the quasi-octave 243/121 is too complex, being at -18 generators.

			=== Deneb ===
			{{Main\|Deneb}}
	[[Deneb]] (b11 & b28) tempers out the comma [[6655/6561]] and equates three [[11/9]]'s with [[9/5]]. It is supported by [[11edt]] and the 11-note {{mos scalesig\|6L 5s<3/1>\|link=1}} scale can be used with diatonic-like melodic properties. In some tunings, 5 generators up is an accurate [[8/3]]–if this is included, rendering it a 3.5.8.11 or 8.3.5.11 temperament, then it is actually related to [[hitchcock]]. [[28edt]] is notable for being the unique intersection of deneb and delta centauri. The 3.5.7.11 temperament [[Mintra]], although it's mainly known as a strong extension of [[Mintaka]], is additionally a weak extension of Deneb and a weak extension of [[BPS]], combining all three temperaments into an efficient no-2s 11-limit temperament.		[[Deneb]] (b11 & b28) tempers out the comma [[6655/6561]] and equates three [[11/9]]'s with [[9/5]]. It is supported by [[11edt]] and the 11-note {{mos scalesig\|6L 5s<3/1>\|link=1}} scale can be used with diatonic-like melodic properties. In some tunings, 5 generators up is an accurate [[8/3]]–if this is included, rendering it a 3.5.8.11 or 8.3.5.11 temperament, then it is actually related to [[hitchcock]]. [[28edt]] is notable for being the unique intersection of deneb and delta centauri. The 3.5.7.11 temperament [[Mintra]], although it's mainly known as a strong extension of [[Mintaka]], is additionally a weak extension of Deneb and a weak extension of [[BPS]], combining all three temperaments into an efficient no-2s 11-limit temperament.

			=== Polaris ===
	[[Polaris]] (b17 & b32) constructs an ~11/9 from stacking ~5/3 generators, similar to delta centauri, but here ~5/3 is tuned quite sharp so that -6 5/3s stacked (or 6 [[9/5]]s stacked) tritave-reduced is ~11/9.		[[Polaris]] (b17 & b32) constructs an ~11/9 from stacking ~5/3 generators, similar to delta centauri, but here ~5/3 is tuned quite sharp so that -6 5/3s stacked (or 6 [[9/5]]s stacked) tritave-reduced is ~11/9.

CompactStar: /* Rank-2 temperaments */

2026-11-02T05:06:19Z

Rank-2 temperaments

← Older revision		Revision as of 05:06, 2 November 2026
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	=== Rank-2 temperaments ===		=== Rank-2 temperaments ===
	[[Delta centauri]] (b28 & b71) is a [[microtemperament]] that is generated by [[~]][[5/3]], and 9 generators stacked represent [[11/9]]. It tempers out the [[4-cent comma]] (1953125/1948617). The best tuning is to flatten 5/3 by around half a cent, though just 5/3 or slightly sharp 5/3 will work fine too. For a microtemperament, it is quite low in [[complexity]], and can be used with a 13-note {{MOS scalesig\|2L 11s<3/1>\|link=1}} or 15-note {{MOS scalesig\|13L 2s<3/1>\|link=1}} (these are melodically comparable to 8-10 note octave-equivalent scales), making it a very [[efficiency\|efficient]] temperament. Due to the ~3/1 period and ~5/3 generator, there is one-to-one correspondence between the [[3.5 subgroup]] and mapped intervals ~~(and in fact the untempered 3.5 subgroup could be used as a tuning similar to using [[Pythagorean tuning]] as a [[schismic]] tuning)~~. It also has the advantage (or disadvantage, depending on your point of view) that 11/9 is simple enough to appear in the 13-note and 15-note MOS scales but the quasi-octave 243/121 is too complex, being at -18 generators.		[[Delta centauri]] (b28 & b71) is a [[microtemperament]] that is generated by [[~]][[5/3]], and 9 generators stacked represent [[11/9]]. It tempers out the [[4-cent comma]] (1953125/1948617). The best tuning is to flatten 5/3 by around half a cent, though just 5/3 or slightly sharp 5/3 will work fine too. For a microtemperament, it is quite low in [[complexity]], and can be used with a 13-note {{MOS scalesig\|2L 11s<3/1>\|link=1}} or 15-note {{MOS scalesig\|13L 2s<3/1>\|link=1}} (these are melodically comparable to 8-10 note octave-equivalent scales), making it a very [[efficiency\|efficient]] temperament. Due to the ~3/1 period and ~5/3 generator, there is one-to-one correspondence between the [[3.5 subgroup]] and mapped intervals. It also has the advantage (or disadvantage, depending on your point of view) that 11/9 is simple enough to appear in the 13-note and 15-note MOS scales but the quasi-octave 243/121 is too complex, being at -18 generators.

	[[Deneb]] (b11 & b28) tempers out the comma [[6655/6561]] and equates three [[11/9]]'s with [[9/5]]. It is supported by [[11edt]] and the 11-note {{mos scalesig\|6L 5s<3/1>\|link=1}} scale can be used with diatonic-like melodic properties. In some tunings, 5 generators up is an accurate [[8/3]]–if this is included, rendering it a 3.5.8.11 or 8.3.5.11 temperament, then it is actually related to [[hitchcock]]. [[28edt]] is notable for being the unique intersection of deneb and delta centauri. The 3.5.7.11 temperament [[Mintra]], although it's mainly known as a strong extension of [[Mintaka]], is additionally a weak extension of Deneb and a weak extension of [[BPS]], combining all three temperaments into an efficient no-2s 11-limit temperament.		[[Deneb]] (b11 & b28) tempers out the comma [[6655/6561]] and equates three [[11/9]]'s with [[9/5]]. It is supported by [[11edt]] and the 11-note {{mos scalesig\|6L 5s<3/1>\|link=1}} scale can be used with diatonic-like melodic properties. In some tunings, 5 generators up is an accurate [[8/3]]–if this is included, rendering it a 3.5.8.11 or 8.3.5.11 temperament, then it is actually related to [[hitchcock]]. [[28edt]] is notable for being the unique intersection of deneb and delta centauri. The 3.5.7.11 temperament [[Mintra]], although it's mainly known as a strong extension of [[Mintaka]], is additionally a weak extension of Deneb and a weak extension of [[BPS]], combining all three temperaments into an efficient no-2s 11-limit temperament.

CompactStar: /* Rank-2 temperaments */

2026-11-02T05:04:43Z

Rank-2 temperaments

← Older revision		Revision as of 05:04, 2 November 2026
Line 20:		Line 20:

	=== Rank-2 temperaments ===		=== Rank-2 temperaments ===
	[[Delta centauri]] (b28 & b71) is a [[microtemperament]] that is generated by [[~]][[5/3]], and 9 generators stacked represent [[11/9]]. It tempers out the [[4-cent comma]] (1953125/1948617). The best tuning is to flatten 5/3 by around half a cent, though just 5/3 or slightly sharp 5/3 will work fine too. For a microtemperament, it is quite low in [[complexity]], and can be used with a 13-note {{MOS scalesig\|2L 11s<3/1>\|link=1}} or 15-note {{MOS scalesig\|13L 2s<3/1>\|link=1}} (these are melodically comparable to 8-10 note octave-equivalent scales), making it a very [[efficiency\|efficient]] temperament. Due to the ~3/1 period and ~5/3 generator, there is one-to-one correspondence between the [[3.5 subgroup]] and mapped intervals (and in fact the untempered 3.5 subgroup could be used as a tuning similar to using [[Pythagorean tuning]] as a schismic tuning). It also has the advantage (or disadvantage, depending on your point of view) that 11/9 is simple enough to appear in the 13-note and 15-note MOS scales but the quasi-octave 243/121 is too complex, being at -18 generators.		[[Delta centauri]] (b28 & b71) is a [[microtemperament]] that is generated by [[~]][[5/3]], and 9 generators stacked represent [[11/9]]. It tempers out the [[4-cent comma]] (1953125/1948617). The best tuning is to flatten 5/3 by around half a cent, though just 5/3 or slightly sharp 5/3 will work fine too. For a microtemperament, it is quite low in [[complexity]], and can be used with a 13-note {{MOS scalesig\|2L 11s<3/1>\|link=1}} or 15-note {{MOS scalesig\|13L 2s<3/1>\|link=1}} (these are melodically comparable to 8-10 note octave-equivalent scales), making it a very [[efficiency\|efficient]] temperament. Due to the ~3/1 period and ~5/3 generator, there is one-to-one correspondence between the [[3.5 subgroup]] and mapped intervals (and in fact the untempered 3.5 subgroup could be used as a tuning similar to using [[Pythagorean tuning]] as a [[schismic]] tuning). It also has the advantage (or disadvantage, depending on your point of view) that 11/9 is simple enough to appear in the 13-note and 15-note MOS scales but the quasi-octave 243/121 is too complex, being at -18 generators.

	[[Deneb]] (b11 & b28) tempers out the comma [[6655/6561]] and equates three [[11/9]]'s with [[9/5]]. It is supported by [[11edt]] and the 11-note {{mos scalesig\|6L 5s<3/1>\|link=1}} scale can be used with diatonic-like melodic properties. In some tunings, 5 generators up is an accurate [[8/3]]–if this is included, rendering it a 3.5.8.11 or 8.3.5.11 temperament, then it is actually related to [[hitchcock]]. [[28edt]] is notable for being the unique intersection of deneb and delta centauri. The 3.5.7.11 temperament [[Mintra]], although it's mainly known as a strong extension of [[Mintaka]], is additionally a weak extension of Deneb and a weak extension of [[BPS]], combining all three temperaments into an efficient no-2s 11-limit temperament.		[[Deneb]] (b11 & b28) tempers out the comma [[6655/6561]] and equates three [[11/9]]'s with [[9/5]]. It is supported by [[11edt]] and the 11-note {{mos scalesig\|6L 5s<3/1>\|link=1}} scale can be used with diatonic-like melodic properties. In some tunings, 5 generators up is an accurate [[8/3]]–if this is included, rendering it a 3.5.8.11 or 8.3.5.11 temperament, then it is actually related to [[hitchcock]]. [[28edt]] is notable for being the unique intersection of deneb and delta centauri. The 3.5.7.11 temperament [[Mintra]], although it's mainly known as a strong extension of [[Mintaka]], is additionally a weak extension of Deneb and a weak extension of [[BPS]], combining all three temperaments into an efficient no-2s 11-limit temperament.

CompactStar: /* Rank-2 temperaments */

2026-11-02T05:04:24Z

Rank-2 temperaments

← Older revision		Revision as of 05:04, 2 November 2026
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	=== Rank-2 temperaments ===		=== Rank-2 temperaments ===
	[[Delta centauri]] (b28 & b71) is a [[microtemperament]] that is generated by [[~]][[5/3]], and 9 generators stacked represent [[11/9]]. It tempers out the [[4-cent comma]] (1953125/1948617). The best tuning is to flatten 5/3 by around half a cent, though just 5/3 or slightly sharp 5/3 will work fine too. For a microtemperament, it is quite low in complexity, and can be used with a 13-note {{MOS scalesig\|2L 11s<3/1>\|link=1}} or 15-note {{MOS scalesig\|13L 2s<3/1>\|link=1}} (these are melodically comparable to 8-10 note octave-equivalent scales), making it a very [[efficiency\|efficient]] temperament. Due to the ~3/1 period and ~5/3 generator, there is one-to-one correspondence between the [[3.5 subgroup]] and mapped intervals (and in fact the untempered 3.5 subgroup could be used as a tuning similar to using [[Pythagorean tuning]] as a schismic tuning). It also has the advantage (or disadvantage, depending on your point of view) that 11/9 is simple enough to appear in the 13-note and 15-note MOS scales but the quasi-octave 243/121 is too complex, being at -18 generators.		[[Delta centauri]] (b28 & b71) is a [[microtemperament]] that is generated by [[~]][[5/3]], and 9 generators stacked represent [[11/9]]. It tempers out the [[4-cent comma]] (1953125/1948617). The best tuning is to flatten 5/3 by around half a cent, though just 5/3 or slightly sharp 5/3 will work fine too. For a microtemperament, it is quite low in [[complexity]], and can be used with a 13-note {{MOS scalesig\|2L 11s<3/1>\|link=1}} or 15-note {{MOS scalesig\|13L 2s<3/1>\|link=1}} (these are melodically comparable to 8-10 note octave-equivalent scales), making it a very [[efficiency\|efficient]] temperament. Due to the ~3/1 period and ~5/3 generator, there is one-to-one correspondence between the [[3.5 subgroup]] and mapped intervals (and in fact the untempered 3.5 subgroup could be used as a tuning similar to using [[Pythagorean tuning]] as a schismic tuning). It also has the advantage (or disadvantage, depending on your point of view) that 11/9 is simple enough to appear in the 13-note and 15-note MOS scales but the quasi-octave 243/121 is too complex, being at -18 generators.

	[[Deneb]] (b11 & b28) tempers out the comma [[6655/6561]] and equates three [[11/9]]'s with [[9/5]]. It is supported by [[11edt]] and the 11-note {{mos scalesig\|6L 5s<3/1>\|link=1}} scale can be used with diatonic-like melodic properties. In some tunings, 5 generators up is an accurate [[8/3]]–if this is included, rendering it a 3.5.8.11 or 8.3.5.11 temperament, then it is actually related to [[hitchcock]]. [[28edt]] is notable for being the unique intersection of deneb and delta centauri. The 3.5.7.11 temperament [[Mintra]], although it's mainly known as a strong extension of [[Mintaka]], is additionally a weak extension of Deneb and a weak extension of [[BPS]], combining all three temperaments into an efficient no-2s 11-limit temperament.		[[Deneb]] (b11 & b28) tempers out the comma [[6655/6561]] and equates three [[11/9]]'s with [[9/5]]. It is supported by [[11edt]] and the 11-note {{mos scalesig\|6L 5s<3/1>\|link=1}} scale can be used with diatonic-like melodic properties. In some tunings, 5 generators up is an accurate [[8/3]]–if this is included, rendering it a 3.5.8.11 or 8.3.5.11 temperament, then it is actually related to [[hitchcock]]. [[28edt]] is notable for being the unique intersection of deneb and delta centauri. The 3.5.7.11 temperament [[Mintra]], although it's mainly known as a strong extension of [[Mintaka]], is additionally a weak extension of Deneb and a weak extension of [[BPS]], combining all three temperaments into an efficient no-2s 11-limit temperament.