3.5.7 subgroup: Difference between revisions

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~~#redirect~~ [[~~Just~~ intonation subgroup #7~~-limit subgroups~~]]

The '''3.5.7 subgroup''' is a [[non-octave]] [[just intonation subgroup]] consisting of [[rational interval]]s where 3, 5, and 7 are the only allowable [[prime factor]]s, so that every such interval may be written as a ratio of integers which are products of 3, 5, and 7. This is an infinite set. Some examples of intervals in this subgroup are [[5/3]], [[7/3]], [[7/5]], [[25/21]], [[35/27]], and so on.

[[Category:~~Subgroup~~]]

The 3.5.7 subgroup is a [[retraction]] of the [[7-limit]], obtained by removing prime 2. Its simplest [[expansion]] is the [[3.5.7.11 subgroup]], which adds prime [[11/1|11]].

[[Category:Rank-3 temperaments]]

[[Category:7-limit]]

The approaches to composition in this subgroup can be categorized as follows:

# By not having an [[interval of equivalence]], by composing in the 7-limit but without octavation;

# By treating the [[3/1|3rd harmonic]] as the interval of equivalence.

Since octavation is usually trivial to implement, this article focuses on the second approach.

== Introduction to tritave equivalence ==

See [[EDT #Introduction to tritave equivalence]].

== Chords and harmony ==

If factors of 2 are eliminated, the search for consonant intervals begins with the odd harmonic series, 1:3:5:7:9:…. We can take the second tritave of the series, 3:5:7:9, and find within it the two [[isoharmonic]] triads [[3:5:7]] and [[5:7:9]]; the analogy here is with the third octave of the full harmonic series, 4:5:6:7:8, and the isoharmonic triad [[4:5:6]], the foundation of triadic harmony in [[5-limit]] theory. Hence, 3:5:7 or 5:7:9 can be viewed as the fundamental consonant triad of no-2's music, and if we then apply the 5-limit analogy one more time, these triads are bounded by the intervals [[7/3]] or [[9/5]] respectively, either of them filling the role of the "perfect fifth" in diatonicism.

The standard [[Bohlen–Pierce]] theory takes 3:5:7 to be the fundamental triad, and therefore naturally goes together with scales generated by 7/3, or equivalently 9/7 (the latter being convention), against the tritave. 7/3 generates pentatonic ({{mos scalesig|4L 1s<3/1>|link=1}}) and enneatonic ({{mos scalesig|4L 5s<3/1>|link=1}}) [[mos]] scales, and therefore the enneatonic, known as the ''Lambda'' scale, can be seen as the analog of the [[5L 2s|diatonic]] scale.

''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.

== Temperaments ==

=== Equal temperaments ===

=== Rank-2 temperaments ===

{{Todo|cleanup|improve readability|inline=1|text=Rewrite for clarity}}

As generators of the Lambda scale run from [[9edt|7\9edt]] to [[4edt|3\4edt]], [[13edt]] is the smallest equal tuning [[support]]ing it, and can be seen as an equivalent of [[12edo]]. However, 13edt's accuracy in the 3.5.7 subgroup is much better than 12edo's in the 5-limit, more comparable to that of [[31edo]]. Therefore, higher multiples of 13edt remain excellent 3.5.7 subgroup tunings as well, and can be used to introduce higher harmonics – [[39edt]] is especially notable in this regard, with a good representation of both the [[11/1|11th]] and [[13/1|13th]] harmonics.

The rank-2 temperament generated by 7/3 that is satisfied in 13edt's 3.5.7 subgroup representation is [[BPS]], which tempers out the comma [[245/243]] and thereby equates the interval [[5/3]] to two generators down ([[81/49]] considering tritave-reduction), therefore flattening 7/3 by a fraction of this comma. It is also the {{nowrap| 4 & 9 }} temperament in the 3.5.7 subgroup, and for these reasons serves a function very analogous to that of [[meantone]] in the 5-limit.

If we instead take [[9/5]], or more simply [[5/3]], as a generator, the temperament supported by 13edt is [[arcturus]], which equates 7/3, two tritaves up (i.e. [[21/1]]) to six steps of 5/3. Naïvely, 5/3 as generator would be the most natural application of the [[Pythagorean tuning|Pythagorean]] principle of using the next higher prime harmonic (5) as a generator against the tritave. However, a larger mos scale is needed to get full use out of the 7th harmonic, and due to the proximity of 5/3 to half the tritave, most simple mos scales of arcturus are quite hard. It is advisable to use ({{mos scalesig|2L 9s<3/1>|link=1}}) or ({{mos scalesig|2L 11s<3/1>|link=1}}) scales – and therefore, higher edt tunings such as [[28edt]] or [[43edt]].

Moving through the scales of 13edt, we find the temperament [[sirius]], defined so that two generator steps represent [[7/5]] and three represent [[5/3]]; therefore the comma [[3125/3087]] is tempered out and the generator represents [[25/21]]. The smallest complete proper mos of sirius is the hard {{mos scalesig|6L 1s<3/1>|link=1}}, though some say that it is disadvantageous to have a non-octave scale with a single step of distinct size from the others, because it creates a strong sense of a second equal division of a ''y'' (in this case, [[25/9]]) less than 3/1 and therefore competes with the relatively fragile tritave equivalence. If this "problem" has not been named yet, "cross-equivalence artifacting" would be a perfect name for it. Otherwise, the 13-note mos of Sirius, {{mos scalesig|6L 7s<3/1>|link=1}}, is usable although this is very soft and close to 13edt in good tunings.

At higher accuracies, the rank-2 temperament tempering out [[16875/16807]], called [[canopus]], begins to predominate. This has a mapping {{mapping| 1 3 3 | 0 -5 -4 }} and a pure-tritaves TE generator a slightly flat 7/5 at 581.512{{c}}; the temperament's property is that four 7/5's reach [[27/7]]. This can be extended, notably, to include the fractional harmonic [[11/4]], which is quite close to three 7/5's. This temperament has mos of size 3, 4, 7, 10, 13, 23, 36, etc., with the 36-note mos being particularly even.

The final interval which 13edt can reasonably use to generate a rank-2 temperament is its false 3/2 of 5 degrees. By a weird coincidence, it will generate the {{mos scalesig|5L 3s<3/1>|link=1}} unfair [[father]] octatonic scale just as if it were an interval of an edo, except that the scale will not always contain a false 4/3 as it must in an edo. This means, most importantly, that 16/15 cannot be assumed to be a "comma" tempered out by this false father temperament when it is taken as a temperament of full just intonation. By a second, and totally separate, weird coincidence, the well-known Bohlen–Pierce temperament is its index-2 subtemperament.

Due to its 9/7 generator, the temperament which is to BP what neutral temperaments are to syntonic temperaments does not become intelligibly a division of the tritave until extended to 17 tones whereas edos supporting various neutral temperaments have an "ordinary" heptatonic scale which is intelligibly a division of the octave. Additionally, 7 and 9 being consecutive odd numbers means that trying to force this temperament into a no-2's subgroup induces very poor "approximations" of less intelligible higher harmonics. To avoid this, this temperament should be assumed to be a temperment of the 8.3.5.7 subgroup tempering out 245/243 and [[64/63]], the familiar comma from edos supporting the [[superpyth]]agorean or [[parapyth]]agorean diatonic scale.

Among the edts tempering out 245/243, 13edt stands out. An apt analogy can be drawn with edos supporting meantone: 4edt and 9edt are to BP what 5edo and 7edo are to meantone. However, in contrast to meantone, the simplest edt supporting the BP enneatonic scale – 13edt, the traditional tempered BP scale – is the most accurate and lowest in tuning error until 56edt. However, there are many edts supporting BP temperament which also support extensions to higher-limit temperaments; in particular 2, 3, and 4 times 13 in the form of {{EDTs| 26, 39, and 52 as well as 56EDT.}} For tempering out 16875/16807, 13edt again stands out, though much better accuracy can be found in more complex divisions such as 114edt or 127edt. All of this explains the focus on 13edt to the exclusion of other edts among practitioners of the art of non-octave composition, but it must be noted that the analysis is only valid if consideration is confined to 3.5.7, which is exactly analogous to confining it to the 5-limit with edos. There's a whole other world out there which has not been much explored.

For example, {{u|CompactStar}} suggested the alternative fundamental chord [[11:13:15]] to avoid the highly-dissonant [[7/5]] tritone present in the simpler 3:5:7 chord, with the best temperament for this being [[electra]]. 15edt very well approximates the 5th and 13th harmonics, and 12edt, the 13th and 17th. 39edt makes for a fine 3.5.7.11.13 system, tempering out 245/243, 275/273, 847/845 and 1331/1343, and so supporting among other things the {{mapping| 13 19 23 0 2 | 0 0 0 1 1 }} temperament supported by the whole suite of 13''n''-edts: 13, 26, 39, 52, 65, 78, etc.

[[Category:Just intonation subgroups|#]]

[[Category:Rank-3 temperaments|#]]

[[Category:7-limit|#]]

@@ Line 1: / Line 1: @@
-#redirect [[Just intonation subgroup #7-limit subgroups]]
+The '''3.5.7 subgroup''' is a [[non-octave]] [[just intonation subgroup]] consisting of [[rational interval]]s where 3, 5, and 7 are the only allowable [[prime factor]]s, so that every such interval may be written as a ratio of integers which are products of 3, 5, and 7. This is an infinite set. Some examples of intervals in this subgroup are [[5/3]], [[7/3]], [[7/5]], [[25/21]], [[35/27]], and so on.
-[[Category:Subgroup]]
+The 3.5.7 subgroup is a [[retraction]] of the [[7-limit]], obtained by removing prime 2. Its simplest [[expansion]] is the [[3.5.7.11 subgroup]], which adds prime [[11/1|11]].
-[[Category:Rank-3 temperaments]]
-[[Category:7-limit]]
+The approaches to composition in this subgroup can be categorized as follows:
+# By not having an [[interval of equivalence]], by composing in the 7-limit but without octavation;
+# By treating the [[3/1|3rd harmonic]] as the interval of equivalence.
+Since octavation is usually trivial to implement, this article focuses on the second approach.
+== Introduction to tritave equivalence ==
+See [[EDT #Introduction to tritave equivalence]].
+== Chords and harmony ==
+If factors of 2 are eliminated, the search for consonant intervals begins with the odd harmonic series, 1:3:5:7:9:…. We can take the second tritave of the series, 3:5:7:9, and find within it the two [[isoharmonic]] triads [[3:5:7]] and [[5:7:9]]; the analogy here is with the third octave of the full harmonic series, 4:5:6:7:8, and the isoharmonic triad [[4:5:6]], the foundation of triadic harmony in [[5-limit]] theory. Hence, 3:5:7 or 5:7:9 can be viewed as the fundamental consonant triad of no-2's music, and if we then apply the 5-limit analogy one more time, these triads are bounded by the intervals [[7/3]] or [[9/5]] respectively, either of them filling the role of the "perfect fifth" in diatonicism.
+The standard [[Bohlen–Pierce]] theory takes 3:5:7 to be the fundamental triad, and therefore naturally goes together with scales generated by 7/3, or equivalently 9/7 (the latter being convention), against the tritave. 7/3 generates pentatonic ({{mos scalesig|4L 1s<3/1>|link=1}}) and enneatonic ({{mos scalesig|4L 5s<3/1>|link=1}}) [[mos]] scales, and therefore the enneatonic, known as the ''Lambda'' scale, can be seen as the analog of the [[5L 2s|diatonic]] scale.
+''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.
+== Temperaments ==
+=== Equal temperaments ===
+A list of edts with progressively better tunings for the 3.5.7 subgroup (decreasing TE error, bold ones do particularly well in this subgroup): [[4edt|'''b4''']], [[6edt|b6]], [[7edt|b7]], [[9edt|b9]], [[13edt|'''b13''']], [[56edt|b56]], [[69edt|b69]], [[75edt|b75]], [[88edt|b88]], [[101edt|b101]], [[114edt|b114]], [[144edt|b144]], [[157edt|b157]], [[245edt|b245]], [[258edt|'''b258''']], [[271edt|'''b271''']], ….
+=== Rank-2 temperaments ===
+{{Todo|cleanup|improve readability|inline=1|text=Rewrite for clarity}}
+As generators of the Lambda scale run from [[9edt|7\9edt]] to [[4edt|3\4edt]], [[13edt]] is the smallest equal tuning [[support]]ing it, and can be seen as an equivalent of [[12edo]]. However, 13edt's accuracy in the 3.5.7 subgroup is much better than 12edo's in the 5-limit, more comparable to that of [[31edo]]. Therefore, higher multiples of 13edt remain excellent 3.5.7 subgroup tunings as well, and can be used to introduce higher harmonics – [[39edt]] is especially notable in this regard, with a good representation of both the [[11/1|11th]] and [[13/1|13th]] harmonics.
+The rank-2 temperament generated by 7/3 that is satisfied in 13edt's 3.5.7 subgroup representation is [[BPS]], which tempers out the comma [[245/243]] and thereby equates the interval [[5/3]] to two generators down ([[81/49]] considering tritave-reduction), therefore flattening 7/3 by a fraction of this comma. It is also the {{nowrap| 4 & 9 }} temperament in the 3.5.7 subgroup, and for these reasons serves a function very analogous to that of [[meantone]] in the 5-limit.
+If we instead take [[9/5]], or more simply [[5/3]], as a generator, the temperament supported by 13edt is [[arcturus]], which equates 7/3, two tritaves up (i.e. [[21/1]]) to six steps of 5/3. Naïvely, 5/3 as generator would be the most natural application of the [[Pythagorean tuning|Pythagorean]] principle of using the next higher prime harmonic (5) as a generator against the tritave. However, a larger mos scale is needed to get full use out of the 7th harmonic, and due to the proximity of 5/3 to half the tritave, most simple mos scales of arcturus are quite hard. It is advisable to use ({{mos scalesig|2L 9s<3/1>|link=1}}) or ({{mos scalesig|2L 11s<3/1>|link=1}}) scales – and therefore, higher edt tunings such as [[28edt]] or [[43edt]].
+Moving through the scales of 13edt, we find the temperament [[sirius]], defined so that two generator steps represent [[7/5]] and three represent [[5/3]]; therefore the comma [[3125/3087]] is tempered out and the generator represents [[25/21]]. The smallest complete proper mos of sirius is the hard {{mos scalesig|6L 1s<3/1>|link=1}}, though some say that it is disadvantageous to have a non-octave scale with a single step of distinct size from the others, because it creates a strong sense of a second equal division of a ''y'' (in this case, [[25/9]]) less than 3/1 and therefore competes with the relatively fragile tritave equivalence. If this "problem" has not been named yet, "cross-equivalence artifacting" would be a perfect name for it. Otherwise, the 13-note mos of Sirius, {{mos scalesig|6L 7s<3/1>|link=1}}, is usable although this is very soft and close to 13edt in good tunings.
+At higher accuracies, the rank-2 temperament tempering out [[16875/16807]], called [[canopus]], begins to predominate. This has a mapping {{mapping| 1 3 3 | 0 -5 -4 }} and a pure-tritaves TE generator a slightly flat 7/5 at 581.512{{c}}; the temperament's property is that four 7/5's reach [[27/7]]. This can be extended, notably, to include the fractional harmonic [[11/4]], which is quite close to three 7/5's. This temperament has mos of size 3, 4, 7, 10, 13, 23, 36, etc., with the 36-note mos being particularly even.
+The final interval which 13edt can reasonably use to generate a rank-2 temperament is its false 3/2 of 5 degrees. By a weird coincidence, it will generate the {{mos scalesig|5L 3s<3/1>|link=1}} unfair [[father]] octatonic scale just as if it were an interval of an edo, except that the scale will not always contain a false 4/3 as it must in an edo. This means, most importantly, that 16/15 cannot be assumed to be a "comma" tempered out by this false father temperament when it is taken as a temperament of full just intonation. By a second, and totally separate, weird coincidence, the well-known Bohlen–Pierce temperament is its index-2 subtemperament.
+Due to its 9/7 generator, the temperament which is to BP what neutral temperaments are to syntonic temperaments does not become intelligibly a division of the tritave until extended to 17 tones whereas edos supporting various neutral temperaments have an "ordinary" heptatonic scale which is intelligibly a division of the octave. Additionally, 7 and 9 being consecutive odd numbers means that trying to force this temperament into a no-2's subgroup induces very poor "approximations" of less intelligible higher harmonics. To avoid this, this temperament should be assumed to be a temperment of the 8.3.5.7 subgroup tempering out 245/243 and [[64/63]], the familiar comma from edos supporting the [[superpyth]]agorean or [[parapyth]]agorean diatonic scale.
+Among the edts tempering out 245/243, 13edt stands out. An apt analogy can be drawn with edos supporting meantone: 4edt and 9edt are to BP what 5edo and 7edo are to meantone. However, in contrast to meantone, the simplest edt supporting the BP enneatonic scale – 13edt, the traditional tempered BP scale – is the most accurate and lowest in tuning error until 56edt. However, there are many edts supporting BP temperament which also support extensions to higher-limit temperaments; in particular 2, 3, and 4 times 13 in the form of {{EDTs| 26, 39, and 52 as well as 56EDT.}} For tempering out 16875/16807, 13edt again stands out, though much better accuracy can be found in more complex divisions such as 114edt or 127edt. All of this explains the focus on 13edt to the exclusion of other edts among practitioners of the art of non-octave composition, but it must be noted that the analysis is only valid if consideration is confined to 3.5.7, which is exactly analogous to confining it to the 5-limit with edos. There's a whole other world out there which has not been much explored.
+For example, {{u|CompactStar}} suggested the alternative fundamental chord [[11:13:15]] to avoid the highly-dissonant [[7/5]] tritone present in the simpler 3:5:7 chord, with the best temperament for this being [[electra]]. 15edt very well approximates the 5th and 13th harmonics, and 12edt, the 13th and 17th. 39edt makes for a fine 3.5.7.11.13 system, tempering out 245/243, 275/273, 847/845 and 1331/1343, and so supporting among other things the {{mapping| 13 19 23 0 2 | 0 0 0 1 1 }} temperament supported by the whole suite of 13''n''-edts: 13, 26, 39, 52, 65, 78, etc.
+[[Category:Just intonation subgroups|#]]
+[[Category:Rank-3 temperaments|#]]
+[[Category:7-limit|#]]