3.5.7 subgroup: Difference between revisions

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== 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 ==

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

If factors of two 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-twos 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 "fifth" in diatonicism.

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 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. 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.

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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.

'''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 terminology.

[[Category:Just intonation subgroups|#]]

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

[[Category:7-limit|#]]

@@ Line 11: / Line 11: @@
 == 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 ==
@@ Line 19: / Line 26: @@
 {{Todo|cleanup|improve readability|inline=1|text=Rewrite for clarity}}
-If factors of two 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-twos 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 "fifth" in diatonicism.
+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 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. 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.
@@ Line 37: / Line 42: @@
 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.
-'''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 terminology.
 [[Category:Just intonation subgroups|#]]
 [[Category:Rank-3 temperaments|#]]
 [[Category:7-limit|#]]