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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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 38: / Line 43: @@
 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|#]]