EDT: Difference between revisions

Line 18:

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.

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 ({{sl|4L 1s}}) and enneatonic ({{sl|4L 5s}}) [[MOS]] scales, and therefore the enneatonic, known as the "Lambda" scale, can be seen as the analog of the diatonic scale. As generators of the Lambda scale run from [[9edt|7\9]] to [[4edt|3\4]], [[13edt]] is the smallest equal temperament supporting 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 [[11/9|11th]] and [[13/9|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 ({{sl|4L 1s}}) and enneatonic ({{sl|4L 5s}}) [[MOS]] scales, and therefore the enneatonic, known as the "Lambda" scale, can be seen as the analog of the diatonic scale. As generators of the Lambda scale run from [[9edt|7\9]] to [[4edt|3\4]], [[13edt]] is the smallest equal temperament supporting 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 [[11/9|11th]] and [[13/9|13th]] harmonics).

The linear temperament generated by 7/3 that is satisfied in 13edt's 3.5.7 subgroup representation is [[Bohlen–Pierce–Stearns]], 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 30:

The final interval which 13EDT can reasonably use to generate a rank two temperament is its false 3/2 of 5 degrees. By a weird coincidence, it will generate the {{sl|5L 3s}} 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 ~~the fact of~~ 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-twos subgroup induces very poor "approximations" of less intelligible higher harmonics. To avoid this, this temperament should be assumed to be a temperment of the 3.5.7.8 subgroup tempering out 245/243 and 64/63, the familiar comma from EDOs supporting the [[Superpyth]]agorean or [[Parapyth]]agorean diatonic scale.

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-twos subgroup induces very poor "approximations" of less intelligible higher harmonics. To avoid this, this temperament should be assumed to be a temperment of the 3.5.7.8 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 nonatonic 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 nonoctave composition, but it must be noted that the analysis is only valid if consideration is confined to the 7-limit, 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.

Line 36:

For example, [[User:CompactStar|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]] temperament. 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 [{{val|13 19 23 0 2}}, {{val|0 0 0 1 1}}] temperament supported by the whole suite of 13nEDTs: 13, 26, 39, 52, 65, 78 etc.

'''One should bear in mind that, ~~assuming~~ tritave equivalence, ~~when determining which harmonics are represented, the~~ ratios of 3 in the denominator are fungible instead of those of 2.''' For example ~~making~~ the fifth harmonic 5:3 a "major sixth" by conventional pitch class terminology.

'''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.

There are other uses, or conceptualizations, of tritave-based tunings. Purely intuitive use of these myriad, assuredly xenharmonic structures comes to mind (see "EDO" versus "equal temperament"). Another intent might be to find or define temperaments (such as Magic, Hanson, etc.), or to provide exact formulae for stretching/compressing what would musically be used as an "ordinary" octave of ~2:1. (And given the stable nature of octave-based systems, some aesthetic overlap even in the most tritave-equivalent of music, would be forseeable.) For instance, the Bernhard-Stopper (19edt) temperament, might for instance be found useful in tuning pianoforti, being equivalent to 12edo, except for a 1.2 cents sharp octave which is relevant to inharmonicity.

@@ Line 18: / Line 18: @@
 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.
-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 ({{sl|4L 1s}}) and enneatonic ({{sl|4L 5s}}) [[MOS]] scales, and therefore the enneatonic, known as the "Lambda" scale, can be seen as the analog of the diatonic scale. As generators of the Lambda scale run from [[9edt|7\9]] to [[4edt|3\4]], [[13edt]] is the smallest equal temperament supporting 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 [[11/9|11th]] and [[13/9|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 ({{sl|4L 1s}}) and enneatonic ({{sl|4L 5s}}) [[MOS]] scales, and therefore the enneatonic, known as the "Lambda" scale, can be seen as the analog of the diatonic scale. As generators of the Lambda scale run from [[9edt|7\9]] to [[4edt|3\4]], [[13edt]] is the smallest equal temperament supporting 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 [[11/9|11th]] and [[13/9|13th]] harmonics).
 The linear temperament generated by 7/3 that is satisfied in 13edt's 3.5.7 subgroup representation is [[Bohlen–Pierce–Stearns]], 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 &amp; 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 30: / Line 30: @@
 The final interval which 13EDT can reasonably use to generate a rank two temperament is its false 3/2 of 5 degrees. By a weird coincidence, it will generate the {{sl|5L 3s}} 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 the fact of 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-twos subgroup induces very poor "approximations" of less intelligible higher harmonics. To avoid this, this temperament should be assumed to be a temperment of the 3.5.7.8 subgroup tempering out 245/243 and 64/63, the familiar comma from EDOs supporting the [[Superpyth]]agorean or [[Parapyth]]agorean diatonic scale.
+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-twos subgroup induces very poor "approximations" of less intelligible higher harmonics. To avoid this, this temperament should be assumed to be a temperment of the 3.5.7.8 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 nonatonic 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 nonoctave composition, but it must be noted that the analysis is only valid if consideration is confined to the 7-limit, 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.
@@ Line 36: / Line 36: @@
 For example, [[User:CompactStar|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]] temperament. 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 [{{val|13 19 23 0 2}}, {{val|0 0 0 1 1}}] temperament supported by the whole suite of 13nEDTs: 13, 26, 39, 52, 65, 78 etc.
-'''One should bear in mind that, assuming tritave equivalence, when determining which harmonics are represented, the ratios of 3 in the denominator are fungible instead of those of 2.''' For example making the fifth harmonic 5:3 a "major sixth" by conventional pitch class terminology.
+'''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.
 There are other uses, or conceptualizations, of tritave-based tunings. Purely intuitive use of these myriad, assuredly xenharmonic structures comes to mind (see "EDO" versus "equal temperament"). Another intent might be to find or define temperaments (such as Magic, Hanson, etc.), or to provide exact formulae for stretching/compressing what would musically be used as an "ordinary" octave of ~2:1. (And given the stable nature of octave-based systems, some aesthetic overlap even in the most tritave-equivalent of music, would be forseeable.) For instance, the Bernhard-Stopper (19edt) temperament, might for instance be found useful in tuning pianoforti, being equivalent to 12edo, except for a 1.2 cents sharp octave which is relevant to inharmonicity.