EDT: Difference between revisions

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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 ({{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 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.

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. Naively, 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 ({{sl|2L 9s}}) or ({{sl|2L 11s}}) scales—and therefore, higher EDTs such as [[28edt]] or [[43edt]].

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. Naively, 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 EDTs 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 [[6L 1s (3/1-equivalent)|6L 1s]], 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, [[6L 7s (3/1-equivalent)|6L 7s]], is usable although this is very soft and close to 13edt in good tunings.

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At higher accuracies, the rank two 3.5.7 temperament tempering out 16875/16807 called [[Canopus]] begins to predominate. This has a mapping [{{val|1 3 3}}, {{val|0 -5 -4}}] and a pure-tritaves TE generator a slightly flat 7/5 at 581.512 cents; the temperament's property is that four 7/5s reach [[27/7]]. This can be extended, notably, to include the fractional harmonic [[11/4]], which is quite close to three 7/5s. This 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 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.

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 {{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-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.

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It is useful to consider EDTs that both ''closely'' and ''poorly'' approximate EDOs. The former are usable as stretches and compressions of EDOs with strong flat or sharp tendencies, while the latter allow for no-twos harmony without the distraction of octaves appearing. It is possible to define "dual-octave" EDTs similar to dual-fifth EDOs, as those whose closest approximation of 2 is more than 1/3 of a step off (so in other words, they have a better closest approximation of the 4th harmonic than the 2nd).

Otherwise, one can speak of EDTs that correspond to a diatonic [[val]] (i.e. the EDT's size is some EDO added to an approximation of [[3/2]] in that EDO that is a [[5L 2s|diatonic]] generator), which is equivalent to the EDT's approximation of [[2/1]] generating the {{sl|8L 3s}} scale against the tritave, therefore being between 5\8edt and 7\11edt.

Otherwise, one can speak of EDTs that correspond to a diatonic [[val]] (i.e. the EDT's size is some EDO added to an approximation of [[3/2]] in that EDO that is a [[5L 2s|diatonic]] generator), which is equivalent to the EDT's approximation of [[2/1]] generating the {{mos scalesig|8L 3s<3/1>|link=1}} scale against the tritave, therefore being between 5\8edt and 7\11edt.

EDTs with this property include {{EDTs| 19, 27, 30, 35, 38, 41, 43, 46, 49, 51, 52, 54, 57, 59, 60, 62, 63, 65, 67, 68, 70, 71, 73 to 76, 78, 79, 81 to 87, and all greater than 88.}}

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 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 ({{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 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.
-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. Naively, 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 ({{sl|2L 9s}}) or ({{sl|2L 11s}}) scales—and therefore, higher EDTs such as [[28edt]] or [[43edt]].
+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. Naively, 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 EDTs 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 [[6L 1s (3/1-equivalent)|6L 1s]], 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, [[6L 7s (3/1-equivalent)|6L 7s]], is usable although this is very soft and close to 13edt in good tunings.
@@ Line 28: / Line 28: @@
 At higher accuracies, the rank two 3.5.7 temperament tempering out 16875/16807 called [[Canopus]] begins to predominate. This has a mapping [{{val|1 3 3}}, {{val|0 -5 -4}}] and a pure-tritaves TE generator a slightly flat 7/5 at 581.512 cents; the temperament's property is that four 7/5s reach [[27/7]]. This can be extended, notably, to include the fractional harmonic [[11/4]], which is quite close to three 7/5s. This 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 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.
+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 {{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-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.
@@ Line 136: / Line 136: @@
 It is useful to consider EDTs that both ''closely'' and ''poorly'' approximate EDOs. The former are usable as stretches and compressions of EDOs with strong flat or sharp tendencies, while the latter allow for no-twos harmony without the distraction of octaves appearing. It is possible to define "dual-octave" EDTs similar to dual-fifth EDOs, as those whose closest approximation of 2 is more than 1/3 of a step off (so in other words, they have a better closest approximation of the 4th harmonic than the 2nd).
-Otherwise, one can speak of EDTs that correspond to a diatonic [[val]] (i.e. the EDT's size is some EDO added to an approximation of [[3/2]] in that EDO that is a [[5L 2s|diatonic]] generator), which is equivalent to the EDT's approximation of [[2/1]] generating the {{sl|8L 3s}} scale against the tritave, therefore being between 5\8edt and 7\11edt.
+Otherwise, one can speak of EDTs that correspond to a diatonic [[val]] (i.e. the EDT's size is some EDO added to an approximation of [[3/2]] in that EDO that is a [[5L 2s|diatonic]] generator), which is equivalent to the EDT's approximation of [[2/1]] generating the {{mos scalesig|8L 3s<3/1>|link=1}} scale against the tritave, therefore being between 5\8edt and 7\11edt.
 EDTs with this property include {{EDTs| 19, 27, 30, 35, 38, 41, 43, 46, 49, 51, 52, 54, 57, 59, 60, 62, 63, 65, 67, 68, 70, 71, 73 to 76, 78, 79, 81 to 87, and all greater than 88.}}