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If factors of two are eliminated, the simplest possible triad is (1):(3):5:7:(9), with 1, 3 and 9 in parentheses as they're all tritave-equivalent to 1. Hence, 3:5:7 can be viewed as the fundamental consonant triad of no-twos music. The linear temperament that best approximates these chords in the moderate complexity range is the Bohlen-Pierce linear temperament eliminating 245/243, which has a no-twos mapping of [<1 1 2|, <0 2 -1|] and a pure-tritaves TE generator which is a sharp 9/7 of 440.488 cents. It possesses [[MOSScales|MOS]] of the forms 4L1s (pentatonic) and 4L5s (nonatonic), and larger MOS of size 13, 17, 30, 43, 56, 69 and 82. This temperament serves a function analogous to meantone in the 5-limit.
If factors of two are eliminated, the simplest possible triad is (1):(3):5:7:(9), with 1, 3 and 9 in parentheses as they're all tritave-equivalent to 1. Hence, 3:5:7 can be viewed as the fundamental consonant triad of no-twos music. The linear temperament that best approximates these chords in the moderate complexity range is the Bohlen-Pierce linear temperament eliminating 245/243, which has a no-twos mapping of [<1 1 2|, <0 2 -1|] and a pure-tritaves TE generator which is a sharp 9/7 of 440.488 cents. It possesses [[MOSScales|MOS]] of the forms 4L1s (pentatonic) and 4L5s (nonatonic), and larger MOS of size 13, 17, 30, 43, 56, 69 and 82. This temperament serves a function analogous to meantone in the 5-limit.


At higher complexities, the rank two 3.5.7 temperament tempering out 16875/16807 called [[Canopus|Canopus]] begins to predominate. This has a mapping [<1 3 3|, <0 -5 -4|] and a pure-tritaves TE generator a slightly flat 7/5 at 581.512 cents. This has MOS of size 3, 4, 7, 10, 13, 23, 36, etc, with the 36 note MOS being particularly even.
At higher complexities, the rank two 3.5.7 temperament tempering out 16875/16807 called [[Canopus]] begins to predominate. This has a mapping [<1 3 3|, <0 -5 -4|] and a pure-tritaves TE generator a slightly flat 7/5 at 581.512 cents. This has MOS of size 3, 4, 7, 10, 13, 23, 36, etc, with the 36 note MOS being particularly even.


The other no twos rank two temperament which 13edt "supports" is [[Arcturus|Arcturus]], which takes an ~5:3 as a generator. I speak advisedly of 13edt supporting this temperament because the lowest-error proper MOS of it is a 2L 11s triskaidecatonic scale. However, if you do not mind having a smeary 5, you will need only a 2L 7s (nonatonic) scale to make an understandable rendition of it.
The other no twos rank two temperament which 13edt "supports" is [[Arcturus]], which takes an ~5:3 as a generator. I speak advisedly of 13edt supporting this temperament because the lowest-error proper MOS of it is a 2L 11s triskaidecatonic scale. However, if you do not mind having a smeary 5, you will need only a 2L 7s (nonatonic) scale to make an understandable rendition of it.


The named but not necessarily no twos rank two temperament which 13edt "supports" is [[Sirius|Sirius]], which takes a generator between ~7:6 and ~6:5. Like Arcturus, I speak advisedly of 13edt supporting it because the most proper small MOS of it is triskaidecatonic. Unlike Arcturus, there is a smaller MOS of it than this which is technically proper. However, this MOS is the Grumpy heptatonic scale the use of which is made problematic by the uniqueness of the step of the second size. It is problematic to have the step of the second size be unique in a subscale of an edx because it creates a strong sense of a second equal division of a y strictly less than x, and this sense of two different equal divisions trying to happen in the same scale causes ordinary concepts of equivalence to break down in spectacular ways. If this "problem" has not been named yet, "cross-equivalence artifacting" would be a perfect name for it.
The named but not necessarily no twos rank two temperament which 13edt "supports" is [[Sirius]], which takes a generator between ~7:6 and ~6:5. Like Arcturus, I speak advisedly of 13edt supporting it because the most proper small MOS of it is triskaidecatonic. Unlike Arcturus, there is a smaller MOS of it than this which is technically proper. However, this MOS is the Grumpy heptatonic scale the use of which is made problematic by the uniqueness of the step of the second size. It is problematic to have the step of the second size be unique in a subscale of an edx because it creates a strong sense of a second equal division of a y strictly less than x, and this sense of two different equal divisions trying to happen in the same scale causes ordinary concepts of equivalence to break down in spectacular ways. If this "problem" has not been named yet, "cross-equivalence artifacting" would be a perfect name for it.


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 [[5L_3s|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 [[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 Superpythagorean or Parapythagorean diatonic scale.
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 Superpythagorean or Parapythagorean diatonic scale.
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==0...99==
==0...99==
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==100...199==
==100...199==
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''(many pages do not exist yet)''


{| class="wikitable"
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Also may be found convenient: http://www.nonoctave.com/tuning/twelfth.html
Also may be found convenient: http://www.nonoctave.com/tuning/twelfth.html


=EDO-EDT correspondences=
=EDT-EDO correspondences=


{| class="wikitable"
{| class="wikitable"
|-
|-
! | EDT
! | EDO
! | EDO
! | EDT
! | Comments
! | Comments
|-
|-
| | [[5edo|5edo]]
| | [[8edt]]
| | [[8edt|8edt]]
| | [[5edo]]
| | 8edt is equivalent to 5edo with ~11 cent octave compression.
| | 8edt is equivalent to 5edo with ~11 cent octave compression.
Equivalently, 5edo is 8edt with ~18 cent stretched tritaves.
Equivalently, 5edo is 8edt with ~18 cent stretched tritaves.
[[Patent_val|Patent vals]] match through the 13 limit.
[[Patent_val|Patent vals]] match through the 13 limit.
|-
|-
| | [[9edt]]
| |  
| |  
| | [[9edt|9edt]]
| |  
| |  
|-
|-
| | [[6edo|6edo]]
| | [[10edt]]
| |  
| |  
| |  
| |  
|-
|-
| |
| | [[11edt]]
| | [[10edt|10edt]]
| | [[7edo]]
| |
|-
| | [[7edo|7edo]]
| | [[11edt|11edt]]
| | 11edt is equivalent to 7edo with ~10 cent stretched octaves.
| | 11edt is equivalent to 7edo with ~10 cent stretched octaves.
Patent vals differ in the 7 limit, but neither can really be said
Patent vals differ in the 7 limit, but neither can really be said
to represent the 7th harmonic with a straight face.
to represent the 7th harmonic with a straight face.
|-
|-
| | [[12edt]]
| |  
| |  
| | [[12edt|12edt]]
| |
| |
|-
| | [[8edo|8edo]]
| |
| |
|-
|-
| | [[13edt]]
| |  
| |  
| | [[13edt|13edt]]
| | The equal-tempered BP scale cannot be considered equivalent to [[8edo]].
| | The equal-tempered BP scale cannot be considered equivalent to 8edo.
|-
|-
| | [[9edo|9edo]]
| | [[14edt]]
| | [[14edt|14edt]]
| | [[9edo]]
| | There is a lot of mismatch between the pure-octave and pure-tritave tunings,
| | There is a lot of mismatch between the pure-octave and pure-tritave tunings,
but the patent vals match through the 13 limit. Great for stretched-octave pelog!
but the patent vals match through the 13 limit. Great for stretched-octave pelog!
|-
|-
| | [[15edt]]
| |  
| |  
| | [[15edt|15edt]]
| |  
| |  
|-
|-
| | [[10edo|10edo]]
| | [[16edt]]
| | [[16edt|16edt]]
| | [[10edo]]
| | Similar situation to 5edo~8edt. Patent vals match through the 17 limit.
| | Similar situation to 5edo~8edt. Patent vals match through the 17 limit.
|-
|-
| |
| | [[17edt]]
| | [[17edt|17edt]]
| |
|-
| | [[11edo|11edo]]
| |  
| |  
| |  
| |  
|-
|-
| | [[18edt]]
| |  
| |  
| | [[18edt|18edt]]
| |  
| |  
|-
|-
| | [[12edo|12edo]]
| | [[19edt]]
| | [[19edt|19edt]]
| | [[12edo]]
| | 19edt is 12edo with ~1.2 cent octave stretch. Patent vals match
| | 19edt is 12edo with ~1.2 cent octave stretch, and consistent through the 10-integer-limit.
only through the 7 limit, but neither can be said to include 11 at all.
|-
|-
| | [[20edt]]
| |  
| |  
| | [[20edt|20edt]]
| |  
| |  
|-
|-
| | [[13edo|13edo]]
| | [[21edt]]
| |  
| |  
| |  
| |  
|-
|-
| | [[22edt]]
| |  
| |  
| | [[21edt|21edt]]
| |  
| |  
|-
|-
| | [[23edt]]
| |  
| |  
| | [[22edt|22edt]]
| |  
| |  
|-
|-
| | [[14edo|14edo]]
| | [[24edt]]
| |
| | [[15edo]]
| |
|-
| |
| | [[23edt|23edt]]
| |
|-
| | [[15edo|15edo]]
| | [[24edt|24edt]]
| | This is only a rough correspondence, as the (5n)edo ~ (8n)edt sequence
| | This is only a rough correspondence, as the (5n)edo ~ (8n)edt sequence
begins to break down. The patent vals match only through the 5 limit.
begins to break down. 24edt is only consistent through the 6-integer-limit, with discrepancy for the 7th harmonic.
|-
|-
| | [[16edo|16edo]]
| | [[25edt]]
| | [[25edt|25edt]]
| | [[16edo]]
| | Also only a rough correspondence; 25edt corresponds to 16edo
| | Also only a rough correspondence; 25edt corresponds to 16edo
with ~17 cent octave stretch. Patent vals match through the 5 limit.
with ~17 cent octave stretch, and consistent through the 6-integer-limit.
|-
|-
| | [[26edt]]
| |  
| |  
| | [[26edt|26edt]]
| | Double BP
| | Double BP
|-
|-
| | [[17edo|17edo]]
| | [[27edt]]
| | [[27edt|27edt]]
| | [[17edo]]
| | 27edt is 17edo with ~2.5 cent compressed octaves. With the exception of
| | 27edt is 17edo with ~2.5 cent compressed octaves. With the exception of
5 (which neither represents well), patent vals match through the 13 limit.
5 (which neither represents well), patent vals match through the 16-integer-limit.
|-
|-
| | [[28edt]]
| |  
| |  
| | [[28edt|28edt]]
| |  
| |  
|-
|-
| | [[18edo|18edo]]
| | [[29edt]]
| |  
| |  
| |  
| |  
|-
|-
| |
| | [[30edt]]
| | [[29edt|29edt]]
| | [[19edo]]
| |
| | 30edt is 19edo with ~5 cent stretched octaves, and consistent through the 10-integer-limit.
|-
| | [[19edo|19edo]]
| | [[30edt|30edt]]
| | 30edt is 19edo with ~5 cent stretched octaves.
Patent vals match through the 7 limit.
|-
|-
| | [[31edt]]
| |  
| |  
| | [[31edt|31edt]]
| |  
| |  
|-
|-
| | [[20edo|20edo]]
| | [[32edt]]
| |  
| |  
| |  
| |  
|-
|-
| | [[33edt]]
| |  
| |  
| | [[32edt|32edt]]
| |  
| |  
|-
|-
| | [[34edt]]
| |  
| |  
| | [[33edt|33edt]]
| |  
| |  
|-
|-
| | [[21edo|21edo]]
| | [[35edt]]
| |  
| | [[22edo]]
| |  
| | 35edt is 22edo with ~4 cent compressed octaves, and consistent through the 12-integer-limit.
|-
|-
| | [[36edt]]
| |  
| |  
| | [[34edt|34edt]]
| |  
| |  
|-
|-
| | [[22edo|22edo]]
| | [[37edt]]
| | [[35edt|35edt]]
| | 35edt is 22edo with ~4 cent compressed octaves.
Patent vals match through the 11 limit.
|-
| |  
| |  
| | [[36edt|36edt]]
| |  
| |  
|-
|-
| | [[23edo|23edo]]
| | [[38edt]]
| |  
| | [[24edo]]
| |  
| | Same ~1.2 cent octave stretch as 12edo~19edt. 38edo is consistent through the 6-integer-limit.
Patent vals match through the 19-limit.
|-
|-
| | [[39edt]]
| |  
| |  
| | [[37edt|37edt]]
| |
|-
| | [[24edo|24edo]]
| | [[38edt|38edt]]
| | Same ~1.2 cent octave stretch as 12edo~19edt.
Patent vals match through the 19 limit.
|-
| |
| | [[39edt|39edt]]
| | [[Triple_BP|Triple BP]]
| | [[Triple_BP|Triple BP]]
|-
|-
| | [[25edo|25edo]]
| | [[40edt]]
| |  
| |  
| |  
| |  
|-
|-
| |
| | [[41edt]]
| | [[40edt|40edt]]
| | [[26edo]]
| |
| | 41edt is 26edo with ~6 cent stretched octaves, but only consistent through the 10-integer-limit, with discrepancy for the 11th harmonic.
|-
| | [[26edo|26edo]]
| | [[41edt|41edt]]
| | 41edt is 26edo with ~6 cent stretched octaves.
Patent vals match through the 7 limit.
|-
|-
| | [[42edt]]
| |  
| |  
| | [[42edt|42edt]]
| |  
| |  
|-
|-
| | [[27edo|27edo]]
| | [[43edt]]
| | [[43edt|43edt]]
| | [[27edo]]
| | 43edt is 27edo with ~6 cent compressed octaves.
| | 43edt is 27edo with ~6 cent compressed octaves, and consistent through the 10-integer-limit.
Patent vals match through the 7 limit.
|-
| |
| | [[44edt|44edt]]
| |
|-
|-
| | [[28edo|28edo]]
| | [[44edt]]
| |  
| |  
| |  
| |  
|-
|-
| | [[45edt]]
| |  
| |  
| | [[45edt|45edt]]
| |  
| |  
|-
|-
| | [[29edo|29edo]]
| | [[46edt]]
| | [[46edt|46edt]]
| | [[29edo]]
| | 46edt is 29edo with ~0.9 cent compressed octaves.
| | 46edt is 29edo with ~0.9 cent compressed octaves, and consistent through the 16-integer-limit.
Patent vals match through the 89 limit. (Really! I checked!)
Patent vals match through the 89-limit.
|}
|}


=Multiples of 13EDT which approximate EDO=
=Multiples of 13EDT which approximate EDO=
Also, on the topic of multiples of 13edt, 26 (double) and 39 (triple) offer very good harmonic approximations, the former of the 8th, 13th and 17th partials, and the latter of the 11th and 13th. However, quadruple and quintuple, ie. 52 and 65edt, also exist offering good approximations of the octave. 52edt is very nearly [[33edo|33edo]], and 65edt is practically identical to [[41edo|41edo]].
Also, on the topic of multiples of 13edt, 26 (double) and 39 (triple) offer very good harmonic approximations, the former of the 8th, 13th and 17th partials, and the latter of the 11th and 13th. However, quadruple and quintuple, ie. 52 and 65edt, also exist offering good approximations of the octave. 52edt is very nearly [[33edo]], and 65edt is practically identical to [[41edo]].


=See also=
=See also=
Retrieved from "https://en.xen.wiki/w/EDT"