EDT: Difference between revisions
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EDT: Equal Division of the Tritave (3/1, perfect twelfth). Also sometimes written as ed3. | EDT: Equal Division of the Tritave (3/1, perfect twelfth). Also sometimes written as ed3. | ||
=Introduction= | == Introduction == | ||
Western music generally revolves around the principle of '''octave equivalence''': notes an octave apart are often perceived in western music as being the same ''chroma'' but differing in pitch height. As the octave corresponds to a 2/1 frequency ratio, it has been proposed that the next-simplest after the octave, the 3/1, can also be used to evoke a sense of chroma equivalence. This interval corresponds to a perfect twelfth in the diatonic scale, but when used to refer to an equivalence interval it is often called the "tritave". | Western music generally revolves around the principle of '''octave equivalence''': notes an octave apart are often perceived in western music as being the same ''chroma'' but differing in pitch height. As the octave corresponds to a 2/1 frequency ratio, it has been proposed that the next-simplest after the octave, the 3/1, can also be used to evoke a sense of chroma equivalence. This interval corresponds to a perfect twelfth in the diatonic scale, but when used to refer to an equivalence interval it is often called the "tritave". | ||
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The [[BP|Bohlen-Pierce (BP) scale]], most commonly consisting of 13 equal divisions of the tritave (although a justly-intoned version exists as well), seems to have been the first such arrangement to be seriously studied and made into music. The BP scale was independently discovered by Heinz Bohlen, John Pierce and Kees Van Prooijen. Bohlen found it while looking for triads with equal-difference tones, Prooijen uncovered it while searching for equally-tempered scales with accurate higher harmonics, and Pierce stumbled upon it trying to find consonant chords other than 4:5:6. Though they all started with different goals in mind, each of them amazingly ended up at the same destination. | The [[BP|Bohlen-Pierce (BP) scale]], most commonly consisting of 13 equal divisions of the tritave (although a justly-intoned version exists as well), seems to have been the first such arrangement to be seriously studied and made into music. The BP scale was independently discovered by Heinz Bohlen, John Pierce and Kees Van Prooijen. Bohlen found it while looking for triads with equal-difference tones, Prooijen uncovered it while searching for equally-tempered scales with accurate higher harmonics, and Pierce stumbled upon it trying to find consonant chords other than 4:5:6. Though they all started with different goals in mind, each of them amazingly ended up at the same destination. | ||
=Rank two temperaments= | == Rank two temperaments == | ||
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. | ||
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=Individual pages for EDTs= | == Individual pages for EDTs == | ||
==0...99== | === 0...99 === | ||
''(some pages do not exist yet)'' | ''(some pages do not exist yet)'' | ||
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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 | ||
=EDT-EDO correspondences= | == EDT-EDO correspondences == | ||
{| class="wikitable" | {| class="wikitable" | ||
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=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 through sextuple, ie. 52, 65 and 78EDT, also exist offering good approximations of the octave. 52EDT is very nearly [[33edo|33EDO]] and 78EDT is very nearly [[49edo|49EDO]], while 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 through sextuple, ie. 52, 65 and 78EDT, also exist offering good approximations of the octave. 52EDT is very nearly [[33edo|33EDO]] and 78EDT is very nearly [[49edo|49EDO]], while 65EDT is practically identical to [[41edo|41EDO]]. | ||
=See also= | == See also == | ||
* [[Minimal consistent EDTs]] | * [[Minimal consistent EDTs]] | ||
* [[Consistency levels of small EDTs]] | * [[Consistency levels of small EDTs]] | ||
* [[Relative Errors of Small EDTs]] | |||
* [[Tritave Reduced Harmonics]] | * [[Tritave Reduced Harmonics]] | ||
* Heinz Bohlen's work: http://www.huygens-fokker.org/bpsite/otherscales.html | * Heinz Bohlen's work: http://www.huygens-fokker.org/bpsite/otherscales.html | ||
[[Category:Edt]] | |||
[[Category:3/1]] | [[Category:3/1]] | ||
[[category:nonoctave]] | [[category:nonoctave]] | ||