20edo
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... 20-tone equal temperament, or 20edo, divides the octave into exactly 20 equal steps of 60 cent…
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20-tone equal temperament, or 20edo, divides the octave into exactly 20 equal steps of 60 cents each. It contains smaller edos 2, 4, 5, and 10 and is part of the 5n Family of equal divisions of the octave. 20 edo fairly approximates the harmonics 7 (from 5 edo), 11, 13 & 15 (from 10 edo), 19 & 27 (from 4 edo), 29 and 31; as well as the other harmonics more loosely (though to some people, still functionally) approximated. Thus, 20-EDO does a reasonably convincing approximation of harmonics 4:7:11:13:15.
Compositions
Underwater Spontaniety (lo-fi ambient) by Stephen Weigel
Pulling Weeds in the Dark Ages (xen-pop) by Stephen Weigel
God's Favorite Tuning (xen-pop) by Stephen Weigel
You Lied, and I'm not Mentioning your Name (xen-pop) by Stephen Weigel
20edo
edited
... 20-tone equal temperament, or 20edo, divides the octave into exactly 20 equal steps of 60 cent…
...
20-tone equal temperament, or 20edo, divides the octave into exactly 20 equal steps of 60 cents each. It contains smaller edos 2, 4, 5, and 10 and is part of the 5n Family of equal divisions of the octave. 20 edo fairly approximates the harmonics 7 (from 5 edo), 11, 13 & 15 (from 10 edo), 19 & 27 (from 4 edo), 29 and 31; as well as the other harmonics more loosely (though to some people, still functionally) approximated. Thus, 20-EDO does a reasonably convincing approximation of harmonics 4:7:11:13:15.
Compositions
God's Favorite Tuning (xen-pop) by Stephen Weigel
You Lied, and I'm not Mentioning your Name (xen-pop) by Stephen Weigel
Etude in 20-tone equal tuning play by Herman Miller
20ET — Prelude & Fugue a3 by Aaron Hunt
Fractional monzos
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Deutsch
A fractional monzo is like an ordinary monzo except that coefficients have been extended…
Deutsch
A fractional monzo is like an ordinary monzo except that coefficients have been extended to allow them to be rational numbers. If |e2 e3 ... ep> is a fractional monzo, then it represents 2^e2 3^e3 ... p^ep just as with an ordinary monzo. Hence, for instance, |1/13 -1/13 7/26> represents the interval 2^(1/13) 3^(-1/13) 5^(7/26). By taking the least common multiple of the denominators, intervals represented by a fractional monzo can always be written as an nth root of a positive rational number; for instance from our example, (312500/9)^(1/26). By taking a dot product with <cents(2) cents(3) ... cents(p)| the value in cents of a monzo or fractional monzo may be obtained. For instance, in the above example (1/13)*1200.0 - (1/13)*cents(3) + (7/26)*cents(5) = 696.1648 cents.
Vectors in interval space, where the coefficients are allowed to be real numbers, do not uniquely correspond to intervals, whereas monzos do. Fractional monzos do also; for each fractional monzo there is one and only one nth root of a positive rational number which corresponds to it.
Arabic, Turkish, Persian
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Other languages: Deutsch Deutsch
Along with the Indian tradition, the music of the Middle and N…
Other languages: DeutschDeutsch
Along with the Indian tradition, the music of the Middle and Near East (Arabic, Turkish, and Persian) is one of the important microtonal music traditions.
A central concept is "maqam" (pl. maqamat), which corresponds somewhat (but not exactly) to the Western "mode". An introduction to maqam theory can be found on http://www.maqamworld.com. The Arabic maqam and Turkish makam systems differ to some degree from the related Persian system of dastgah.
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Systems that just fulfill level 1 are still generally considered bad compromises. An additional quite basic requirement for a more accurate maqam system is the availability of two kinds of neutral seconds, a smaller one for the Bayati tetrachord and a larger one for the Rast tetrachord. Moreover, the major second whould preferably be a "major" wholetone, while the minor second should be a "small" semitone - as a consequence, the wholetone is to be divided into a smaller semitone (limma) and a larger one (apotome). I.e. minor seconds will be available in two varieties, too.
Important EDOs that meet these requirements are 53edo and 72edo. Both of these have found a certain dissemination in middle-eastern music.
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also proposes 34edo and34edo, 41edo and 46edo, within limits also 29edo, as acceptable compromises. These29edo, 34edo and 41edo have the
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is lower. 41edo and41edo, 34edo and 29edo can thus
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many. There areare, by the way, not just
Level 3
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and 72edo , although already containing much more pitches than many westerners can imagine, do not
A system that meets the tuning needs to a satisfactory degree is proposed in Ozan Yarman's dissertation (also summarized in the mentioned paper ): a 79-tone MOS subset of 159edo. A short description (quoting a post to the tuning list) is also here.
Tsaharuk, as proposed by Jacques Dudon, can be realized in 77edo, 94edo, 111edo, 128edo, 145edo, 171edo, 359edo.
Arabic, Turkish, Persian
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... Turkish
Tetrachords and Makams of Turkey - another theory site
Eric Ederer: Makam and Beyon…
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Turkish
Tetrachords and Makams of Turkey - another theory site
Eric Ederer: Makam and Beyond
http://www.turkishmusicportal.org - listening
http://www.turkishmusic.org - listening
60cddd*
diminished, dominant, august
{mp7r2ms_cd.png}
The bubble chart above (source) arranges the temperaments by complexity (x-axis) vs. damage (y-axis). Colors are arbitrary. The size of the bubble is proportional to (complexity*damage). Note that the "bonus temperament" Ennealimmal is not shown (a much larger range would be needed to make it visible).
{mp7r2ms_cd_d1.png}
The same chart, only "zoomed in".