“Das N -EDOs Klavier” Shunya Kiyokawa 16/3/14
1. Abstract I have taken an interest in composing music using “n Equal Divisions of the Octave (nEDO)” or “n 平均律” which is a tuning divided an octave by various numbers, and making music collection with high quality piano sound. Sometimes, this tuning is called “n Equal Temperament (nET). But, I had three problems. 1st is “There is not usually known musical theory”. So I decided to make my own musical theory. I think new musical theory needs uniqueness as EDO. Therefore I searched past microtonal “operation”s, specifically Tristan Murail as “overtone”, Bela Bartók, Charles Ives, Ivan Wyschnegradsky, Alois Hába, and Toru Takemitsu as “quarter tone”, Harry Partch’s “Monophonique” system and Bohlen-Pierce scale as “just interval”, Turk, Indonesia and others as “ethnic music”. That result shows the purpose of “quartertone” is expanding 12-tone music; “overtone” and “just interval” is a simple frequency ratio; “ethnic music” is accenting character of ethnic music. I decided the uniqueness as EDO is getting away from these things. I decided the uniqueness as EDO is including intervals that have 20 cent distance from 12 tone music (each 100 cent). For example, 60 cent, 170 cent, 423 cent, 620 cent, 980 cent is permission, 81 cent, 286 cent, 490 cent and 719 cent is not permission. I call these intervals “Superset” and I picked out some tones “Number Intervals” from these intervals for uniformity as musical collections. And also, I got uniqueness as EDO musical format by analyzing Aaron Hunt and Chris Vasivil. Both use specific code in their works. So, I use a lot of codes on same musical algorithm in contrast. Because I think people change to receive feeling if they have already known sound of nEDO. First, I think composer should make music whose purpose is teaching sound of nEDO to listener. Finally, I decided my own musical theory. 2nd is “There is not notation for nEDO”. I made new microtonal notation written by only Number Intervals. 3rd is “It is difficult to make high quality piano sound”. I developed “F0 Tuner” which is able to make high quality piano sound. This Windows software is able to play optional frequency of the sampler detecting each sampler’s fundamental frequency by pitch bending. I made “Das N -EDOs Klavier” by integrating these states. This music collection is written from 5EDO to 24EDO excluding 6EDO and 12EDO. This script describes making my own music theory after “Superset”, and notation.
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2. Superset “Superset” is intervals that have 20 cent distance from 12 tone music (each 100 cent). For example, 60 cent, 170 cent, 423 cent, 620 cent, 980 cent is permission, 81 cent, 286 cent, 490 cent and 719 cent is not permission. The list is below. Each number shows step (In 12ET, step means semitone). 5EDO―(0, 1, 2, 3, 4, 5) 7EDO―(0, 1, 2, 5, 6, 7) 8EDO―(0, 1, 3, 5, 7, 8) 9EDO―(0, 1, 2, 4, 5, 7, 8, 9) 10EDO―(0, 2, 3, 7, 8, 10) 11EDO―(0, 3, 4, 5, 6, 7, 8, 11) 13EDO―(0, 3, 4, 5, 6, 7, 8, 9, 10, 13) 14EDO―(0, 2, 3, 4, 5, 9, 10, 11, 12, 14) 15EDO―(0, 1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 15) 16EDO―(0, 1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 16) 17EDO―(0, 1, 2, 5, 6, 8, 9, 11, 12, 15, 16, 17) 18EDO―(0, 1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 18) 19EDO―(0, 1, 2, 4, 6, 7, 9, 10, 12, 13, 15, 17, 18, 19) 20EDO―(0, 1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 20) 21EDO―(0, 1, 3, 4, 6, 8, 10, 11, 13, 15, 17, 18, 20, 21) 22EDO―(0, 1, 3, 5, 6, 8, 10, 12, 14, 16, 17, 19, 21, 22) 23EDO―(0, 1, 3, 5, 7, 9, 10, 11, 12, 13, 14, 16, 18, 20, 22, 23) 24EDO―(0, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 24)
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3. Number Intervals I picked out some tones “Number Intervals” from Superset for uniformity as musical collections. If numbers of superset is less of equal 7, Number Intervals becomes directly superset. But, if numbers of superset is over 7, Number Intervals pick out 7 tones from superset by inversion of intervals in order to leave a lot of intervals. Number Interval list is below. Each number shows step. EDO i1 i2 5 1 2 7 1 2 8 1 3 9 1 2 10 2 3 11 3 4 13 3 4 14 2 3 15 2 3 16 3 6 17 2 5 18 2 4 19 2 4 20 2 4 21 3 4 22 3 5 23 5 7 24 3 5
i3 3 5 5 4 7 5 5 4 4 9 6 5 7 7 6 6 10 7
i4 4 6 7 5 8 6 7 5 6 11 8 8 9 11 11 8 12 11
i5
i6
7
8
7 8 9 8 13 11 11 13 14 13 12 14 15
8 9 10 11 14 15 14 17 17 18 16 20 19
i7
10 12 14 15 16 17 18 19 20 21 22 23
i1, i2, i3, i4, i5, i6, i7 are notation intervals. My score notates only these numbers. And also, method of thinking i8, i9, i10.... is below.
11EDO Number Intervals ((3)) Notation intervals i1, i7, i13 13EDO Number Intervals ((3)) Notation intervals i1, i8, i15
((4)) i2, i8, i14 ((4)) i2, i9 , i16
((5)) i3, i9, i15
((6)) ((7)) ((8)) i4, i10, i16 i5, i11, i17 i6, i12, i18
((5)) ((7)) ((8)) ((9)) ((10)) i3, i10, i17 i4, i11, i18 i5, i12, i19 i6, i13, i20 i7, i14, i21
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4. Notation -Each color shows musical part. -Horizon line shows melody, vertical line shows code.
Melody
Code
Phrase 1
間
Phrase 2
間
Phrase 3, 4, 5, 6, 7
A part of “Das N -EDOs Klavier” score -Phrase 1’s red under melody line (red part) of 13 EDO is 3(key)-6-9-5-2 step. Key (lowest number of scale, 3) +i1(3) +i1(3) –i2(4) –i1(3). i1 and i2 are Notation Interval. Phrase 1’s red up melody line (purple part) of 13 EDO is 6-9-12-8-5 (each number is i1 (3 steps) higher than low melody line). Phrase 2’s red part line is 6(this step refers to phrase 1’s blue circle) -10-14-9-5 step. 6 +i2(4) +i2(4) –i3(5) –i2(4). Phrase 3, 4, 5, 6, 7 is repeated same algorithm. If numbers of Number Intervals is 5 or 6, phrase 6 or 7 is not played. -Each tone doesn’t have time value. The composer or player decides each time value. -「間」(Ma) is nearly Blank. The beginning of music is no sound, but「間」have gradually small time value tones and show musical progression.
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5. Score “Das N -EDOs Klavier” score is below.
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-After 4th section, 1st section appears again.
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