P12 in Tunelab Pro / P12 theoretical basics

[Bernhard Stopper]

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OOPs, send click was done to fast, the first version has to be corrected =
by some * in the math and the final statement....
  ----- Original Message -----=20
  From: Bernhard Stopper=20
  To: Pianotech=20
  Sent: Wednesday, June 02, 2004 9:37 PM
  Subject: P12 in Tunelab Pro / P12 theoretical basics


  Ric,
  I agree that the tuning itself was tendencially practiced more or less =
by good tuners already long time ago.=20

  What i claim as new is mainly the theory behind the P12 tuning. =
Especially the transformation of the standard  12-5th circle that has to =
be closed with 7 octaves into a 12-12th circle that is closed with 19 =
octaves. And the direct transformation of the pythagorean tuning into =
equal P12 by replacing simply the mathmatical 2/1 octave ratio with the =
"acoustic octave" (later explained).=20
  (The effect of inharmonicity can be divided out at this point, it is =
added later as instrument immanent factor that stretches all ratios =
according to the instrument inharmonicity curve)

  For those who are not familiar with any maths the traditional fifth =
circle can be expressed in words as

  Twelve fifths =3D seven octaves + pythagorean comma

  Mathematically this can be written as ( 3/2 for 5th, pc for =
pythagorean comma and 2 for octave ratio):

  (3/2)^12 =3D (2)^7 * pc

  Now comes the transformation trick:

  Dividing out the twelve fifths give:

  (3^12) / (2^12) =3D 2^7 * pc

  Sorting 3 (Pure 12th) and 2 (Pure octave) give:

  3^12 =3D 2^12 * 2^7 * pc=20

  =3D> 3^12 =3D 2^19 * pc

  This describes a 12-P12 circle that is closed by 19 Octaves + =
pythagorean comma=20
  (what itself is the base for a new musical system i called =
"Dodecachord"  not mentioned here in detail).

  In P12 tuning, the pythagorean comma is divided in 19 parts and added =
evenly to the octaves that become 1,2 cent wider than pure (in the =
instrument, inharmonicity must be added here).

  The factor for one keystep is the 19th root of 3.

  Every of 1/19 pythagorean comma stretched octave can be now rewritten =
as 2^(pc/19) or 3^(12/19) instead of the 2. I call this "acoustic =
octave" since pure octaves with a ratio of 2 don=B4t sound "just" when =
played melodically. (what has been proven in many investigations)

  Now let=B4s look at the pythagorean intervals:

  2/1 (Octave)
  3/1 (P12)
  3/2 (5th)
  4/3 (4th)
  9/8 (M sec.)
  81/64 (M3rd)
  256/243 (m sec.)

  All can be split down into ratios of octaves 12ths and rewritten as=20

  2^1/1
  3^1/1
  3^1/2^1
  2^2/3^1
  3^2/2^3
  3^4/2^6
  2^8/3^5

  Substituting the mathematical octave ratio 2 with the P12=B4s =
"acoustic octave" 2^(pc/19) or 3^(12/19)

  Now results in

  2^1/1
  3^1/1
  3^1/3^(12/19)^1
  3^(12/19)^2/3^1
  3^2/3^(12/19)^3
  3^4/3^(12/19)^6
  3^(12/19)^8/3^5

  So in consequence can be said:
  Equal temperament based on Pure 12ths* is the direct transformation of =
the pythagorean tuning by simply replacing the mathematical octave ratio =
of 2 with the acoustic ocatve ratio of 3^(12/19).
  I find this philosphically somehow interesting/important.

  *published as "Stopper Tuning" in euro-piano 3/1988

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