P12 in Tunelab Pro / P12 theoretical basics

[Bernhard Stopper]

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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* directly transforms the =
pythagorean tuning simly by 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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