P12 in Tunelab Pro / P12 theoretical basics

[Bernhard Stopper]

This is a multi-part message in MIME format.

---------------------- multipart/alternative attachment
some additional thoughts....

Twelve fifths - pythagorean comma =3D seven octaves =20
mathematically (3/(2^(pc/12)))^12 =3D (2)^7=20
is traditional equal temperament

Twelve fifths=3D seven octaves  +  pythagorean comma=20
mathematically (3/2)^12 =3D (2)^(pc/7)^7
is equal pure 5th (Cordier)

Twelve 12ths =3D 19 Octaves + pythagorean comma
mathematically 3^12 =3D 2^(pc/19)^19
or
3^12 =3D 3^(12/19)^19
is equal pure 12th (Stopper)=20
or "acoustic octave transformed pythagorean"


the list of the "pythagorean" given below is not complete, it can be =
extended to any interval combination of 3 and 2 ratio of the keyboard

regards,

Bernhard Stopper
  ----- Original Message -----=20
  From: Bernhard Stopper=20
  To: Pianotech=20
  Sent: Wednesday, June 02, 2004 9:50 PM
  Subject: Re: P12 in Tunelab Pro / P12 theoretical basics


  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

---------------------- multipart/alternative attachment
An HTML attachment was scrubbed...
URL: https://www.moypiano.com/ptg/pianotech.php/attachments/aa/21/b4/76/attachment.htm

---------------------- multipart/alternative attachment--