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<DIV><FONT face=Arial size=2>
<DIV>Ric,</DIV>
<DIV>I agree that the tuning =
itself was tendencially practiced
more or less by good tuners already long time ago. </DIV>
<DIV> </DIV>
<DIV>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). </DIV>
<DIV>(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)</DIV>
<DIV> </DIV>
<DIV>For those who are not familiar with any maths the traditional fifth =
circle can be expressed in words as</DIV>
<DIV> </DIV>
<DIV>Twelve fifths = seven octaves + pythagorean comma</DIV>
<DIV> </DIV>
<DIV>Mathematically this can be written as ( 3/2 for 5th, pc for =
pythagorean
comma and 2 for octave ratio):</DIV>
<DIV> </DIV>
<DIV>(3/2)^12 = (2)^7 + pc</DIV>
<DIV> </DIV>
<DIV>Now comes the transformation trick:</DIV>
<DIV> </DIV>
<DIV>Dividing out the twelve fifths give:</DIV>
<DIV> </DIV>
<DIV>(3^12) / (2^12) = 2^7 + pc</DIV>
<DIV> </DIV>
<DIV>Sorting 3 (Pure 12th) and 2 (Pure octave) give:</DIV>
<DIV> </DIV>
<DIV>3^12 = 2^12 * 2^7 + pc </DIV>
<DIV> </DIV>
<DIV>=><STRONG> 3^12 = 2^19 + pc</STRONG></DIV>
<DIV> </DIV>
<DIV>This describes a <STRONG>12-P12 circle that is closed by 19 =
Octaves +
pythagorean comma </STRONG></DIV>
<DIV>(what itself is the base for a new musical system i called
"Dodecachord" not mentioned here in detail).</DIV>
<DIV> </DIV>
<DIV>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).</DIV>
<DIV> </DIV>
<DIV>The factor for one keystep is the 19th root of 3.</DIV>
<DIV> </DIV>
<DIV>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 donīt sound "just" when played =
melodically.
(what has been proven in many investigations)</DIV>
<DIV> </DIV>
<DIV>Now letīs look at the pythagorean intervals:</DIV>
<DIV> </DIV>
<DIV>2/1 (Octave)</DIV>
<DIV>3/1 (P12)</DIV>
<DIV>3/2 (5th)</DIV>
<DIV>4/3 (4th)</DIV>
<DIV>9/8 (M sec.)</DIV>
<DIV>81/64 (M3rd)</DIV>
<DIV>256/243 (m sec.)</DIV>
<DIV> </DIV>
<DIV>All can be split down into ratios of octaves 12ths and =
rewritten as
</DIV>
<DIV> </DIV>
<DIV>2^1/1</DIV>
<DIV>3^1/1</DIV>
<DIV>3^1/2^1</DIV>
<DIV>2^2/3^1</DIV>
<DIV>3^2/2^3</DIV>
<DIV>3^4/2^6</DIV>
<DIV>2^8/3^5</DIV>
<DIV> </DIV>
<DIV>Substituting the mathematical octave ratio 2 with the P12īs =
"acoustic
octave" 2^(pc/19) or 3^(12/19)</DIV>
<DIV> </DIV>
<DIV>Now results in</DIV>
<DIV> </DIV>
<DIV>
<DIV>2^1/1</DIV>
<DIV>3^1/1</DIV>
<DIV>3^1/3^(12/19)^1</DIV>
<DIV>3^(12/19)^2/3^1</DIV>
<DIV>3^2/3^(12/19)^3</DIV>
<DIV>3^4/3^(12/19)^6</DIV>
<DIV>3^(12/19)^8/3^5</DIV></DIV>
<DIV> </DIV>
<DIV>So in consequence can be said:</DIV>
<DIV><STRONG>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)</STRONG>.</DIV>
<DIV>I find this philosphically =
somehow interesting/important.</DIV>
<DIV> </DIV>
<DIV>*published as "Stopper Tuning" in euro-piano 3/1988</DIV>
<DIV> </DIV>
<DIV> </DIV>
<DIV> </DIV>
<DIV> </DIV>
<DIV> </DIV></FONT></DIV></BODY></HTML>