<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN"> <HTML><HEAD> <META http-equiv=Content-Type content="text/html; = charset=iso-8859-1"> <META content="MSHTML 6.00.2800.1400" name=GENERATOR> <STYLE></STYLE> </HEAD> <BODY bgColor=#ffffff> <DIV><FONT face=Arial size=2>some additional = thoughts....</FONT></DIV> <DIV><FONT face=Arial size=2></FONT> </DIV> <DIV><FONT face=Arial size=2> <DIV><STRONG>Twelve fifths - pythagorean comma = seven octaves</STRONG>  </DIV> <DIV>mathematically </FONT><FONT face=Arial size=2><STRONG>(3/(2^(pc/12)))^12 = (2)^7</STRONG> </FONT></DIV> <DIV><FONT face=Arial size=2>is <STRONG>traditional equal temperament</STRONG></FONT></DIV> <DIV><FONT face=Arial size=2></FONT> </DIV> <DIV><FONT face=Arial size=2> <DIV><STRONG>Twelve fifths= seven octaves  +  pythagorean comma</STRONG> </DIV> <DIV>mathematically <FONT face=Arial size=2><STRONG>(3/2)^12 = = (2)^(pc/7)^7</STRONG></FONT></DIV> <DIV>is <STRONG>equal pure 5th (Cordier)</STRONG></DIV> <DIV> </DIV> <DIV><STRONG>Twelve 12ths = 19 Octaves + pythagorean = comma</STRONG></DIV> <DIV>mathematically <STRONG>3^12 = 2^(pc/19)^19</STRONG></DIV> <DIV><STRONG>or</STRONG></DIV> <DIV><STRONG>3^12 = 3^(12/19)^19</STRONG></DIV> <DIV><STRONG>is equal pure 12th = <STRONG>(Stopper)</STRONG> </STRONG></DIV> <DIV><STRONG>or "acoustic octave transformed = pythagorean"</STRONG></DIV> <DIV> </DIV> <DIV> </DIV> <DIV>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</DIV> <DIV> </DIV> <DIV>regards,</DIV> <DIV> </DIV> <DIV>Bernhard Stopper</DIV></DIV></DIV></FONT> <BLOCKQUOTE dir=ltr style="PADDING-RIGHT: 0px; PADDING-LEFT: 5px; MARGIN-LEFT: 5px; = BORDER-LEFT: #000000 2px solid; MARGIN-RIGHT: 0px"> <DIV style="FONT: 10pt arial">----- Original Message ----- </DIV> <DIV style="BACKGROUND: #e4e4e4; FONT: 10pt arial; font-color: = black"><B>From:</B> <A title=b98tu@t-online.de = href="mailto:b98tu@t-online.de">Bernhard Stopper</A> </DIV> <DIV style="FONT: 10pt arial"><B>To:</B> <A = title=pianotech@ptg.org href="mailto:pianotech@ptg.org">Pianotech</A> </DIV> <DIV style="FONT: 10pt arial"><B>Sent:</B> Wednesday, June 02, 2004 = 9:50 PM</DIV> <DIV style="FONT: 10pt arial"><B>Subject:</B> Re: P12 in Tunelab Pro = / P12 theoretical basics</DIV> <DIV><BR></DIV> <DIV> <DIV> <DIV><FONT face=Arial size=2>OOPs, send click was done to = fast, the first version has to be corrected by some * in the math and the final statement....</FONT></DIV> <BLOCKQUOTE dir=ltr style="PADDING-RIGHT: 0px; PADDING-LEFT: 5px; MARGIN-LEFT: 5px; = BORDER-LEFT: #000000 2px solid; MARGIN-RIGHT: 0px"> <DIV style="FONT: 10pt arial">----- Original Message ----- </DIV> <DIV style="BACKGROUND: #e4e4e4; FONT: 10pt arial; font-color: = black"><B>From:</B> <A title=bernhard.stopper@piano-stopper.de href="mailto:bernhard.stopper@piano-stopper.de">Bernhard = Stopper</A> </DIV> <DIV style="FONT: 10pt arial"><B>To:</B> <A = title=pianotech@ptg.org href="mailto:pianotech@ptg.org">Pianotech</A> </DIV> <DIV style="FONT: 10pt arial"><B>Sent:</B> Wednesday, June 02, = 2004 9:37 PM</DIV> <DIV style="FONT: 10pt arial"><B>Subject:</B> P12 in Tunelab Pro / = P12 theoretical basics</DIV> <DIV><BR></DIV> <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 don愒 = sound "just" when played melodically. (what has been proven in many = investigations)</DIV> <DIV> </DIV> <DIV>Now let愀 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 = P12愀 "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* 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)</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></FONT></DIV></BLOCKQUOTE></DIV></DIV></BLOCKQUOTE></BOD= Y></HTML>