devil in the music

[Glenn Scheper]

> The interval which our awkwardly unflatted B makes with F
> was known to the ancients as the 'devil in the music.'

I don't know music, just math, but let me attempt this.

I wrote a C program to show the 12th roots of 2, and see
that they agree with "well tempering" per a piano tuning
web page (in C code below). Examining the output lines,
for (not actual, but philosopher's) F and B...

  5:  1.3348 ratio, or  170.86 Hz.  vs. F  170.86
 11:  1.8877 ratio, or  241.63 Hz.  vs. B  241.63

That's six steps, or the same as these, the sqrt of two:

  0:  1.0000 ratio, or  128.00 Hz.  vs. C  128.00
  6:  1.4142 ratio, or  181.02 Hz.  vs. F# 181.02

Whereas other harmonious sounds are approximately small
integer ratios apart, this pair is an irrational number.

I uploaded two screen shots from Adobe Audition, one of
the result of mixing 1000 Hz with 1260 Hz, ~ 4:5 ratio,
and one of mixing 1000 Hz with 1414 Hz, ~ sqrt( 2 ).

http://home.earthlink.net/~glenn_scheper/1000_1260.gif

http://home.earthlink.net/~glenn_scheper/1000_1414.gif

The envelope for the simple ratio repeats quickly, so
the neurons only have to report, more of the same, no
change; whereas the other pair never repeats, so the
poor neurons must work constantly to derive meaning.

l/*
    Tone.cpp
    Feb 09 2007 Glenn Scheper

    This proves "well tempering" just means using the twelth roots of 2.
    output of program: Computed roots on left, Web page nominal on right:

  0:  1.0000 ratio, or  128.00 Hz.  vs. C  128.00
  1:  1.0595 ratio, or  135.61 Hz.  vs. C# 135.61
  2:  1.1225 ratio, or  143.68 Hz.  vs. D  143.68
  3:  1.1892 ratio, or  152.22 Hz.  vs. D# 152.22
  4:  1.2599 ratio, or  161.27 Hz.  vs. E  161.27
  5:  1.3348 ratio, or  170.86 Hz.  vs. F  170.86
  6:  1.4142 ratio, or  181.02 Hz.  vs. F# 181.02
  7:  1.4983 ratio, or  191.78 Hz.  vs. G  191.78
  8:  1.5874 ratio, or  203.19 Hz.  vs. G# 203.19
  9:  1.6818 ratio, or  215.27 Hz.  vs. A  215.27
 10:  1.7818 ratio, or  228.07 Hz.  vs. A# 228.07
 11:  1.8877 ratio, or  241.63 Hz.  vs. B  241.63
 12:  2.0000 ratio, or  256.00 Hz.  vs. C  256.00

*/

#include <stdio.h>
#include <math.h>

// These are not the actual values, but the "Philosophical Standard" from:
// http://www.balaams-ass.com/piano/Web%20Pages/pianotuningfischer.htm

char * notes[] = {
   "C  128.00",
   "C# 135.61",
   "D  143.68",
   "D# 152.22",
   "E  161.27",
   "F  170.86",
   "F# 181.02",
   "G  191.78",
   "G# 203.19",
   "A  215.27",
   "A# 228.07",
   "B  241.63",
   "C  256.00",
};

main ()
{
    double ln2 = log(2.0);
    double lnht = ln2 / 12.0;
    double halftone = exp(lnht);
    double freq = 1.0;
    double C = 128.00;
    int i;
    for(i=0; i<=12; ++i)
    {
        printf(" %2d: %7.4lf ratio, or %7.2lf Hz.  vs. %s\n",
            i, freq, freq * 128.0, notes[i] );
        freq = freq * halftone;
    }
}


Yours truly,
Glenn Scheper
http://home.earthlink.net/~glenn_scheper/
glenn_scheper + at + earthlink.net
Copyleft(!) Forward freely.