FIGURE 18. THE DIVINE MONOCHORD. This particular monochord is tuned in the key of G, while the examples [below] and in the introduction use the key of C. The three top notes on this monochord are incorrectly placed.

APPENDIX III: THE FORMATION AND RATIOS OF THE PYTHAGOREAN SCALE
BY DAVID R. FIDELER

FIGURE 19. THE RATIOS OF THE PYTHAGOREAN SCALE

AS NOTED in the introductory essay, the structure of the musical scale possesses a great deal of significance in Pythagorean thought as it is an excellent example of the principle of mathematical harmonia at work. In the case of the scale, the "opposites" of the high (2) and the low (1) -- the two extremes of the octave -- are united in one continuum of tonal relationships through the use of a variety of forms of proportion which actively mediate between these two extremes.

The best way to understand the mathematical principles of harmonic mediation involves actually charting out and playing out the ratios of the scale on the monochord. In constructing a monochord, it is best to make it as long as possible, perhaps in the region of 4-5 feet, as that makes it easier to differentiate between the harmonic nodal points at the high end of the spectrum (see fig. 6).

It is useful at first to play out the harmonic overtone series. Measure the exact length of the string and then mark off the overtone intervals: 1/2 the string length, 1/3 the string length, 1/4 the string length, etc. It is possible to play out the overtone series without the use of the bridge; simply pluck the string about 1 inch from either end while simultaneously touching the nodal point with the other hand. It will be noted that there is an inverse relationship between the vibrational frequency of the tone and the string length. This is also illustrated in the above chart: hence a tone with a vibration of 2 is associated with a string division of .5 or 1/2. It is also useful at this point to play out the harmonic "Tetraktys," or the perfect consonances: 1:2 (octave), 2:3 (perfect fifth), and 3:4 (perfect fourth). Listen carefully to these ratios and reflect on the fact that you are actually hearing the relationships between these primary whole numbers.

To "tune" the monochord to the ratios of the Pythagorean scale use the string length ratios in the above chart, multiplying these ratios by the length of the string. Mark off these intervals, along with the corresponding notes, on the sounding board as they are carefully measured out.

Having marked out the Pythagorean scale, it might be useful at this point to review the material in the introductory essay relating to the harmonic proportion and then to play out these relations:

1) Play out the relationship of the octave (1:2). These are the two tonal extremes which must be united.

2) Play out the arithmetic mean linking together the extremes: C-G-c, or 6-9-12. This is the perfect fifth, the strongest musical relationship (2:3).

3) Play out the harmonic mean linking together the two extremes: C-F-c, or 6-8-12. This is the perfect fourth, the next strongest musical relationship (3:4).

4) Now play out the harmonic or musical proportion which is the basis of the musical scale: C-F-G-c, or 6:8::9::12. Play this out as a continued proportion and then the individual parts. Play out the two perfect fifths 6:9 and 8:12. Play out the two perfect fourths 6:8 and 9:12. Then play out the whole tone 8:9.

5) Having played out the harmonic foundation of the scale, now "fill in" the remaining 8:9 whole tone intervals. Play out C-D, D-E, G-A, and A-B. Along with F-G, these are all in the 8:9 ratio.

6) Play out the ratio of the leimma or the semitone: E-F and B-c. The leimma is the relationship between the perfect fourth and three whole tones.

7) Finally play out the entire scale: C-D-E-F-G-A-B-c. Through the use of arithmetic, harmonic and geometric proportion the two extremes have been successfully united.

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