the-rule-of-31

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octave, or 38.71 cents. Similarly, the Major 3rd is 387.1 cents wide, compared to the just Major 3rd which is 386.3 cents 17 . The octave in 31-division can be visualized as a wheel with 31 spokes. Note that all pure Major 3rds are 10 steps apart. Figure 4. Pitch Wheel of 31-division octave with meantone congruities. This fine-grained equally-divided octave closes the infinite loop of ascending fifths, albeit rather late in the cycle (F-double-sharp is at step 25, and F-triple-sharp at step 27 which finally lines up with its true enharmonic equivalent A-double-flat). These obviously ridiculous extensions into chromatic deep-space combined with the psychoacoustic indistinguishability between the 31-division Major 3rd 17 ibid. p. 163.
and the Just Major 3rd, make this system a perfect candidate to be the basis of a 1/4 comma fretting scheme. In a way, it’s not unlike Vincenzo Galilei’s equal tempered system, only that the gradations are much finer, and many of the intervening “steps” are skipped to generate the major and minor semitones. Step-By-Step Guide for Meantone Fretting First the player must find the string length from which to calculate the divisions. Treatises from the sixteenth and seventeenth centuries instruct one to use the complete distance from the nut to the bridge. There is a serious problem with this method however. As one frets notes onto the fingerboard, the string is stretched slightly, causing an increase in tension, and thus, an increase in pitch. The increase is a non-trivial amount and can throw the series of geometrically calculated major and minor semi-tones into complete disarray. To compensate for this, I recommend applying a technique proposed by Eugen Dombois 18 . Dombois’ technique is to find what he calls the “playing scale”. This is a length slightly shorter than the complete length of the string. Using this slightly shorter length for the calculations will cause the fret placement to be shifted minutely in the direction of the nut (away from the bridge). Thus, when fretting a note, the increase in pitch from depressing the string will be compensated for by the subtly longer sounding length. Unfortunately, Dombois’ recommendation entails a trial-and-error method which requires one to calculate all the fret positions, place them, and then judge the quality of the temperament by ear. This laborious process can be simplified by the following method: play the 3rd partial on the 6th course. This harmonic can be found at 1/3 of the string length from the nut and will produce a pitch one octave and a fifth above the open string. Slide your VIIth fret to a position directly under where the harmonic is found, but then adjust it so that the pitch of the fretted note matches the pitch of the harmonic (though an octave lower). You will find that the position of this fret is slightly closer to the nut than the position of the harmonic. Multiply the distance from the nut to this repositioned VIIth fret by 3 to get the “sounding length”. This distance will be slightly shorter than the true distance from the nut to the bridge, though how much shorter will vary from one instrument to the next. 18 Dombois, Eugen. “Correct and easy fret placement.” Journal of the Lute Society of America. vi (1973). p. 31.
Page 1 and 2: The Rule of Thirty-One: Realizing 1
Page 3 and 4: fifths which actually “overshoots
Page 5 and 6: fourths to avoid a discordantly wid
Page 7: implying, perhaps, that someone had
Page 11 and 12: Note Name Step Number (n) Frequency
Page 13 and 14: Figure 6. Fret positions for “A
Page 15 and 16: Notice how the marked “fa” and
Page 17 and 18: Figure 13. Template aligned to 6th
Page 19 and 20: Once the frets are placed, one can
Page 21: Freis, Wolfgang. “Perfecting the

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