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Using the adjusted measurement, multiply the “sounding length” by each “string length factor” in Figure 5. The factors are derived by a similar formula as that which finds 12-tone equal tempered intervals, except instead of finding the 12th root of 2, we find the 31st root of 2. F = (2 1/31 ) n ÷ 220hz In this formula, “F” is the frequency of the note, “n” is the degree of the scale you are looking for (0-31), and 220hz is the “pitch center” of your quarter-comma system (see “Figure 1). This can be any frequency, but in the example, 220hz is used. The resultant frequencies for all the degrees of the 31-note division can then be translated into string length factors by dividing the fundamental frequency by the frequency of the pitch above it: “string length factor” = 220hz ÷ F Then simply multiply each “string length factor” by the total “sounding length” to get the distance from the bridge to the nut. “distance from bridge to fret” = “string length factor” × “sounding length” Here it is important, though, not to place frets based on the measurement from the bridge, because we are working from an imaginary bridge (to compensate for pitch sharpening). You must subtract the new sounding length from the total sounding length to get the distance from the nut. “distance from nut to fret” = “sounding length” − “distance from bridge to fret” The data from these calculations is summarized in the following chart. A lute with a “sounding length” of 650mm was used, though the true distance from nut to bridge was in fact 655mm. Also, the open 3rd course (pitch “A” 220hz) was used as a reference, but due to the equal-division system, any pitch may be used with position “0” as the root. Note Name Step Number (n) Frequency (F) (in Hertz) String length factor Distance from bridge to nut Distance from nut (D) A 0 220.0 1 650 0 1 225.0 0.9778 635.6 14.4 A# 2 230.1 0.9561 621.5 28.5 Bb 3 235.3 0.9349 607.7 42.3
Note Name Step Number (n) Frequency (F) (in Hertz) String length factor Distance from bridge to nut Distance from nut (D) 4 240.6 0.9141 594.2 55.8 B 5 246.0 0.8943 581.3 68.7 6 251.6 0.8744 568.4 81.6 7 257.3 0.8550 555.8 94.2 C 8 263.1 0.8362 543.5 106.5 9 269.0 0.8178 531.6 118.4 C# 10 275.1 0.7997 519.8 130.2 Db 11 281.3 0.7821 508.4 141.6 12 287.7 0.7647 497.1 152.9 D 13 294.2 0.7478 486.1 163.9 14 300.9 0.7311 475.2 174.8 D# 15 307.7 0.7150 464.8 185.2 Eb 16 314.6 0.6993 454.5 195.5 17 321.7 0.6839 444.5 205.5 E 18 329.0 0.6687 434.7 215.3 19 336.5 0.6538 425.0 225.0 20 344.1 0.6393 415.5 234.5 F 21 351.8 0.6254 406.5 243.5 22 359.8 0.6115 397.5 252.5 F# 23 367.9 0.5980 388.7 261.3 Gb 24 376.3 0.5846 380.0 270.0 25 384.8 0.5717 371.6 278.4 G 26 393.5 0.5591 363.4 286.6 27 402.4 0.5467 355.4 294.6 G# 28 411.5 0.5346 347.5 302.5 Ab 29 420.8 0.5228 339.8 310.2 30 430.3 0.5113 332.3 317.7 A 31 440.0 0.5000 325.0 325.0 Figure 5. Chart of 31-division data, based on a 650mm sounding length.
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 and 8: implying, perhaps, that someone had
Page 9: and the Just Major 3rd, make this s
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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