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ut not fully described until the publication of Pietro Aron’s Toscanello in musica (Venice, 1523) 1 . This tuning system is a modification of the earlier system of pythagorean tuning, in which a chain of pure fifths is stacked to generate the pitches of the diatonic scale. While this tuning was appropriate for the consonant 4ths, 5ths, and octaves of 15th century music, the resultant “wide” major third sounds harsh in the context of 16th century polyphony with its pervasive use of 3rds and 6ths. The difference between a pythagorean third (the result of four ascending perfect 5ths) and an acoustically pure major third is called the syntonic comma, and is equivalent to 21.51 cents 2 . To generate euphonious thirds (5:4), the meantone temperament alters the pythagorean system by “narrowing” the fifths, each by 1/4 of the syntonic comma. Dy definition, the narrowing of each of these fifths by 1/4 of the syntonic comma causes the fifth note of the chain (the fourth interval) to fall exactly on an acoustically pure third. In Figure 1, this adaptation of the pythagorean chain of fifths can be seen referenced from a pitch center of “A”, extending outward into sharps on the right, and flats on the left. For compactness, the chain of fifths is here seen as fifths alternating with fourths. Note that an ascending fifth narrowed from pure by a 1/4 comma is equivalent to a descending fourth widened from pure by a 1/4 comma. Figure 1. Chain of 1/4 comma fifths and fourths. In this figure one may take any succession of four neighboring intervals to find a pure major third (C-flat to E-flat, A-flat to C, A to C-sharp, F-sharp to A-sharp, etc.). One important property of this system is that, unlike 12-tone Equal Temperament, the chain does not “close” after 12 pitches. In fact, because of the flattening of the fifth, the series flattens as it progresses, and “undershoots” the octave (C-flat to B is less than an octave). This is in contrast to the pythagorean chain of pure 1 Barbour, J. Murray. Tuning and Temperament: A Historical Survey. East Lansing: Michigan State Press, 1951. p. 25-26. 2 Greated, Clive. “Comma.” Grove Music Online. Oxford Music Online.
fifths which actually “overshoots” the octave (C-flat to B is greater than an octave). Of the three most common “regular” temperaments (temperaments in which all of the fifths are the same size), only Equal Temperament “closes” with 12 and only 12 unique pitches. This is significant for performers because an “open” chain of fifths can (theoretically) extend outward into sharps, double-sharps, triple sharps, etc. in one direction, and flats, double flats, triple flats, etc. in the other direction ad infinitum. While the physical limitations of acoustic instruments make this infinite spiral impossible to realize in sound, it should not then be assumed that realizing at least some of the so-called “enharmonic equivalents” is impossible. In fact, our earliest extant fretting document comes from the ninthcentury theorist Al-Kindī. His instructions for fretting the ‘ud is pythagorean, and generates sounding pitches beyond a chain of twelve fifths 3 . Even seven centuries later, the vihuelist Juan Bermudo called for split frets in his detailed instructions for a pythagorean fretting scheme 4 . This subtle difference in pitch between between the “enharmonic equivalents” generates another surprising phenomenon: since there are (at least) two unique pitches between a given whole-tone, the semi-tones must therefor be unequal. In a pythagorean chain of fifths, the “overshooting” of the octave causes the “sharped” notes to sound higher in pitch than their enharmonics on the flatside of the chain. Conversely, in the chain of meantone fifths, the “undershooting” of the octave causes the “sharped” notes to sound lower in pitch compared to their flat-keyed neighbors. In the meantone system, the smaller semitone is called the minor semitone (or chromatic semitone), while the larger is called the major semitone (or diatonic semitone). Primary Sources for Fretting Schemes on the Lute and Viola da Gamba While tuning instructions abound for keyboard instruments from the 16th century, treatises on fretting schemes are relatively rare. One of the first comprehensive guides comes from the aforementioned Juan Bermudo (1555). The musicologist Wolfgang Freis notes that Bermudo’s Declaración de instrumentos musicales of 1555 represents the work of a theoretician more than a that of a practical musician 5 . His pythagorean tuning scheme, as well as his invention of a completely new 3 Lindley, Mark. Lutes, Viols and Temperaments. Cambridge: Cambridge University Press, 1984. p. 9. 4 ibid. p. 17-18 5 Freis, Wolfgang. “Perfecting the Perfect Instrument: Fray Juan Bermudo on the Tuning and Temperament of the ‘vihuela de mano’.” Early Music, Vol. 23, No. 3, Iberian Discoveries III (Aug., 1995). p. 421-435.
Page 1: The Rule of Thirty-One: Realizing 1
Page 5 and 6: fourths to avoid a discordantly wid
Page 7 and 8: implying, perhaps, that someone had
Page 9 and 10: and the Just Major 3rd, make this s
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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