1. References to the Lucidarium follow Herlinger's numbering of its tractata, chapters, and clauses (1985). The | sign indicates links to the edition at the TML website http://www.music.indiana.edu/tml/14th. [EDITORIAL NOTE: The URL leads to fourteenth-century texts at the TML site., which has both the Gerbert version (MARLU) and the Herlinger edition (MARLUC). The Ludicarium tractata are in separate HTML files, in the form MARLUC#_TEXT.html, where # is the number of an individual tractate in the treatise (e.g. MARLUC8_TEXT.html).]

2. Below, a tone designated by letter-name within | signs, e.g., |c|, can be heard, in the html version, by clicking on the letter-name, or getting the corresponding RealAudio (ra) file, e.g., http://mto/audio/4.6/c.ra. [Editorial Note: Users may need to adjust the volume when playing some audio files.] For sound files, bass-clef staff letter-names are capitalized and tripled (GGG for 1st-line), or doubled (AA, BB, ... GG for 1st space to top space). Upper registers are designated by single letters (from a, just below middle c to g--with a numeral 1 for an 8ve higher: a1, b1, ...). By contrast, single capitals (A, B, ...G) designate any member of a pitch class. Flats are designated by b (e.g., Bb for B-flat; Marchetto's high sharps, by their characteristic numbers (e.g., C74); non-Marchettan sharps, by # (e.g., C#). In the sonic exx., c is middle c (256Hz); unless otherwise indicated, tuning is Pythagorean (e.g., g = ( 3/2 ) * 256 = 384Hz).

3. Oliver B. Ellsworth (1987: 340) gives a clear account of how Marchetto adapted earlier semitone terms by shifting each "up 1 notch."

4. In my study of melodic interval perception, all ogives (cf. Guilford 1954) were significantly close to (i.e., diverged non-significantly from) the students' responses at the p <= .05 level in a standard chi-squared test for grouped data (on which see Smith 1985: 319-414). For intervals of 1 to 12 semitones, responses were grouped into 5-cent increments within the medial, ( 100n + -30 )% ) range; for the unison/semitone pair, within the ( 5 - 45 )% range.

5. Karol Berger (1987: 16-29) comprehensively surveys successors' responses to Marchetto's terms and notational signs. As Berger (26) and Ellsworth (1987: 337-38) emphasize, the Lucidarium survives in 15 full and 3 incomplete copies (~1317-1500), and the Marchettan orthodoxy of Bonaventura da Brescia's Breviloquium enjoyed 19 early editions (1497-1570).

6. Mieczyslaw Kolinski's Pythagorean formulation of the 22 srutis of Ancient South Asian tuning (1961) would also result in a displaced cycle and satisfy the scale analysis of Clough et al. (1993), the latter re-framed as comprising 2 kinds of intervals: a = 90% and b = 24%. In such a construal, a sruti could be understood, like Marchetto's diesis, as the difference between consecutive strings, marks, or frets on a tuning instrument (e.g., of the vina variety).

End of Footnotes