## Software

Ratio to Cents

PC Polygon Assistant:Allows students to display the polygon notation for a pc set and explore basic transformations, properties and relations in pc spacePC Polygon Assistant is a touch-optimized Web app that uses SVG to render the

Current version: 2.03polygon notation(Toussaint 2013) for any pc set. Moreover, it reports the basic properties of the T_{n}/I_{n}set class (Straus 2016) and cyclic interval array of the prime form (Morris 1987), and allows the student to explore transformations (T_{n}& I) and relations (C) in pc space. Playback is available via Web Audio API. This program is currently in beta testing. For more information, see my ATMI 2016 presentation.

Twelve-Tone Assistant

Ratio to Cents:Allows students to convert an interval frequency ratio to centsRatio to Cents is a simple mobile Web app that allows students to convert an interval frequency ratio to cents on their smartphone. Built with jQuery Mobile, the program utilizes the following conversion formula:

where

cis cents, andf/_{2}fis a pitch interval expressed as a frequency ratio. Students typically_{1}enter a frequency ratioderived from the harmonic series, or a tuning system like Pythagorean intonation,e.g., 1/1, 2/1, 3/2, 4/3, 5/3, 5/4, 6/5, 9/8, 7/4, 81/64, 22/7, 81/80, etc.

and the program reports the ratio'ssize in centsanddecimal equivalent. Decimal values (e.g., 1.618/1, 3.14/2, etc.), reciprocal frequency ratios (e.g., 1/2, 2/3, 3/4, etc.), and frequency values in Hz (e.g., 466/440, 660,440, 880/440, etc.) may also be entered. All values are rounded to 3 decimal places. For example, enter the just (5-limit) perfect fifth 3/2 and the program will report its decimal equivalent as 1.5 and its size in cents as 701.955 cents. For more information, see my ATMI 2012 presentation.Used in the following courses:

- MUSC 725 Contemporary Styles II
- MUSC 726B Mathematics & Music

MUSC 726T Tuning TheoryCurrent version: v2.03

The Harmonic Series

Twelve-Tone Assistant:A twelve-tone theory calculator and matrix makerLike its companion program Atonal Assistant, this pedagogically-oriented JavaScript application was written in the late 1990s for maximal compatibility with Joseph N. Straus's

Introduction to Post-Tonal Theory. It performs basic 12-tone row operations and generates matrices useful in the composition and analysis of twelve-tone music.Current version: v2.4

Used in the following courses:

- MUSC 525 Post-Tonal Theory

MUSC 725 Contemporary Styles II

The Harmonic Series: A Web-based multimedia introduction to the harmonic (overtone) series for musicians.Used in the following courses:

- MUSC 215 Music Theory III
- MUSC 726B Mathematics & Music

MUSC 726T Tuning TheoryCurrent version: v2.0

News: I am in the process of developing a mobile-friendly version of these resources

**SLAPI**: *A harmonic series interval player for Mac OS
and Windows.*

String Length and Pitch Interval (SLAPI) is a harmonic series interval player. It provides my students with a convenient way to compare the size of pitch intervals within the octave (Gann 1997) based on integer frequency ratios. The student enters a ratio (e.g., 3/2) and SLAPI calculates the frequency above A4 that corresponds with that ratio and plays the interval (e.g., 440 & 660 Hz) using a Karplus-Strong plucked string tone sample. The program reports the ratio's decimal equivalent (1.5), size in cents (702), and difference from the nearest equal-tempered interval (+2). The equal-tempered counterpart is played using a sampled piano tone.

Used in the following courses:

- MUSC 215 Music Theory III
- MUSC 726T Tuning Theory
Current version: v2.0

News: I am in the process of developing a new version of this program using Max 8.

### References

Campbell, Murray and Clive Greated. 1987/2001. *The
Musician's Guide to Acoustics*. New York: Oxford
University Press.

Gann, Kyle. 2019. *The Arithmetic of Listening: Tuning
Theory and History for the Impractical Musician*. Urbana,
IL: University of Illinois Press.

Morris, Robert. 1987. *Composition with Pitch Classes*.
New Haven: Yale University Press.

Rahn, John. 1908. *Basic Atonal Theory*. New York:
Longman.

Straus, Joseph N. 2016. *Introduction to Post-Tonal Theory*,
4th ed. New York: Norton.

Toussaint, Godfried T. 2019. *The Geometry of Musical
Rhythm: What Makes a "Good" Rhythm Good?*, 2nd ed. Boca
Raton, FL: CRC Press.

Tymoczko, Dmitri. 2011. A Geometry of Music: Harmony and Counterpoint in the Extended Common Practice. New York: Oxford University Press.