## 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

polygon 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.Used in the following courses: Current version: 1.91d

Atonal 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:Current version: v2.03

Twelve-Tone Assistant

Atonal Assistant:An atonal theory calculatorWritten with the post-tonal theory novice in mind, this pedagogically-oriented Web-based software application runs in a Web browser (under JavaScript 1.2+). It was designed in the late 1990s for maximal compatibility with Joseph N. Straus's

Introduction to Post-Tonal Theory. Students use the program to check homework exercises, to aid in analysis and composition, and to freely explore the pitch-class set universe. Atonal Assistant can list a a pc set's normal form (Rahn algorithm), set class, prime form, literal complement, abstract complement, Z-correspondent, interval-class vector and index vector. It can also perform the following twelve-tone operations T_{n}, T_{n}I, and M_{n}, and includes a Mod 12 and Interval Calculator (opi, upi, opci & ic). The results of all calculations are generated as ASCII text so that students may copy and paste the output into a word processer.Used in the following courses:Current version: v2.20

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.Used in the following courses:Current version: v2.5

The Harmonic Series: A Web-based multimedia introduction to the harmonic (overtone) series for musicians.Used in the following courses:Current version: v1.0

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

**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.Download: SLAPI v1.1 for Mac OS | SLAPI v1.1 for Windows | Documentation (pdf)

Used in the following courses:Current version: v1.1

### References

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

Gann, Kyle. 1997. "Anatomy of An Octave," in *Information on Alternative Tunings*. Available online at: http://www.kylegann.com/microtonality.html.

Morris, Robert. 1987. Composition with Pitch Classes. New Haven: Yale University Press. {GBd}

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

Toussaint, Godfried T. 2013. *The Geometry of Musical Rhythm: What Makes a "Good" Rhythm Good?* Boca Raton, FL: CRC Press. {GB}

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