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The Harmonic Series vs.
its approximation in 12TET

by REGINALD BAIN


The animation in Fig. 1 is designed to allow you to compare the first twelve partials of a harmonic series for C2 (a just tuning) with the same 12 pitches approximated in twelve-tone equal temperament (12TET). In the animation, each just partial is immediately followed by its approximation in 12TET. Just partials appear in in a red boxes. 12TET approximations appear in (gray boxes). The deviation of each 12TET approximation from a just tuning is indicated in cents (+/-). Pitch pairs are allowed to overlap so that beating may be heard. A chart summarizing the frequency calculations used to create Fig. 1 appears in Fig. 2.

Fig. 1. A comparison of the first twelve partials of a harmonic series for C2 vs. their approximation in 12TET.

Fig. 2. Frequency values and differentials for comparison.
Partial Number123456789101112
US Standard Pitch NameC2C3G3C4E4G4(Bb4)C5D5E5(F#5)G5
Freq. in 12TET (Hz.)
Based on A4 = 440 Hz.
65.41130.81196.00261.63329.63392.00466.16523.25587.33659.26739.99783.99
Differential in cents00+20-14+2-310+4-14-49-2
Freq. in a Just Tuning (Hz.)

Based on C2 = 65.41 Hz.
65.41130.82196.23261.64327.05392.46457.87523.28588.69654.1719.51784.92
C4 = middle-C
All frequency calculations have been rounded to the nearest 1/100 Hz.
The differential in cent indicates the discrepancy between to two systems.

Technical note:
This animation in Example 1 has a MIDI sound track. In order for it to sound correctly, QuickTime's pitch bend range must be set to its default of +/- 2 semitones. If you are interested in learning how to create simple musical QuickTime with MIDI animations like these click here.

Updated: September 21, 2002

Reginald Bain | University of South Carolina | School of Music | Disclaimer
http://www.music.sc.edu/fs/bain/atmi02/
A Web-based Multimedia Approach to the Harmonic Series: Beats

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