Flexible Tuning Software: Beyond Equal Temperament

The premise of this creative Capstone project was to develop a computer instrument that is capable of tuning itself flexibly in such a way as to not require a tempering of the Western scale, as is necessary for fixedly tuned instruments. The difficulty in creating such a system of tuning arises from the mathematical paradox of the musical harmonic series, which is the sequence of frequencies that sound naturally as overtones over a fundamental pitch. They follow a proportional pattern of 1:2, 2:3, 3:4, etc. These small-integer ratios of frequencies represent the consonant (in-tune) harmonic intervals. When an interval does not have a small-integer ratio between its notes, it is perceived as “out-of-tune,” and the two frequencies will compete with each other. The problem becomes apparent when the ratios are used to create the Western scale on a fixed instrument. When tuning such an instrument (i.e. a piano), the intervals inherently cannot all be tuned justly, or according to the appropriate proportions. The nature of the ratios does not allow larger intervals to be explained exactly by the smaller intervals, though they are expected to coexist and be used simultaneously in Western music. For example, an octave in music is the equivalent of three major thirds; however, the justly tuned octave (2:1) is not equal to the sum of three major thirds (5:4); 5:43=125:64, not 2:1. The difference, then, between notes tuned in terms of different intervals is known as a comma, and the attempts to distribute this comma are called temperaments. To overcome the impossibility of perfectly consonant temperaments, we have created a computer program that can function as a self-tuning instrument by mathematically calculating the frequency of each pitch played in relation to the pitch preceding it. Though this allows every interval between sequentially played notes to be in tune, it does mean that pitches are not fixed, and rather are flexible and changing as a piece progresses. The program possesses the capability to play several historical fixed temperaments, namely Pythagorean, Quarter-Comma Meantone, Werckmeister III, and Equal Temperament. However, it also is capable of playing in two non-fixed tuning systems, distinguished as Sequential Tuning and Flexible Tuning. The Sequential Tuning system uses the ratios in the harmonic series to tune each note proportionally to the most recent note pressed. It is ideal for educational purposes, clearly demonstrating the flexibility of the program, but is less practical for tuning purposes, as it disregards held notes and tunes solely based on the notes pressed while the note is held, creating obtrusive dissonances between any held note and the notes tuned sequentially before it is released. The Flexible Tuning system remedies this issue by first tuning notes based on a held note, and if there is no note being held, then on the most recent note pressed. This eliminates the noticeable dissonances of the Sequential Tuning system, while still tuning each note flexibly and in real time based on its interval relationships to other notes played before and at the same time as it. In this way, every interval sounding is the appropriate small integer ratio for that interval in the harmonic series