Narrow Results

We find a fifth approximation of the Just Intonation which generalizes Equal Temperament. The intervals causing a dilemma are the second and the minor seventh and the tritone because they are unambiguous in Just Intonation (the relative frequencies 10/9, 9/8, 8/7 and 7/4, 16/9, 18/10 and 45/32, 64/45, respectively). If we do not consider the second and seventh with the relative frequencies 8/7 and 7/4, respectively, all the music intervals in this approximation either coincide with he Just Intonation interval values (the octave, fifth, fourth, second (9/8) and the minor seventh (16/9)) or are exactly the one comma distant from the corresponding Just Intonation intervals. This comma is 32 805/32 768 ≈ 1.00112915, which is less than the ratio of frequencies of the perfect and the equal tempered fifths (≈ 1.00112989).

We already proved the existence of an orthonormal basis of wavelets having an irrational dilation factor with an infinite number of wavelet shapes, and based on its theory, we proposed an orthonormal basis of wavelets with an arbitrary real dilation factor. In this paper, with the development of these fundamentals, we propose a new type of orthonormal basis of wavelets with customizable frequency bands. Its frequency bands can be freely designed with arbitrary bounds in the frequency domain. For example, we show two types of orthonormal bases of wavelets. One of them has an irrational dilation factor, and the other is designed based on the major scale in just intonation.