### continuous chord progression

Following up on a side note from this post, it is possible to construct continuous changes from one chord to the other by linear interpolation of their frequency envelopes (i.e., chord interpolation).

Here is an example in Mathematica.

A[f_] := Sin[2*Pi*t*f*220];

For example, one of the major triads is 4:5:6, and the harmonic minor triad is 6:7:9.

First, the two chords:

`Play[{A[1260/1260] + A[1575/1260] + A[1890/1260]}, {t, 0, 1}]`

`Play[{A[1260/1260] + A[1470/1260] + A[1890/1260]}, {t, 0, 1}]`

Normally, if you were to move from one to the other (e.g. on an intrument), you might bend the middle note from a major 3rd until it becomes a septimal minor 3rd with respect to the low note. However, this (inner) interpolation produces dissonant beats during the entire process:

`Play[{A[1260/1260] + A[(t*1470 + (1 - t)*1575)/1260] + A[1890/1260]}, {t, 0, 1}]`

On the other hand, by frequency blending instead (and keeping the implied harmonic structure), the (outer) interpolation produces a very agreeable progression*:

`Play[{A[1260/1260] + t*A[1470/1260] + (1 - t)*A[1575/1260] + A[1890/1260]}, {t, 0, 1}]`

This was just one simple example.

*Indeed, this may very well be the implicit aim in many works with multiple voices.

Now this is very interesting. No, not the isomorphic keyboard, which is old, but the odd harmonic based scale and compositions. I’m not entirely convinced of the merits of odd harmonics (on grounds of psychoacoustics), but in my opinion, this is a much more productive direction to take than say, mechanical serialism and aleatoricism.

[...] To generalize even further, omitting harmonics is a special case of applying weighting (or a spectral filter) to all the harmonics. This is perhaps what people aptly call “color” of the sound. Thus all chords and chord colors can actually be identified entirely by the weight vector on the harmonic series. (This may not be as trivial as it sounds, because it implies additional degrees of freedom in musical activity, by varying weights dynamically and continuously, for example, and generalizing discrete progression of chords.) [...]