## Best introductory materials for Harmonic Analysis?

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lunar
Posts: 5
Joined: Fri May 18, 2012 2:18 am UTC

### Best introductory materials for Harmonic Analysis?

I'm a comsci major but after finishing up Calc III in about a month I will likely not be taking more Math courses due to degree constraints.

That said, I believe I will need to teach myself some basics regarding Harmonic Analysis to support a proof I am working on for the feasibility of a computational solution to the Equal Temperament vs non-transposable Just Intonation problem that has persisted for centuries (these terms would only be familiar to those versed in Music Theory or possibly digital audio).

What are some good introductory materials on the subject?

I already have some familiarity with Fourier transformations due to a background in digital audio.

Thanks!

f5r5e5d
Posts: 104
Joined: Tue May 08, 2012 3:22 am UTC

### Re: Best introductory materials for Harmonic Analysis?

it doesn't seem there is a mathematically tractable problem here - musical temperment is intimately tied to human perception, cultural fashions and tied to physical instruments contruction, builders choices over centuries

lunar
Posts: 5
Joined: Fri May 18, 2012 2:18 am UTC

### Re: Best introductory materials for Harmonic Analysis?

Indeed, at first glance I would also be inclined to agree.

However, this was not the way music theorists were thinking at the time of the introduction of Equal Temperament (ET) roughly 500 years ago. At that time, ET was seen as a necessary compromise, which continues to be the case for acoustic instruments to this day.

This is not necessarily applicable to electronics instruments though, and this is where I begin to base my case.
To clarify, Just Intonation (JI) is a tuning system which prefers the lowest possible ratios for all intervals from a given base pitch. The problem with it is that although it works well for the base scale and primary (I, IV, V) chords, it has an undesirable byproduct of putting other intervals within the system out of tune with each other.

In an acoustic instrument there is no way to accommodate for these discrepancies. However, in a digital instrument, if an early warning is provided (such as, the tapping of a note on a one octave foot pedal alike the ones organists use slightly before playing the notes in the new key) it is possible to retune the pitches within the new system so that it takes on the same harmonious qualities relative to the new (either temporary, or permanent) base pitch. This would not be unlike pressing the clutch before changing gear in a manual transmission vehicle.

What I've alluded to directly above is the essence of the system I propose. A single tap on a note on the foot pedal could represent a temporary modulation that will ultimately return to the original base pitch (say, II - I with no change in sense of what I is). A double tap could represent a permanent modulation to a new key (such as I - II7(V7/V pivot to new key) - I (new key)).

Right now I am working out the rules for the retunings based on the sets and relations between sets, but it is a fairly large data set (12^3 - 12 original timings for each pitch, another 12 for temporary modulations, and another 12 for permanent modulations). After a permanent modulation, the third set becomes the first set of the new system.

The reason I believe this work calls for support from Harmonic Analysis is due to the hypothesis I am holding at the moment that suggests there may be a way to classify the retunings so as to reduce the complexity of the data set by finding the commonalities amongst certain modulations. This would be variable depending on the harmonic "distance" between the base tuning and target retuning relative to the "closeness" of the keys involved.

For example, a modulation to a relatively primary key would require less and/or less extreme retunings than a modulation to a relatively secondary key, and conversely modulations to the most foreign keys would require the most retunings of the most pitches.

Another factor is the constraint of needing to not retune common pitches that are to be played both immediately before and immediately after the retuning.

That all said, it is very much still something I am in the process of working out, but I have gathered enough evidence so far, I believe, to feel that this is very much possible and also very much worth pursuing.

As for my original request, I have hunted around and now suspect that thorough study of the Fourier series is prerequisite to harmonic analysis, and have found a good introductory book on the subject.

I invite any and all discussion regarding both topics as well as my ambition!