_______ 1. It Goes Like This PDF Print E-mail
Written by D. Bond   
Saturday, 31 October 2009 10:34

1. It Goes Like This



it goes like this:

the fourth, the fifth,

the minor fall,

the major lift.


            Leonard Cohen




In the computer age, a study of musical harmony can now be conducted with a precision unknown to previous ears.[1]


In this discussion we will revisit concepts of pitch, intonation, consonance and dissonance, with the aim at advancing a cohesive theory of harmony.


Naturally occurring acoustic phenomena, such as the harmonic series, summation and difference tones, beats, et al, will serve as our reference point, providing the theoretical and experiential foundation for the discussion, and provide us with a motive for engaging in this exploration.


These three - Just Intonation, Pythagorean tuning and Equal Temperament [ET] - will be examined. Their histories will be analyzed insofar as the fundamental principles can be demonstrated. These principles will form the basis of a generalized system of intonation and harmony.


This effort serves to expand the musical resources of composers beyond the traditional 12 equally tempered pitches, to include the realm of harmonic and acoustically sound relationships founded upon the Pythagorean and just intonational systems, and beyond. The theory presented here will be shown to be flexible enough to accommodate alternate, irregular, irrational systems of arbitrary complexity.


Over the course of this discussion we will describe a variety of sets, each consisting of an infinite number of pitches organized with respect to their relationships. These sets have various properties of symmetry, inversion, repetition and self-similarity. We will begin to use the basic properties of these harmonic sets and in a preliminary fashion, generalize and expand upon them.


Along the way, aspects of the aesthetics and performance of music using this system will be investigated with reference to human perception, the limits of the ear, and instrumental practice. Whether the ear is capable of discriminating the enormous variety of relationships described here, and whether one could expect traditional performers to consistently realize these in live concert situations will be addressed. Challenges involved in the notation of these discreet relationships will be touched upon briefly.


In future writings and compositions, I plan to take up the vast topic of suggesting various ways of combining and manipulating these sets in compositionally interesting ways and the design of instruments and interfaces for the management and control of these resources. For now we keep it simple, using only a traditional electronic keyboard.


To begin, we set out on this exploration with a computer running the swiss-army knife of music gear, Max5/MSP/Jitter, a MIDI keyboard input and 4 octaves of keys. Example patches are included in a folder, but also compressed-text files will be included with this document.[2]

next 2. Thus Have I Heard

[1] John Cage once wished us all ‘Happy New Ears’ and lifting a glass in return, we begin.

[2] This will not be done - yet. People wishing to inquire about these patches can write to This e-mail address is being protected from spambots. You need JavaScript enabled to view it . I’d be happy to share them.

Last Updated on Tuesday, 25 May 2010 22:15