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The Dictionary of Harmonic Functions endeavours to lay out the materials (interval relations, harmonic patterns) for the 12-E.T. tuning system in an elegant as well as practical manner. The relative simplicity of the tuning system, as opposed to the more complex musical cycles of 31 or 53, makes such an inventory manageable. Moreover, the 12 cycle is not dependant on a layout involving the matrix associated with the more complex cycles. Unlike the 19, 31, 43, and 53 cycles which are Prime, the 12 cycle is a Composite System, donating special properties to this much-used framework.
Those of us who own tuneable synthesizers posses an amazing freedom. For not only may we quit Equal Temperament and explore Just Intonation, but also we may use any frequency as our Pitch Standard. We are no longer tied to “A”=440 Hz. as our base frequency. But why choose one base frequency rather than another? Is the choice of a tuning Standard arbitrary? This article represents my research for a Standard which is based upon resonances found in the natural world. The key to unlocking these natural resonances is the Law of the Octave.
The study of harmonics is usually and rightfully associated with musical tuning theory. However, another appropriate context for this branch of arithmetic is the establishment of a calendar. In both situations, the procedure rests fundamentally on the measurement of a time period for a vibratory event. In the case of tuning theory, these cyclical interactions happen very quickly–hundreds, even thousands of periods per second. Consequently it is convenient to use the concept of frequency (cycles per second) as a descriptive device. In the case of calendrics, we are still measuring a physical vibratory event, but the time periods are very long; consequently, the frequency numbers are very low. It becomes more convenient to express measurements by period. For example, the year cycle is about 365 days long. We could convert this number of days to seconds (a big number!), and the invert the ratio to give us the frequency. In other words, frequency and period stand in inverse relation to each other.
The Pythagorean theory of music was centred around the concept of means. This is understandable, given that all of the ancient Asian cultures saw life as a play of opposites. The Chinese postulated that life is created and balanced through the polarity of the yin-yang. The Babylonians, Egyptians and Greeks also saw life as a dynamically balanced “middle path” between opposites.
We usually understand the ratios of JI as derived from the Harmonic Series. Any Just interval can be viewed as a frequency ratio between two whole numbers, numbers which refer directly to the Series itself. However, it may not be generally appreciated that there is also an alternative way to generate the Just ratios. This method has great historical importance as well as being of intrinsic interest in itself. I refer to the procedure of examining the 
The Tetractys symbol is an exquisite example of how a simple visual pattern can have multi-varied meanings and ‘levels’ of interpretive significance. The famous Pythagorean oath refers to the Tetractys, praising it as the source “which contains the fount and root of eternal nature”. A profusion of insights can be derived from the image, concepts which are relevant to arithmetic (number relations), harmonics (musical tuning theory), and geometry (number relations in space). Only a pitifully small amount of ancient commentary has been perserved; but the study of the above disciplines uncovers more and more relevance for the symbol. In this short paper, I will briefly review some historical interpretations and offer a few insights of my own that come from extensive work in harmonics and sacred geometry.
An explanation of how the author derived a better notation for 53-E.T. from the practical experience of tuning stringed instruments in Just Intonation. The necessity of an organised efficient procedure in tuning by Harmonics led to the conception of a “field-map” which became the seed-bed for the new pitch notation and its functional counterpart.
The two polychords are large steel guitars which were inspired by the ancient chinese instrument called the ch’in or ‘philosopher’s lute’. I wanted an instrument with a long scale length, yet short enough to be playable as a steel guitar (using a sliding bar or ‘steel’). Polychord 1 has two banks of strings; in other words, it is ‘double-necked’. Polychord 2 has a single bank of strings. These two instruments are meant to be together, and to compliment each other. Polychord 1 has a ‘chordal’ tuning; it is a ‘rhythm guitar’ in its orientation. Polychord 2 has a ‘scalar’ tuning; it is a ‘lead’ guitar. This is reflected in the tuning of each instrument.