The graphics presented here are the result of 25 years of playing around with different ways of representing temperaments. Initially, I only wanted to represent the qualities of the major and minor triads in the various keys, and for this purpose alone, a simple “deviation from pure” chart is quit adequate. However, what I was really trying to achieve was something far more instructive, a true graphical representation of the multidimensional interrelationships which both dictate and result from the structure of a temperament. In other words, both the causes and the symptoms. My self-imposed criteria were that the graphical method should be intuitive in that the visual representation mimics reality in the following ways:

1. The “raw material” of any temperament structure, which are the variations in the precise height of each note, should be indicated by higher or lower positions on the graph.

2. The variations in interval size which result from these height variations, that is, wider or narrower, should be represented by wider or narrower distances between two defining points on the graph, and not by some quantitative indication of abstraction of the interval size, such as a mere bar or column representing cents from pure or degree of tempering in comma fractions.

3. The graphical relationships between the notes themselves should suggest their actual relationships, for example, the major third C-E should be represented as a vertical structure with C below and E above, and the Eb of minor third C-Eb should be located between them.

4. The inevitable circulating nature of any dodecatonic (i.e. limited to a gamut of 12 tones) temperament should be implied if not illustrated, even when the logic of the temperament itself is not circulating. On the keyboard, there is no physical barrier to circularity, and musicians can – and did – easily find themselves stumbling across the “gap” (both in thirds and fifths), either intentionally, for dramatic effect, or by necessity, due to inescapable practical requirements (transposed organ parts, for example).

5. The graphical system should be capable of being manipulated in a way which mimics the task of the tuner, i.e., one should be able to grab a given note wherever it appears and move it up or down, and the result should be instantly transferred to all harmonic relationships involving that note.

After reading the wonderful books of Edward Tufte (an absolute must for anyone designing any type of graphical interface!), through which I became familiar with the brilliant and inspiring work of Charles Joseph Minard (more info here and here), the method presented here finally emerged as the fruit of a long process of trial and error. I would hope that viewers would take the time to let the logic of its structure sink in a bit, for even after dealing with temperaments for more than 30 years, I myself have gained additional insight into the topic by manipulating these graphs. They are now my standard tool for both the design and evaluation of dodecatonic temperaments.