Shortly after starting this blog about a year ago, I realized that a much more efficient use of space could be achieved in the graphics (what Tufte calls increasing the data density) by eliminating redundant traces and condensing the amount of dead white-space. Since the traces for the pure major and minor thirds were literal copies of the main deviation trace, I decided to eliminate them and use the main trace to fulfill their functions of indicating pure thirds. This was easily accomplished by placing the minor third deviation trace blow the central trace and the major third deviation trace directly above it. This is an indication of the normal state of affairs, since the notes which make minor thirds are almost always too low to be pure, and those which make major thirds are almost always too high to be pure. The amount of space between each of these traces and the central deviation trace now represents the degree of tempering of the third; no space means no tempering, i.e. pure. As with the original version, small dots between the traces indicate the amount of tempering, with each dot representing 1/4 syntonic comma. Here’s how equal temperament looks in the new format. Notice that all the notes which make major thirds are too high (third too wide) by about 5/8 comma, and all the notes which make minor thirds are too low (third too narrow) by 3/4 comma:


Additional Changes

Cents Deviation and Fifth Type are now listed below the graphic in order to reduce visual clutter.

I’ve added some thin vertical lines to divide the chain of fifths into three “trients” (the only reasonable trifold version of the word “quadrant”), since each of these three zones often contain three different degrees of the fifth tempering. Sorge used the three trients as three “marking stones” in order to define the basic outlines of the entire system, and D’Alembert also use the notes E and G# as pivot points in his logic. In his case, the logic of the first trient is to descend steeply using narrow fifths in order to arrive at an E which is low enough to make a pure major third above. In the second trient, he makes the fifths less narrow, or in other words, he descends with a shallower slope, making the third E-G# less pure but still good while simultaneously reducing the amount by which the notes must ascend to return to C. In the third trient, this ascent is smoothly accomplished by using a series of wide fifths all of the same size, distributing the traditional wolf fifth evenly over all of them so that no one fifth is unusable. This conceptual division of the circle in three blocks each of four adjacent fifths is often found in original temperament logic, either implied or explicit.

I’ve also added a small graphic to indicate the functions of the most important half steps, the five accidentals and the two natural half-steps E-F and B-C. To understand this graph, remember that any half-step spelled with two different note names should be diatonic (wide), and any half-step spelled by a natural note and its own accidental should be chromatic (narrow). The evaluation is based on the 18th century standard of a 55 comma octave (codified by Telemann), a chromatic half-step being 4 commas and a diatonic 5. Each type is indicated by a dotted line. Even though this measuring standard is not really applicable to earlier temperaments, it is still a useful measure for comparison. I’ve indicated all the half-steps as ascending intervals according to the traditional meantone distribution, that is, 3 sharps and 2 flats, or in other words, 3 chromatic and 5 diatonic. Here’s how my new recommendation for a “default Baroque” temperament looks; it’s a version of Werckmeister’s continuo mollification of meantone using a 55-comma meantone base:

Werckmeister Continuo v55