Graphical comparison of musical scales and mathematical progressions

This article shows the relationship between harmonic ratios and mathematical progressions.

Geometric and arithmetic progressions

Figure 1 shows a graph of the geometric and arithmetic progressions from 1 to 2 with an additional line based on harmonic ratios for comparison. The geometric progression (black) is based on the twelfth root of two (equal temperament) and is an exponential growth equation of this value over the octave interval. The arithmetic progression (red) is based on 1/12 of the octave interval making each step equal to 8.33% of the interval. The arithmetic progression is a linear growth equation with a slope of 0.08333 in the figure. Notes for just intonation are also shown on the graph and can be chosen to correspond to points of a double-linear progression (green line and data points). Harmonic ratios correspond to arithmetic progressions which are parts of an interval (such as half, thirds, fourths, and eighths; corresponding to the perfect fifth, perfect fourth, major third and major second keys of the musical scale).

Double linear progression

It is shown by the double linear progression (Figure 1) that a tuning sequence can be chosen which closely correlates just intonation tuning to the geometric progression of the Equal Temperament Scale. This is apparently the reason why equal temperament tuning makes an acceptable sound even though the geometric progression is not based on harmonic ratios. The ratio of the slopes of the double linear progression also progress by harmonic ratios {1:1 to 4:3 (x1.333) to 2:1 (x1.5)] as identified in the table of data.

There are several double lines that can be drawn to correspond to the geometric progression and are related to the transposition of musical scales to a different key (referring to the unison, 1:1 ratio, as a different key other than C, for example). Single lines can even be drawn for the twelve-key or seven-key scale of the octave, however, the double linear progression provides a close correlation to the major harmonic ratios of the unison note defined for the octave.

The progressions identified in Figure 1 also displays the difference in determination of the mean. The arithmetic mean between any two notes is part of the linear progression. The geometric mean between any two notes is part of the geometric progression. Between the start and end of the octave interval, the arithmetic mean at step #6 is 1.5, but the geometric mean is 1.4142. The curve connecting the geometric means is always below the line connecting the arithmetic means for this situation. A linear progression can approximate the geometric progression as shown by the lines of the double linear progression but is not equivalent. The more steps that a segment (octave) is broken into the closer the linear progression from step to step approaches the continuous geometric function.

Figure 2 shows the progression of frequency versus the sequential keys on the piano keyboard. The frequency ranges from 27.5 to 4186.01 hertz on the geometric (black) progression corresponding to the Equal Temperament Scale (also see piano key frequencies). The arithmetic (red) progression is shown in comparison and progresses geometrically from octave to octave by a factor of two but progresses linearly within the octave for the interval of the octave. For the defined note of A440 the octave progression is by three factors of 2 (8) above this note (equals a last note A of 3520), and four factors of 2 (16) below this note (equals the first note A of 27.5 Hz). It is difficult to discriminate the values for the different progressions for the graph provided. To discriminate between notes it is necessary to display notes within a single octave as in Figure 1. The progression from note to note within an octave is divided into cents so that the necessary discrimination between notes can occur. A linear extension (green) of the octave from A220 to A440 is shown in order to compare octave intervals that progress equally to those that progress geometrically (increasing each octave interval by a factor of two).

A visual comparison of frequency progression to the logarithmic spiral is shown in Figure 3, Spiral Progression of the Octave. The octave of 12 notes (7 white keys, 5 black keys) can be represented in a circular sequence by increasing the angle between successive notes by 30 degrees. Thus, one octave equals 360 degrees (12 x 30 degrees). The values are normalized to a single note such as A440 equal to one and progress above and below this value. In the figure, the geometric progression of 4 octaves that are shown progresses from a value of 0.5 to 1 to 2 to 4 to 8 for the start (or end) of each octave. The Just Intonation Scale (double linear progression) can be chosen to be very close to the Equal Tempered Scale as can be seen by superposition of the two scales. The arithmetic progression (Archimedean Spiral) over the same number of intervals (notes) progresses from a value of zero to 1 to 2 to 3 to 4 for the start (or end) of each octave. Initially, the arithmetic progression builds up faster but the geometric progression eventually surpasses the increase made by the arithmetic progression (see also, exponential growth).

The magnitude of each note (the distance from the center point) in the spiral progression increases the same as the progression of the defined scale (i.e. 1 to 2 can also be A220 to A440). It can be shown using trigonometry and the Pythagorean Theorem that the magnitude of each note is equal to the square root of the coordinate formula of X-squared + Y-squared. For each point the X-coordinate is defined as the magnitude times the cosine of the sequence angle and the Y-coordinate is defined as the magnitude times the sine of the sequence angle. The math and the resultant plot of the data results in the logarithmic spiral shown. (Reference "e: The Story of a Number"; chapter 11 by Eli Maor for comparison.)See also:
*arithmetic progression
*geometric progression
*equal temperament
*twelfth root of two