Graphical representations can often compress a lot of related information into a single visual whole, making a variety of relationships apparent simultaneously. For composers, a diagram of a set of pitches or intervals (e.g., a scale or chord) can be very helpful in thinking about those pitches and what they can do.
Generally, one is interested in two different kinds of information: melodic relationships (how high or low is each pitch, relatively) and harmonic relationships. These two different aspects of a scale typically result in two different types of diagram: a pitch-line for melodic relations versus a harmonic network for harmonies.
I have used both kinds, but longed for some diagram that would show both harmonic and melodic relationships at once. Then I realized how it could easily be done. The resulting harmonic-melodic diagrams help me to consider scale resources. This article will explain how to make such diagrams, in the hope that you will find them useful too.
Melodic relationships can be illustrated on a pitch-line (analagous to a number line) where the position of each tone on the line, relative to some reference tone, is proportional to the logarithm of the frequency ratio of that tone to the reference. For example, a standard just major scale can be represented by the pitch line in Figure 1:
For harmonic relationships, on the other hand, a scale is often shown as a network, where each link represents a prime harmonic interval, and all links representing a particular prime interval are parallel. Again using our familiar example of Ptolemy's Syntaton Diatonic (and assuming "octave equivalence"), the harmonic network needs links in two different directions for the two prime intervals involved, 3/2 and 5/4, as in Figure 2:
The key to showing both harmonic and melodic relations on one diagram is to let one direction (vertical, say) represent pitch, then to choose the direction and length of each prime harmonic link so that it covers the right vertical distance. I will refer to distances on the diagram in terms of the vertical distance used to represent a double, 2/1. With that as a distance unit, each harmonic link representing the interval 3/2, for example, must extend a vertical distance of log2(3/2) = 0.585 [where log2 means logarithm to the base 2]. The horizontal distance covered is arbitrary, so for a particular type of link one is free to choose any direction other than horizontal, or any length greater than the vertical extent, or any horizontal extent. I have used two different choices, one to diagram arbitrary harmonies, and an alternate choice for strictly two- dimensional harmonies, that is, harmonies involving only two prime intervals.
For two-dimensional harmonies, it is still possible to get the two harmonic directions perpendicular and keep the links the same length. The way it works is: if y1 and y2 are the two vertical extents, then the corresponding horizontal extents are x1 = y2 and x2 = -y1.
Figure 3 (large GIF, 5KB) (or EPS version) diagrams our same boring example. The horizontal lines are for pitch reference, spaced a 2/1 apart. The vertical line links the reference tone 1/1 with its doubles. The harmonic network of the scale is shown three times: the complete version centered on 1/1, and doubles above and below (as much as fit). (One could add to the diagram vertical links connecting each tone with its double(s), but that makes it more cluttered.) The three versions are required to give all the tones of the scale in the middle double, and, as promised, they are in the proper vertical positions.
When more than two harmonic directions are involved, they can't all be perpendicular. To diagram arbitrary harmonies, I chose to have all harmonic links the same length, that of the vertical harmonic link signifying 2/1. (Then every tone one harmonic step away from 1/1, say, is somewhere on the circle centered on 1/1 with its radius equal to one link length.) Then if a particular link covers a vertical distance y, the horizontal distance x can be found from the Pythagorean law: x*x = 1 - y*y. For consistency, I chose to have all links diagonal up to the right, rather than mix left and right.
Yikes! There's that damn example again, lurking in Figure 4 (large GIF, 6KB) (or EPS version), but in the different perspective of this more general representation. Of course, this is not quite as clear as in Figure 3 (large GIF, 5KB), since the harmonic directions are closer together. But it has the advantage that each tone always appears in the same place, regardless of the type of harmony diagrammed. For example, we can see that 3/2 is in the same position in both Figure 4 (large GIF, 6KB) and Figure 5 (large GIF, 5KB) (or EPS version), which shows a very septimal Dorian scale for a change.
Finally, Figure 6 (large GIF, 5KB) (or EPS version) diagrams the 8-tone harmonic scale (8-15), showing how the various prime harmonies give different directions, but all the links are the same length. Of course, a harmonic diagram of an overtone scale is rather superfluous, but it makes the point that this method can diagram harmonies involving any prime intervals, in a consistent way.
Table 1 gives the horizontal and vertical extents of various prime links (in terms of the distance representing 2/1), for the general representation. (Alternatively, to show a 2-D harmony with perpendicular links, use the same y's given here, but use x1 = y2 and x2 = -y1.)
ratio x y 3/2 .811 .585 5/4 .947 .322 7/4 .590 .807 11/8 .888 .459 13/8 .714 .700 17/16 .996 .087 19/16 .969 .248 23/16 .852 .524 29/16 .514 .858 31/16 .299 .954Many variations are possible. Since pitch is shown explicitly, a scale that is not "octave-repeating" can be shown as easily as one that is [see non-octave-repeating example (EPS)]. And a tone not related to any other by a prime harmony could be connected by multiple links, shown dotted. For a more compact diagram, all the horizontal distances could be reduced proportionally. (This would make pitch comparisons easier.) Or as mentioned above, one could choose other harmonic directions (some to the left, perhaps) or varying link lengths, as long as the correct vertical positions are maintained. As for actually drawing these diagrams, I used a computer plotter for accuracy and neatness, but graph paper works well too.
If one were only interested in relative pitches, the pitch-line would be the clearest picture. Or if one only wanted to see harmonic relationships, then a harmonic network using horizontal, vertical, and maybe some diagonal links would be clearest. To see both at once necessitates compromising the clarity of each; the system described here is the best compromise I've found. For purely two-dimensional harmonies, the method with perpendicular harmonic links is clearest. For more general harmonies, the method with equally long links seems the most consistent. I hope that you too find such diagrams interesting and helpful.
Also, some experimentation has shown that these diagrams are most useful for relatively small scales; when there are more than a dozen tones, repeated at several "octaves", the diagram becomes very cluttered [see 7-limit 13-tone example (EPS)].