Diagrammatics of the chromatic scales

The visualizations in the digital Lexicon of moods by Hans Eugen Frischknecht and Jakob Schmid make the "major triad suitability" of scales with 12 notes per octave apparent at a glance, and at the same time make it audible. This special presentation will be discussed below using three selected examples and compared with other ways of visualizing musical scales. This is followed by a few rather unsystematic considerations on the further development of interactive applications for tone systems and tunings.

The diagrams in the lexicon are calculated from the cent deviations from the equal-tempered chromatic scale in 12 semitones (12-EDO = Equal Division of the Octave), which is called the equal-tempered (chromatic) scale in this essay. Different tunings can be compared with each other using cents. The cent scale for pitches and intervals is based on the division of the octave into 1200 equal micro-intervals. The semitones of the equal-tempered chromatic scale therefore measure 100 cents, the major thirds 400 cents and the fifths 700 cents. The major triad in the home position and closest position therefore has the cent values [0 | 400 | 700] if the fundamental is given the value 0 - generally [x | x + 400 | x + 700] if x is the cent value of the chord fundamental in relation to the reference tone c (or a). In contrast, the tuned major triad (rounded to the nearest cent) has the values [0 | 386 | 702] and is in the frequency proportion 4 : 5 : 6. Chords of tones with simple fundamental frequency proportions are perceived as largely fluctuation-free and consonant if the tones involved each have a harmonic overtone spectrum. The figures show that the deviations of the equal-tempered values from the pure values are only 2 cent units for the fifth and 14 cent units for the major third. The different effect of the two triads is mainly due to the different sized thirds and the timbre.

Can the structural principles of a scale be reconstructed from the visualized numerical material? This is not self-evident. As long as the focus is primarily on major triads and their enharmonic equivalents in scales with twelve notes per octave, the representation can be used universally, but at the same time it reduces pitch systems that are conceived in two and higher dimensions to a single dimension.

Leonhard Euler proposed a chromatic scale derived from a two-dimensional pitch network (Euler 1739, pp. 147, 279). Its reproduction in the Lexicon of moods can be seen in Figure 1 above. The keynote symbols, the twelve small red circles, form three sequences of four, each of which lies on slightly inclined parallel straight sections (if the two Cs at the ends of the diagram are mentally placed together):
f-c-g-d / a-e-b-f-sharp / c-sharp-g-sharp-e-flat-b

The characteristic positive slope (2 cents per fifth) of the straight line sections indicates that the fifths in question are pure. The scale is divided into three groups of four tones, each with three pure fifth steps. These are framed in red in the top diagram in Figure 1 for clarity. The vertical positions of the fundamental tone symbols completely determine the interval structure of a scale (this applies to all scales). The major thirds or fifths also completely determine the scale. The third and fifth symbols therefore also form groups of four on slightly inclined straight line segments in Euler. This redundant representation of the information contained in the 12 semitone or fifth steps allows the major triads to emerge as point configurations in the vertical. And thanks to this redundancy, the harmonic construction principles of a scale are revealed.

Fig. 1: In terms of major triadic purity, the universe of 12-tone scales in the lexicon of tunings has the two poles of Leonhard Euler (top) and the equal-tempered scale (middle). Michael Praetorius' almost mean-tone tuning (bottom) is in the middle and reduces Euler's preference for the major-major relationship in favor of quint-related major scales between B flat major and A major with eight almost harmonic major triads.

Euler's scale contains the maximum possible number of six ideal major triads. Because all three tone symbols coincide at F, C, G as well as A, E and B, there is pure intonation [x | x + 386 | x + 702]. Due to its interval structure, it is possible in this scale to make music in the third-related keys of C major and A major with ideal major triads without exception, and the two third-related scales are a perfect major third apart in the frequency ratio 5/4. The other ten "major keys" all have more or less strongly modified triads. In the emphasized B flat major triad, for example, none of the three constituent intervals correspond to the standard of pure tuning, as all three symbols occupy different vertical positions. Triads related to fifths stand directly next to each other in the diagrams. In Euler's scale, the keys of E flat major and A flat/G sharp major stand out most strongly from C major and A major in their interval structure, as none of the triads drawn have the ideal dot shape (the triads of these two scales are framed in purple in the drawing). Focusing on three neighboring chords in each case also shows that G major and B major (highlighted with blue ellipses) have the same interval structure in Euler because the corresponding symbol patterns are congruent. In these two keys, the fifths of the dominant triads are diminished, but their major thirds are pure. The subdominant and tonic have the ideal form.

The scale shown in the middle of Figure 1 is easier to interpret. Here, all twelve triads have the same deviations from pure intonation, and the three lines of symbols run horizontally. This is the equal temperament, the fifth symbols are just under 2 cents below the fundamental symbols (700-702=-2) and the major thirds 14 cents above (400-386=+14). The fifths are therefore only slightly smaller than pure, but the major thirds are noticeably further than their pure-tuned counterparts. Any difference in meaning between the notes b and a sharp cannot be expressed acoustically in this constellation. While enharmonic reinterpretations in equal temperament do not pose an intonation problem, the selection of twelve notes in a purely syntonic (i.e. quint/terce-based) tuning such as Euler's always involves decisions about preferred enharmonic variants. On closer inspection, the somewhat irritating B-flat major triad in Euler's scale turns out to be a sharp-d-f with a diminished fourth a sharp-d and a Pythagorean third d-f, as the representation of Euler's scale in the fifth-octave grid in figure 2 shows. In this representation, pure fifths are arranged horizontally and pure major thirds vertically, so that the ideal major and minor triads correspond to small right-angled triangles, whereas the unusual "B major triad" consists of corner points of the rectangular grid and is constructed quite differently from thirds and fifths.

Fig. 2: Grid representation of the chromatic scale according to Leonhard Euler (left) and Marin Mersenne (right) The ideal major and minor triads of C major and the chord a sharp-d-f or b flat-d-f, which in Mersenne's work consists of a Pythagorean major third and a perfect fifth, are highlighted.

Grid representations were already in use in the 17th and 18th centuries (Muzzulini 2020, 225). They are not directly suitable for tempered tunings. However, they can also be used to represent syntonic scales with more than 12 notes per octave. From the 14th century onwards, various scales with 17 or more notes per octave were proposed. In scales with 17 notes, the five pairs of notes c sharp-d flat, d sharp- flat, f sharp- flat, g sharp-as and a sharp-b are typically realized with two different pitches (cf. the contributions by Martin Kirnbauer, Rudolph Rasch, Denzil Wright and Patrizio Barbieri in the Yearbook of Musicology 2002). Lattice diagrams - including non-rectangular ones - are widely used in the contemporary theoretical literature on tunings, and with additional information can also be used to illustrate mean-tone and other tunings (Lindley, 1987, Yearbook of Musicology, 2002; Lindley, 1993, p. 28).

Unlike in Euler's solution, in Michael Praetorius' scale (fig. 1 below) it is possible to make music in the keys between B flat major and A major with triads that are all closer to the ideal form than their equal-tempered counterparts - at the expense of the remaining four triads. Modulations between major scales that are close to C major and G major in the circle of fifths therefore do not lead to any significant differences in intonation. On the other hand, all fifth steps, with the exception of G sharp flat, are smaller than pure - and noticeably smaller than in the equal-tempered scale, as the fundamental tone symbols form a line descending to the right. The very large diminished sixth g sharp-e flat, which compensates for the other small fifths, is also known as the wolf fifth. The cent scale on the left-hand edge of the picture shows that Praetorius' "wolf" is about a quarter of a semitone larger than the fifth of equal temperament, as the red connecting line between G sharp and E flat rises by around 25 cents.

Fig. 3: Circular diagrams from the 17th century. Left: Diatonic scale with syntonic comma ("schism", 480 : 486 = 80 : 81). The numbers stand for (suitably scaled) string lengths on the monochord. The octave closes at the "Semitonium majus" (288 (576) | 540 (270)) in the ratio 16/15. The rather imprecisely drawn diagram comes from the oldest surviving copy of the now lost original of René Descartes' "Compendium musicæ", which was made for Isaak Beeckman around 1628 (Descartes (1619, fol. 171r). Right: Marin Mersenne's analysis of a chromatic scale in pure tuning (Mersenne 1636, 132).

As in the Lexicon of moods If only octave-periodic scales are considered, their tones are, strictly speaking, octave classes or pitch classes. Circular representations of the tones are also suitable for this purpose. The representation used by Frischknecht and Schmid shows the C major triad at the left and right ends of the diagram. It would also be conceivable to depict it on a cylindrical shell, which would be created by cutting out the diagram and gluing the left edge to the right edge. The chain of fifths then forms a closed line and the proximity of the three major triads in the C major scale becomes apparent. A corresponding transfer of the same information to a circle of fifths "dial", in which the intonation changes are entered in a radial direction, would be a fair, but also unusual representation. In circular arrangements of the chromatic scale, the intervals are often represented as angles. The syntonic comma in the fundamental frequency ratio 81/80 measures just under 22 cents and should correspond to an angle of slightly less than 6° in Descartes' circular diagram in Figure 3 on the left. In contrast to the inconsistent angles in this manuscript, the angles in the Latin first printing correspond quite precisely to the interval sizes, cf. Muzzulini (2015, 197-199), Wardhaugh (2008). Mersenne's circular diagram in figure 3 on the right arranges the twelve notes of the chromatic scale on an almost regular dodecagon. The connections between the notes are labeled with the corresponding frequency ratios. The grid representation in Figure 2 on the right can be derived from this.

In conventional two-dimensional and interactive screen displays, the asymmetry mentioned above could also be compensated for by making the arrangement of the chords cyclically permutable at the touch of a button. Such permutations would also make it easy to determine whether different scales have the same internal structure by directly comparing diagrams, i.e. whether they emerge from each other through transpositions true to the interval if they were displayed on the same webpage. A scale considered by Isaac Newton, for example, emerges from that of Mersenne through a transposition by a pure fifth, which corresponds to a cyclic permutation by one unit. These relationships can be seen directly in a grid representation. If we imagine Euler's chromatic scale transposed downwards by a pure major third, for example, then all twelve points are moved downwards by one grid unit, so that the new bottom row contains the notes D flat, A flat, E flat and B flat. The geometric arrangement of the twelve dots, which represents the inner structure, has not changed. The transposed form makes C major appear as part of a chromatic scale with four B-flats and a sharp, and it differs from Mersenne's solution (figure 2 right) only in the intonation of a single note. In Mersenne, the B flat major triad is in Pythagorean intonation, its fifth is pure, the major third in the ratio 81/64 results from four pure fifths (minus two octaves).

With little programming effort, key figures such as the total mean square deviation of the concrete triads from the tuned triads could be calculated from the rich numerical material of the lexicon. Scales that are separated by transpositions agree in the deviation mentioned, and the different structure of the scale can be deduced with certainty from the difference in the key figures. This allows duplicates and equivalent scales to be determined semi-automatically. The knowledge and visualization of such relationships would be helpful for orientation in the very extensive numerical and pictorial material. James M. Barbour consistently gives the mean deviation and the standard deviation from the same-level scale in cents for his extensive numerical material on twelve-level scales (Barber 1951). It would be more expedient to evaluate the deviations of the twelve major triads from the pure intonation in an analogous way (cf. Hall 1973), in which case small values mean proximity to the Lexicon of moods standing, ideally tuned triads.

As part of the project funded by the Swiss National Science Foundation Sound Color Space of the Zurich University of the Arts were also interactive audiovisual tools to syntonic (i.e. quint/third-octave-based tunings) with grid, circle and spiral arrangements and tested in the context of a virtual museum published, such as the lexicon of moods with synthetic sounds.

Only in recent years has the visualization of music theory and its history as an independent branch of diagrammatology with references to musical iconology increasingly come to the attention of philosophy, aesthetics and musicology (Krämer 2016, 179-193). This essay attempted to point out parallels between historical diagrams of harmonics and contemporary representations, which are only possible in the context of digitalization and the digital humanities. Diagrams reveal the essence of theories and models, and they have didactic potential that seems to do almost without words. The didactic value of visualization has not always been assessed in the same way throughout history. While didactic visualization seemed to play a subordinate role in the 18th century, the idea of the cyclical permutation of tones and harmonies in pitch structures outlined above has its precursors in dynamic didactic tools of the 16th century. For example, elaborately designed books from this period sometimes contain multi-layered diagrams with rotating parts for teaching elementary musical knowledge (cf. Weiss, S. F., 2019).

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