The ThumMusic (Pitch) System

Introduction

The ThumMusic (Pitch) System retains the Common Western Music System's focus on pitch, and is therefore entirely compatible with its staff notation, chord names, etc. It combines two past inventions – one from the early 1700's and the other from the late 1800's – to consistently expose the patterns of intervals among these pitches.

To understand this partial solution, one must first understand the concept of isomorphism, isomorphic keyboards, and the tonnetz, described below.

Isomorphism

The term “isomorphic” is understood, in this document, to mean “being of similar shape, form, or structure”.[8] It is derived from the Greek words “iso-”, meaning “same”, and “morph”, meaning “shape” – hence “same shape”. As previously described, the pattern of intervals that defines a given scale has the same shape – ie, is “isomorphic” – in all keys, as is the pattern of intervals that defines a chord built on a given mode of that scale, an arpeggio of that chord, a melody, etc. Isomorphism is thus a central concept in music (although the term is not often used in this context).

The inherent isomorphism of music is particularly pronounced in equal-temperament tunings, but is also a useful concept in non-equally-tempered tunings (such as meantone and Just Intonation). The concept of isomorphism is also applicable to tunings that divide the octave into more or fewer than twelve semi-tones. The following discussion will, however, assume the use of the 12-tone equal-temperament scale unless specifically stated otherwise.

What Is an Isomorphic Note Layout?

A note-layout is a two-dimensional pattern of notes, presumably associated with a particular arrangement of note-controlling buttons. In an isomorphic note-layout, any two elements that together sound the same musical interval also have the same spatial interval relative to each other.[9] Thus, on an isomorphic keyboard, any given musical interval has the “same shape” wherever it occurs.

If each individual musical interval has a consistent shape, then every given sequence (melody) or combination (harmony) of musical intervals has a consistent shape, too. This means that on an isomorphic keyboard instrument, every given scale, arpeggio, melody, chord, chord progression, or any other sequence and/or combination of intervals has the same fingering in every key.

Figure 1 (below) is an image of the specific isomorphic note-layout used in the ThumMusic System, which henceforth will be called the ThumField^™ layout. The spatial arrangement and shape of the note-controlling buttons in Figure 1 is that of Thumtronics' new electronic musical instrument, the Thummer^™.

Figure 1

A two-octave ThumField also fits conveniently on a standard computer keyboard's button-arrangement, as shown in Figure 2.

Figure 2

The sequence of intervals that defines the major scale is shown in Figure 3 (below) for the keys of C, F, and G. Any sequence of intervals – not just the major scale – is the same in all keys on a ThumField.

Figure 3

The same is true for combinations of intervals – chords and chord progressions. Figure 4 shows the shape of the diatonic tertian triads in the keys of C Major and Gb Major (the tonic is circled in red).

Figure 4

In Figure 4, each diatonic tertian triad is indicated with a triangle (except for vii¡, which is indicated with a black diagonal line). The vertices of the triangles point towards the notes in that triad. Look closely at the triangles labelled I, IV, and V. They all have the same shape: an upward-pointing triangle.

Why? Simple. If you imagine a horizontal line connecting C and E, a diagonal line connecting that E to G, and then another line connecting C and G, then you've just mentally drawn the “shapes” of a major third, a minor third, and a perfect fifth, respectively. The resulting triangle is the “shape” of a major triad, everywhere on a ThumField.

Likewise, if you draw lines between D&F, F&A, and D&A, you'll have drawn the “shapes” of a minor third, major third, and perfect fifth, resulting in the downward-pointing triangular “shape” of the minor triads ii, vi, and iii.

The diminished triad is a stack of two minor thirds, forming a straight diagonal line which bisects the stack of chords. This is shown above with the black line from B through D to F.

Isomorphic Note-Layouts from History

Paul von Janko, a German, patented two such isomorphic keyboards (German patent no. 25282 in 1883, and no. 32138 in 1885). The Chromatic Button Accordion is usually configured with one of two other such layouts, the C-System or the B-System (http://www.thecipher.com/chromatic-accordion-cipher.html).

Kaspar Wicki, a Swiss, patented an isomorphic arrangement of note-controlling devices in 1896 (Swiss patent no. 13329). Not knowing of Wicki's patent, Brian Hayden, an Englishman, re-patented Wicki's note-layout in 1982 (GB Patent no. 2131592) – a patent that was clearly invalid, since the expiration of Wicki's earlier patent released it to the public domain forever afterward.

Wesley, an American, patented yet another isomorphic note-layout as recently as 2002 (US Patent no. 6,501,011).

The world's patent offices should stop accepting patents such as Wesley's on “new” isomorphic note-layouts, because they have all been analysed and their properties are well-known. There's a simple rule for generating all possible isomorphic note-layouts for a given T-tet tuning, the discussion of which is beyond the scope of this paper.[10] Suffice it to say that for an T-tet meantone tuning, there are ((T*2)+1)^2 possible isomorphic layouts in which notes sounded by a given button's rightwardly-adjacent and upandleftwardly-adjacent neighbours are no more than an octave away from the note sounded by the given button. Table 2 shows the number of isomorphic layouts for some meantone tunings.

Table 2: Isomorphic Note Layouts for T-tet Meantone tunings

Of all of these isomorphic note-layouts, the Wicki/Hayden layout is the most suitable for a hand-held electronic musical instrument such as Thumtronics' Thummer. This note-layout, the ThumField layout (shown in Figure 1 and Figure 2 above) is the basis of the ThumMusic System.

Properties of the ThumField

The ThumField note-layout can only be mapped to an arrangement of buttons with are of appropriate size, shape, and spacing. Two such button-arrangements are shown in Figures 1 and 2 above.

Inspection of the ThumField shown in Figure 1 reveals that the natural notes are in the middle, with the flats on the left and the sharps on the right. The notes of the diatonic scale form a dense column, with the pentatonic scale forming a subset thereof. It has compact aspect ratio, allowing three full octaves of notes to be easily spanned by a single hand's fingers. With two such ThumFields – one for each of a musician's hands – the musician can play notes from all six octaves simultaneously.

From any given root, the perfect fourth is up-and-leftwardly adjacent, while the perfect fifth is up-and-rightwardly adjacent. The buttons' size, shape, and spacing are optimized to make it easy to play the buttons that sound these common intervals with a single fingertip. Having the major second also adjacent facilitates playing sus4 and sus2 chords with a single fingertip.

Minor seconds are diagonally-separated, allowing a simple rocking motion of the hand to play a chromatic scale. Major seconds are adjacent. The layout strikes a convenient balance between melody and harmony. The layout is vertically symmetrical around D, which turns out to be a surprisingly important and useful property.

The Tonnetz

Another under-utilized tool of music theory is a geometric construct known as the “harmonic lattice” or “tonnetz,” first described by the mathematician Leonhard Euler c.1730. The tonnetz has one axis along which successive perfect fifths are indicated, and – in standard practice – a substantially orthogonal axis along which major thirds are indicated. Minor thirds can be connected within the plane formed by the first two axes, forming a geometric network of triangles, each representing a major or minor triad. The tonnetz is an excellent tool for visualizing harmonic relationships – triads, chord progressions, key modulations, and the like. However, it is rarely used in music education (at least in English-speaking countries), in part because it is hard to relate the tonnetz to traditional staff notation, chord names, and musical instruments.

The ThumLattice

The ThumMusic System (Pitch-Names) realigns the axes of Euler's tonnetz to match the ThumField. Such a ThumField-aligned tonnetz can be called a ThumLattice™. The ThumLattice is a convenient tool for presenting musical information in a geometrically-structured way.

Chord Progressions on a ThumLattice

In Figure 5 (below), the pitch-labelled circles align with their respective note-controlling buttons on a ThumField. You will note that the shapes of, and geometric relationships between, the chords in Figure 5 precisely match those shown in Figure 4 (above) for C Major. The only difference between the ThumLattice and the ThumField button-pattern is that the buttons in between the button-pairs that sound major thirds are missing. For example, between C and E on the ThumLattice in Figure 5 (below), the ThumField button controlling D is missing from the ThumLattice.

Figure 5 (below) shows the I-IV-V chord progression in C Major.

Figure 5

Figure 6, shows the I-ii-V chord progression, also in C Major. It should be noted that the two chord progressions in Figure 5 and Figure 6 have the same shape on the ThumLattice.

Figure 6

The chord progression shown in Figure 7 (below) also has the same shape as those shown in Figure 5 and Figure 6, although in this case it shows the I-IV-bvii chord progression.

Figure 7

In Figure 8 (below), a related chord progression – the bVII-IV-I – is shown. It has the same shape as those shown in Figure 5 through Figure 7 (above).

Figure 8

The chord progressions in Figure 5 through Figure 8 are all closely related, and this relationship is clearly indicated by their similarity in shape.

A more complex chord progression is shown in Figure 9 (below).

Figure 9

In Figure 9, blue arrows represent major thirds, while green arrows represent minor thirds. Red arrows show root movement in a chord progression in C Major. The chord progression starts with the tonic triad (the I chord, CEG — from the root, follow the blue and green arrows to higher degrees), moves up a fifth to the dominant (V, GBD), up another fifth to the supertonic (ii), and then returns “home” through the V to I. The progression then wanders into subdominant territory, from I to IV, then to ii, vi, iii, V, and then finally back home to I again.

This chord progression is simply a series of loops around the ThumLattice — and so are all other tonal chord progressions.[11]

The rules of tonal chord progressions on ThumLattice are simple: roots may progress only to roots that are either (a) adjacent along the lines of the lattice, or (b) two steps higher up the same line of perfect fifths; also, (c) octaves are equivalent. All rules are made to be broken, of course, and these rules are no exception – as shown by the I-bVII-IV progression in Figure 8, which descends by two perfect fifths, and is rarely used. However, these rules (a) are simple, and (b) can be derived from inspection of the ThumField itself. They are as concrete and tangible as the note-controlling ThumField itself.

Relative Chords on a ThumLattice

It is often difficult for students to grasp the concept of “relative minor,” or similarly, “relative major,” because the concepts are so abstract.

However, these concepts can be shown to arise directly from the concrete reality of a ThumField, using a ThumLattice.[12]

In Figure 10 (below), C is taken as the tonic. To the right of C is found the C Major portion of the tonnetz. Each diatonic tertian chord is shown as a triangle (except for the vii¡, shown as the line BDF). The relationship between C Major's major triads and their relative minors is graphically consistent, and easily understood by inspection. The large dotted red arrows in the C Major region of the ThumLattice point from the “Relative minors” to the chords to which they are related.

Likewise, to the left of C is the C minor portion of the ThumLattice. Here, the arrows point from the relative majors to their related chords. In both cases, the dotted red arrows are pointing back to the line of perfect fifths which contains the tonic.

Figure 10

Summary

The ThumMusic (Pitch) System (1) combines the Wicki/Hayden isomorphic note-layout with (2) an appropriate button-arrangement to produce a ThumField, and (3) aligns a tonnetz with a ThumField to produce the ThumLattice.

Conclusions

The geometric consistency of the ThumField and ThumLattice makes music easier to teach, learn, and play, while being entirely compatible with the Common Western Music System's traditional pitch names, chord symbols, and staff notation. With a standard computer keyboard as its ThumField, the ThumMusic (Pitch) System is a powerful and easily-deployed addition to traditional music education.

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