ThumMusic - ThumMusic PLUS

While the ThumMusic (Pitch) System exposes the geometry of tonal music, it does not resolve any of the five inconsistencies of CWMN. To do that, we must abandon the Common Western Music System's focus on pitch, and focus instead on intervals.

To find the tools that can make this possible, we will need to reach back into the past once again. Combining the ThumField and ThumLattice with three more “innovations” – one from the mid-1700's, one from the mid-1800's, and one from the early 2000's – we will at last have a completely isomorphic system for presenting and controlling musical information: the ThumMusic PLUS System.

The Chromatic Staff

The next “innovation” that we'll borrow from the past is the chromatic staff, first proposed by Roualle de Boisgelou in 1764.[15] It has seven equally-spaced horizontal lines, with six spaces in between them, providing thirteen unique vertical locations altogether. This is precisely enough locations to denote each of the twelve notes of the chromatic scale, plus the octave. De Boisgelou placed C on the top and bottom lines of his chromatic staff, as shown in Figure 17.

Only the outer and middle lines are essential to this system. A three-line variation, with the four non-essential lines erased except for ledger lines, works equally well, and is far easier to read, as is shown in Figure 18.

When the bottom line of one such staff (denoting C) overlies the top line of another such staff (also denoting C), obscuring or replacing the line beneath, the result is two “stacked staves,” as shown in Figure 19.

To transpose a piece written in on a chromatic staff up a minor third (three semi-tones), the whole pattern of notes is simply shifted up by three vertical locations. The pattern's shape stays the same, no matter how many semi-tones it may be shifted under transposition. For example, Figure 20 shows the song “Amazing Grace” notated on a variation of the chromatic staff in the key of C.

Figure 21 shows same song notated on the chromatic staff in the key of F. In this key, all of the notes fit within the range of a single staff. The pattern of notated intervals is consistent under transposition. That is, the chromatic staff is isomorphic.

If the chromatic staff is drawn the same way in all octaves – as it is in Figure 19 – then music notated in any octave will look the same.

Thus, as a direct consequence of its isomorphism, using a chromatic staff overcomes three of the five inconsistencies of traditional notation – inconsistency of clefs, octaves, and interval spacing – leaving inconsistency between keys and tuning unresolved.

Isomorphic Button-Fields and the Chromatic Staff

Isomorphic button-fields are a perfect match with De Boisgelou's chromatic staff. On such a staff, the pattern of intervals does not change when transposing a song from key to key – it just moves to a new position on the staff, the same way the isomorphic button-field player's hand moves to start the same fingering pattern on a different button.

However, there is still a problem. To transpose a song to another key, one must move notes on the staff and one's hand on a button-field. Even with an isomorphic button-field and notation, this impediment – “inconsistency between keys” – remains.

To solve it, we must find a way to name intervals, rather than pitches. Fortunately, such a system has been available for almost a thousand years.

Tonic Solfa

In c.1025 A.D., Gudio d'Arezzo, a Benedictine monk, invented the first musical staff. He also invented a system for naming intervals using short syllables such as Ut, Re, and Mi. This system – solmization – was revised by the English music teacher Sarah Glover in the 1830's, producing a system called Norwich Solfa. John Curwen revised this system further in the 1840's, producing the Tonic Solfa System. Tonic Solfa was widely taught and used in the latter half of the 1800's, especially in teaching congregations to sing hymns.

Tonic solfa[16] is also called “moveable Do,” indicating that the pitch of the note called Do can “move” on transposition. There is a variant called “fixed Do,” in which the same syllable-names are used to denote fixed pitches. Fixed Do is just an alternative convention for naming pitches – not intervals – used exactly as English-speaking countries use A, A#, Bb, etc. to name pitches.

Tonic solfa remains an important part of modern vocal music education, especially when using the methods of Zoltan Kodaly or Emile Jaques-Dalcroze.

Tonics, Modes, Major, and Minor

A brief discussion of modes, major, minor, and their relationship to the tonic is in order, to avoid potential confusion.

Figure 22 shows a circle divided by twelve lines around its perimeter, just as a clock face is divided to indicate the twelve hours of the day, with noon & midnight sharing the top spot.

In Figure 23, the twelve divisions correspond to the division of the octave into the twelve chromatic intervals, of which the diatonic degrees are labelled with their tonic solfa names.

Some of the intervals between diatonic scale degrees are two semi-tones (major second) wide – Do-Re, Re-Mi, Fa-So, So-La, and La-Ti – whereas some are only one semi-tone (a minor second) wide – Mi-Fa and Ti-Do. The particular pattern of intervals shown in Figure 23 is the defining characteristic of the diatonic scale.[17]

Different scales – diatonic, pentatonic, double harmonic, Neapolitan, etc – include or exclude a different subset of the chromatic notes, thus producing a different circular pattern of intervals between included notes.

The “modes” of the diatonic scale always have the diatonic scale's circular sequence of intervals. The only difference between the modes is the note on which the mode starts its journey around the scale's circle of intervals. The starting – and therefore ending – note of a mode in this circular journey is the tonic of that mode.

For example, the red arrow in Figure 24 shows Do-mode (Ionian) starting on Do and sweeping around the cycle.

Likewise, the red arrow in Figure 25 shows La-Mode (Aeolian) starting on La and sweeping around the cycle.

When teaching music using the ThumMusic PLUS System, the various Greek names for the modes should be ignored in favour of Do-mode, Re-mode, Mi-mode, etc. This naming system clarifies the relationship between scales, modes, and chords. The Greek names can always be memorized later, once the theory is understood using the more-mnemonic tonic solfa names.

In this document, the phrase “major key” always means “Ionian mode”, and the phrase “minor key” always means “Aeolian mode”. Other modes – Dorian, Phrygian, and Locrian – are sometimes called “minor” modes because the interval from their root to their third is minor. This “simplification” actually confuses the issue by treating different modes as being the same, when they are not.

The discussion above uses the diatonic scale throughout, but it should be clear that the concepts therein are equally applicable to other tonal scales. These other tonal scales and their modes will be discussed later in this document.

Solfa-Labelled ThumFields

A ThumField can be labelled with tonic solfa interval names as shown in Figure 26 (below), rather than with pitch names as shown in Figure 1 (above).[18]

This places the intervals of the current diatonic scale on the white buttons, and the chromatic notes on the black buttons. Using such a solfa-labelled ThumField, musical structures such as the diatonic scale and the I-IV-V chord progression don't just have the same shape in every key, they are played on the same buttons in every key. Electronic transposition can move the pitches to the correct buttons, maintaining this consistency across all keys.

The ThumFields in Figure 1 and Figure 26 will sound the same pitches, if the lowest Do-button in Figure 26 is associated with Middle C.

Associating solfa names with a ThumField's buttons focuses its player on intervals rather than pitches. Each simple interval has a unique solfa name, with no accidentals, key signatures, or pitch names to confuse matters.

The ThumLine Staff

Combining tonic solfa with a chromatic staff yields the ThumLine^™ staff, shown in Figure 27. To the left of the clef symbol, the staff is fully-lined; to the right of the clef symbol, the staff is in three-line form. The tips of its crescent-shaped clef symbol always point to the staff's Do-line.

Instead of having each vertical location indicate one of the chromatic scale's pitches, as de Boisgelou's chromatic staff does, the vertical locations on the ThumLine staff denote the chromatic scale's simple intervals. For example, the ThumLine staff has a unique vertical location for Do – but not for C, which can be anywhere on the staff, depending on its interval from the tonic of the current key.

In Figure 28, a ThumLine staff is shown with the solfa intervals indicated by each unique vertical location labelled with their solfa names. (The name-labels are not part of the staff.)

The thirteen unique vertical locations of the staff are labelled, from bottom to top, Do, Ra, Re, Me, Mi, Fa, Se, So, Le, La, Te, Ti, and Do. Do is indicated by a solid line, whereas Se is indicated by a dashed line.

Thus the 12-tet ThumLine staff's thirteen unique vertical locations (lines and spaces) from the bottom Do-line to the top Do-line uniquely represent each 100-cent interval from 0 to 1200. The vertical scale is consistent in cents.

ThumLine's crescent-shaped clef symbol has very specific proportions, which can be expressed in cents. A single ThumLine staff is 1200 cents tall, and so is the bounding rectangle into which the ThumLine clef symbol fits. This bounding rectangle is 741.7 cents wide, such that its ratio of height to width is 1.618 (phi). The crescent shape is 200 cents wide at its widest point (which is the vertical centre of the ThumLine staff, as Se), using the same cent-scale horizontally as vertically.

Scale Indicators

The chromatic staff is not biased towards any one scale. It is as convenient to notate music in (for example) the Gypsy Minor scale as it is in the diatonic scale. To clarify the scale that's being used melodically at any given point, a stack of scale indicators is used.[19]

In Figure 29, each note of the diatonic scale is indicated with a small round dot vertically centred in the vertical location associated with that note. These “scale dots” should have a diameter of 75 cents. This proportion ensures that there is a small gap between dots that are centred on adjacent vertical locations (e.g., Ti and Do). Scale dots should be hollow or filled (i.e., white or black) to reflect the coloration of the corresponding note-controlling buttons on a ThumField.

Tonic Indicators

Atonal music, by definition, has no tonic (tonal centre). For atonal music, no tonic indicator is necessary. The use of solfa syllables to name the chromatic intervals need not imply any tonality. A chromatic staff is ideal for 12-tet atonal music. The rest of this discussion presumes that the music being notated is tonal (has a tonic).

As shown in Figure 30 (below), the tonic indicator is 100 cents tall, and its width-to-height ratio is 1.618:1 (phi), so its width is 162 cents (using the same scale horizontally as vertically). The indicator's colour or fill state should be white/hollow if the corresponding note is diatonic, else black/filled (the same rule as with scale dots).

Authentic and Plagal Views

Consider Figure 31 (below). It shows a stack of ThumLine staves with a range of two octaves centred on Se. (The tonic solfa names are included in the figure for the convenience of the reader of this document, but they are not part of the ThumLine staff, and should not generally be shown.)

If one were to erase the top half-octave and the bottom half-octave of the stacked staves in Figure 31 (above), one would be left with a staff that looked exactly like the one in Figure 32 (below).

On the other hand, if one erased the top or bottom octave of the stack of staves shown in Figure 31 (above), one would be left with a staff that looked like that shown in Figure 33.

The essential point here is that these are just different “views” of the same ThumLine staff, with the same crescent-shaped clef symbol in both cases.

Popular melodies often have a range of roughly one octave, and the tonic tends to appear either at the centre of that range or at the outer bounds of it. If the tonic appears in the centre of the melody's range, ThumMusic refers to the melody as being “tonic-centred.” If the tonic appears at the outer bounds of the melody's range, then ThumMusic refers to that melody as being “tonic-bounded.”[20] Thus, the different “views” of the ThumLine staff shown in Figure 32 and Figure 33 commonly result from notating music that is Do-bounded or Do-centred, respectively.

Note that there are 24 possible views of the ThumLine staff in 12-tet: an Xx-bounded view and an Xx-centred view for each chromatic note Xx. However, because most Western music is either major (Do-mode) or minor (La-mode), only four views are common. Furthermore, the Do-centred view is adequate for notating La-bounded music, and the Do-bounded view is adequate for notating La-centred music.

In any case, the different “views” of the ThumLine staff are a convenience only, allowing unused portions of the staff to be erased to maximize the number of staves that can be packed into a given vertical space. This consideration is of great interest to music publishers – especially publishers of “fake books” which have to cram a thousand songs into a single (very thick) book.

Combining Scale and Tonic Indicators

Figure 36 shows the Gypsy Minor Scale (which is actually the So-mode of the Double Harmonic scale).

An experienced musician can derive the scale and tonic of any given song notated in Common Western staff notation by scanning its chords and/or melody. The use of scale and tonic indicators makes this same knowledge available to less-experienced musicians, in a compact, unambiguous, and general-purpose form.

Pitch Class Indicators

Figure 37 shows the song “Amazing Grace” notated in ThumLine's Do-centred view. The small diamond-shaped symbol placed on the Do-line indicates that Do is the tonic. This tells the reader that the song is to be played in Do-mode of whatever key.

Figure 38 shows the same song, notated in exactly the same way, with the note-name C adjacent to the tonic's vertical location, indicating that the pitch class associated with Do is C. This note-name is a “pitch class indicator.” It tells the player that the song is to be played in the Do-mode of C, that is, C Major.

Figure 39 shows the same song; the note-name F placed on the Do-line indicates that the song is to be played in F Major.

Figure 40 shows the same song; the note-name F# placed on the Do-line indicates that the song is to be played in F# major.

The notation of the notes in the song “Amazing Grace” is identical in Figure 37 through Figure 40. The pitch class indicator is the only thing that changes. This shows that ThumLine is consistent across key signatures (“auto-transposing”), in addition to being consistent across clefs, octaves, and intervals as previously discussed.

ThumLine Example: MNMA's Notation Test

Figure 41 (below) shows a La-mode scale, in rhythm, notated on two and a half stacked ThumLine staves.[21] This example is drawn from the second task of the MNMA's Notation Test.[22] The test requires the transcription of the G-Minor scale to the proposed system. This transcription generalizes the result by placing a tonic-indicating diamond in the La-space.

The G minor scale could be specified uniquely by placing a pitch-class indicator to the left of the tonic indicator as shown in the Amazing Grace examples above. Alternatively, the pitch-class indicator G can used as tonic indicator, as shown in Figure 42 (below). This usage is discouraged, however, because it forces the pitch-class name to be quite small.

The octave could be specified by placing octave numerals after the G, indicating that lowest instance of the La-space should indicate the specific pitch G₅ (note 55), as shown in Figure 43.

It should be emphasized that one of the major benefits of ThumLine is in not having to specify a key signature or register. These choices can be made – by the sole musician, by the group, or by the conductor – at the last possible moment, taking into account all of the issues that might affect the choice.[23]

ThumLine Example: Greensleeves

The song “Greensleeves” is shown in ThumLine in Figure 44, in which the tonic indicator in the La-space to the left of the crescent clef indicates that the song is to be played in La-mode (minor) in an unspecified key.

The tonic indicator could be replaced by the letter A to indicate A minor, or A4 to represent a specific note A, etc. Generally speaking, the location of the tonic – note its pitch – is all that should be specified.

Key Change Indicators

In a key change, the pitch associated with each ThumLine staff location changes, up or down, by a specific musical interval. This can be indicated in ThumLine using key change indicators, such as those shown in Figure 45 (below). The first change is down a minor second (-m2); the next, up a major second (+M2); etc.

Figure 45

Each key change indicator consists of:

á a “before” dot, indicting a note's position before the key change

á an “after” dot, indicating the “before” note's position after the key change

á an arrow indicating the direction of pitch movement

á the fewest ledger lines necessary to clarify the positions of the above dots, and

á a pair of parallel vertical lines bounding the above.

It should be understood that the labels appearing below the key change indicators in Figure 45 – e.g., -m2, +M2, etc. – are only present to clarify the meaning of the figure. They are not to be included in key change indicators.

Further, while all of the key change indicators in Figure 45 use Do as their “before” note, this need not necessarily be the case. Generally speaking, in priority order, first, the “before” note should be chosen to ensure that the entire length of the indicator will fit on the staff without “wrapping around” the octave; second, the “before” note should be the tonic, if any; and third, the instance of the “before” note should be chosen to make the direction of the key change as clear as possible.

ThumLine and Solfa-Labelled ThumFields

There is a one-to-one correspondence between ThumLine staff locations and the buttons of ThumField such as that shown in Figure 26. Having transposed such a keyboard into the desired key and octave, each unique vertical location of the ThumLine staff indicates a specific button on the ThumField (although duplicate enharmonic note-controlling buttons may be present, providing multiple fingering options). This one-to-one correspondence is essential for sight-reading.

The ThumLine staff and the ThumField note-layout are both geometric systems for arranging tonal note-patterns. It is therefore reasonable to expect that each has a geometric relationship to the other – and so they do.

Figure 46 shows the geometric relationship between ThumLine and ThumField. The buttons in rows that include Do all fall on staff lines; the buttons in rows that do not include Do all fall on staff spaces.

Figure 46

The staff crosses the keyboard at an angle of about 16¡. This is the angle needed to connect enharmonic notes in 12-tet.

Mirrored ThumFields

Although not shown, it is easy to imagine the mirror-image of a ThumField such as that shown in Figure 26, in which the pitch of major seconds increases from right-to-left instead of left-to right. One can further imagine that the version shown in Figure 26 would be associated with one of the player's hands, and that its mirror-image would be associated with the player's other hand. Since a person's hands are mirror-images of each other, mirroring a pair of ThumFields can provide consistent fingering to each hand.

ThumChord Symbols

The Common Western Music System includes many different chord-naming systems – classical, jazz, Roman Numeral, Nashville Numbering, and more. ThumChord^™ is yet another chord-naming system. Its unique advantage is that it provides a direct visual link between the chord symbol and the shape of the chord on a ThumField.

ThumChord uses a standard ASCII symbol to indicate a given musical interval, and appends a string of these interval-symbols to a root name to describe the stack of intervals above the root. Each interval symbol is chosen to be more or less mnemonic for either (a) the geometric shape of said diatonic interval on the ThumField button-field (as described below), or (b) the number of equally-tempered diatonic minor seconds in said interval. [24]

Table 3 shows the basic ThumChord symbols, from minor second (1) to perfect fourth (5).

Table 3: ThumChord Interval Symbols

1	2	3	4	5
.	:	\	-	\|

A major triad on any root named “Xx” would be indicated with the symbol string “Xx-\”, in which Xx names the root note, “-” indicates that the next note is a major third higher than the previous, and “\” indicates that the next note is a minor third higher than the previous. Examples would include Bb-\ and Do-\. Similar strings can be constructed for all other diatonic tertian chords, added-sixth chords, and sus2 and sus4 chords, among others.

Table 4 (below) shows all possible tertian 7 chords. As should be clear from comparison of the ThumChord and ThumField columns, each chord's shape on the ThumField is strongly suggested by the string of ThumChord symbols. The shape of a given chord is the same on the ThumField for every root in every octave, scale, mode, key, and tuning.

Table 4

Chord Name	ThumChord	ThumField
Diminished	\\\
Half-Diminished (m7b5)	\\-
Minor	\-\
Minor (Major 7)	\--
Dominant	-\\
Major	-\-
Aug. 5th + Major 7th	--\
Aug. 5th + Octave	---

Inversions are indicated by prefixing the root with the degree that is in the bass (e.g., 3Xx-\ for “first inversion” of a major triad, 5Xx-\ for “second inversion”, and so on for extended chords).

Because tertian chords are so common, and the tertian chord on any given degree of a given tonal scale is always the same (by definition), the chord symbol “Xx” – where Xx is a degree of the currently-indicated scale, without any subsequent string of interval symbols – is taken to mean “the usual tertian chord on Xx in the current scale.” For example, in the usual case where a diatonic scale is indicated, the chord symbol “Do” would mean “Do-\” (a major triad on Do), while “So” would mean “So-\\” (a dominant 7 chord on So), and “Re” would mean “Re\-“ (a minor triad on Re). If it is intended to use Re as the dominant of the dominant, or any other such non-diatonic chord, the whole string should be spelled out (“Re-\\”) to remove any ambiguity.

Dyads composed of a root and a higher degree of the root's mode can be indicated with a chord symbol of the form “XxN” where Xx is the root and N is the added degree. Hence “Do5” would indicate a “power chord” on Do, whereas “So5” would indicate a power chord on So. (Note that “Ti5” would indicate an added diminished fifth, since the fifth degree of Ti-mode is a diminished fifth above Ti.)

An arbitrary number of tonic solfa names can be strung together. By convention, the notes are listed in ascending order by pitch. For example, DoMiSo indicates the major triad on Do in root position, whereas MiSoDo indicates the major triad on Do in first inversion.

Functional Harmony

Harmony is functional as well as structural. In both major and minor keys, the tonic chord is a chord of rest; the dominant is a chord of tension. In Do-mode (major), the tonic is always Do, and the dominant is always So. In La-mode (minor), the tonic is always La, and the dominant is always Mi. Thus the solfa names of the chords' roots, combined with the tonic indicator and scale dots, tell the musician something meaningful about their role in functional harmony. For functional analysis, the Common Western Music System requires the use of a separate notation – using Roman numerals for each degree of the scale – because pitch-names tell a musician nothing about their function in a given piece of music. Isomorphic solfa names do.

Figure 47 shows a three-octave ThumField in which the diatonic scale's tertian sequence is extended from the lowest occurrence of Do upwards for two octaves. This tertian sequence – the diatonic “Circle of Thirds” – is the same for all modes of the diatonic scale (although for other modes the cyclic sequence will start in a different place), and shows the order of major and minor thirds in all of the diatonic tertian chords.

Figure 47

Consider, for example, a chord progression that has reached So in the diatonic scale. It doesn't matter whether So is the V of Do-mode or the VII of La-mode – the diatonic tertian sequence rooted on So is always the same (by definition).

á Building a tertian chord rooted on So is a simple matter of stacking up the successive odd-numbered degrees of So-mode, no matter what the melodic mode, key, or tuning.

á Similarly, improvising over a So chord requires playing melodically in So-mode (Mixolydian), no matter what the overall melodic mode, key, or tuning.

Within the diatonic scale, one is rarely, if ever, going to play a dominant 7 (-\\) or half-diminished 7 (\\-) chord on Do – such a chord is contrary to Do's diatonic tertian sequence, which starts with a major 7 chord (-\-). On the other hand, playing a dominant 7 chord on So or a half-diminished 7 chord on Ti would fit the diatonic tertian sequence perfectly, and as such is entirely expected.

On the other hand, despite the fact that the diatonic triad on Re is minor (\-), one might very well play a dominant 7 chord on Re (-\\), because Re is a common “secondary dominant” (V/V, “five of five,” or “dominant of the dominant”). The appearance of a dominant 7 chord on Re, which includes a chromatic note (unlike Re's diatonic tertian chord), indicates to the attentive musician that something “interesting” – i.e., not strictly diatonic – is happening in the music.

Although the above examples are based on the diatonic scale, similar examples can be drawn from any other tonal scale, and would function identically in any meantone tuning.

Solfa-Based ThumLattice

As can be seen in Figure 48, all of the triads of the diatonic scale can be represented on a solfa-based ThumLattice that is the same for all keys.

Figure 48

Figure 49 (below) shows the geometric relationships within and between the elements of the ThumMusic PLUS System. ThumField, ThumLine, ThumChord, and ThumLattice are all unified in this single image. The ThumLattice is drawn for Re-mode (Dorian).

Figure 49

The node-shapes at the vertices of the ThumLattice in Figure 49 are meaningful. [25] The circles fall on the line of perfect fifths that includes the tonic. For the other nodes, the number of sides is equal to the number of major thirds, plus two, that a node is away from the tonic's spine of fifths.

The orientation of the node-shape is also meaningful. If the node-shape has a side facing upwards, then it falls on a spine of perfect fifths that is to the left of the tonic's central spine of perfect fifths. If the note-shape has a vertex pointing up, then it's to the right of the tonic's central spine.

If used as note-head shapes, the node-shapes on this ThumLattice can denote complete 5-Limit Just Intonation information. Common Western staff notation can only denote 3-Limit (Pythagorean) Just Intonation information.

I do not imagine that anyone will use such 5-Limit note-head shapes for live performance purposes, since such note-head shapes are very busy and distracting. However, they could be used for analytical purposes of for programming sequencers.[26]

Triangular Note Heads

Using shaped note-head to display 5-Limit tuning information is visually overwhelming. It's just too much information. However, using shaped note-heads to display 3-Limit information could be quite useful.

In 12-tet, sharp & flat pairs such as G# & Ab are “enharmonic equivalents”, meaning that they are just two different names for the same pitch. However, the notation of a D# vs. Eb can convey useful information, such as whether the notated pitch is the result of diminishing or augmenting a diatonic interval. This information is also useful when intoning notes (with harmonic timbres) during performance.

The same considerations apply when using tonic solfa instead of pitch names. Consider the interval between Do and Me. Is it an augmented second, or a minor third? It would be useful to be able to resolve this ambiguity within ThumLine.

ThumLine satisfies this need through the use of triangular note-heads which indicate which “non-enharmonic” note is intended. Continuing to use the DoMe interval as an example, an upward-pointing triangle in the Me-space would indicate an augmented second (in C Major, a D#; in tonic solfa, Ri), whereas a downward-pointing triangle in the Me-space would indicate a minor third (Eb; Me). It is important to recognize that these triangular note-heads fall on Me's vertical staff location, not on Re's or Mi's. That is, the chromatic note's note-head is modified, not the diatonic note's.

The use of triangular note-heads gives ThumLine precisely the same 3-limit notational power that Common Western music notation has. However, even with this addition, isomorphic solfa staff notation is easier to use than traditional notation.

Consider a beginner, who – like most beginners – just “plays the staff”, ignoring all key signatures, accidentals, shaped note-heads, etc. “Playing the staff” with traditional notation will produce wrong notes if there are any sharps/flats notated at all – whether in the key signature or as accidentals. These notes will be “wrong” in any tuning. However, “playing the staff” with 12-tet ThumLine will always produce the right notes; all that is lost is information about modifying intonation from 12-tet to Just Intonation.

Previous - Inconsistencies in Common Western Music Notation Next - Tonal Scales

[15] See “A Source Book of Proposed Music Notation Reforms,” Gardner Read, 1987, ISBN 0-313-25446-X.

[16] There are 17 note-names in tonic solfa – one for each of the seven diatonic notes (Do Re Mi Fa So La Ti), one for each of the five flat chromatic notes (Ra Me Se Le Te), and one for each of the five sharp chromatic notes (Di Ri Fi Si Li). In order, they are Do Di/Ra Re Ri/Me Mi Fa Fi/Se So Si/Le La Li/Te Ti. The specific syllable-names vary from culture to culture, region to region, and perhaps even among individual teachers.

[17] Figure 23 can be usefully considered to be the diatonic scale's “Circle of Seconds.”

[18] In Figure 26, the flat names have been used for both the flat and sharp notes, which is a convenient simplification if the button-field is to be used solely for 12-tet.

[19] These scale dots could just as conveniently indicate a tone row in serial music.

[20] The traditional names for these ranges are “plagal” and “authentic,” by the way. I do not know what the relationship is between a plagal range and a plagal cadence, or between an authentic range and an authentic cadence. If you can prove an etymological link, I would love to hear about it.

[21] The scale dots in this and related figures should all be white/hollow.

[22] The MNMA's Notation Test is documented in its Music Notation News, Vol 10, No 2, 2nd Qtr 2000, page 6.

[23] Computer scientists may find it convenient to think of this as “late binding.”

[24] In his tutorial materials for the Hayden Layout Concertina (c.1986), Brian Hayden indicated chord shapes on a Wicki/Hayden button-field, but did not represent chords as stacks of intervals using strings of sequential alphanumeric chord-shape symbols as described herein.

[25] So far as I know, W. A. Mathieu was the first to align a tonnetz with a musical staff, in his 1997 book “Harmonic Experience,” which also uses lattice-derived note-head shapes to convey intonation information in a manner very similar to, but not as regular as, the system described herein.

[26] Note also that the described 5-Limit tuning is not meantone, and therefore does not map directly to the ThumField as shown, as a meantone tuning would.

The ThumMusic PLUS System

Introduction