Theory

Mathematical Outline

This Introduction is dedicated to the discussion of two central and mutually related subjects: (Ma-1) to the design of the playing interface and its configurations, and (Ma-2) to the navigation through these configurations.

Ma-1: A Quasi Mechanical Definition of the 84 Configurations

One may divide a circle regularly into seven segments of equal size. Likewise one may divide it regularly into twelve equal segments. But as the number 7 does not divide the number 12 these divisions are incompatible. To put it in another way: No selection of seven vertices from a regular dodecagon yields a regular heptagon.

Regardless of this irrefutable fact, there is still a maximally even selection of seven vertices out of a regular 12-gon. It is the most regular selection, which is possible. What may appear like a rotten compromise at first sight, turns out to be the key to a very interesting mathematical structure.

On the video one may see, how the maximally even selection comes about. The outer circle is divided into seven segments of equal size, the inner circle is divided into twelve segments of equal size. With the help of the numbering 1 till 7 (outside) and 0 till 11 (inside) one may identify every single segment, while the inner circle rotates against the outer. In the separating black circle thee are exactly seven slits, which are regularly distributed. Each one is located precisely in the middle of a corresponding segment from the 7-fold partition of the circle. In each configuration of the rotation we may imagine that the color of each of the seven fixed outer segments flows through the slits into the segments from the inner 12-fold partition, which are passing by in this moment. This determines the selection.

• 1. The seven step positions along this cycle are all uniquely determined by the step interval patterns leading to them and the step interval patterns emanating from them. Each position has a unique character, as some authors say. The seven tone syllables are used in order to denote these step positions:

     1. do: is preceded by TTS ans followed by TTST

     2. re: is preceded by TST ans followed by TSTT

     3. mi: is preceded by STT ans followed by STTT

     4. fa: is preceded by TTS ans followed by TTTS

     5. so: is preceded by TST ans followed by TTST

     6. la: is preceded by STT ans followed by TSTT

     7. ti: is preceded by TTT ans followed by STTS

There are 84 different configuations. One may track them on the basis of the counter in the middle. Each of the inner 12 segments (with white numbers 0 till 11) eventually meets the red outer segment (with the black number 1). This connection persists precisely for 7 configurations. Hence there are 84 = 12 · 7 configurations in total.

An elementary rotation amounts to an rotation angle of 1/84 from the entire circle. Each 12-segment is selected for 7 consecutive configurations and afterwards it is not selected for 5 consecutive configurations. The latter happens when its center is close to the border between two neighboring 7-segments. This observation is the basis for a slightly different visualization. In the video below the entire disk is divided into

• 1. seven colored segments with a circular angle of 7/84 = 1/12 of the entire circle each and

  2. seven shadow segments with a circular angle of 5/84 of the entire circle each.

A 12-segment is selected if and only if its associated number passes through a colored segment. It is not selected if and only if its associated number passes through a shadow segment.

• 1. NB: There are various traditions in basic musical education and ear training, where the tone syllables are used in three main different meanings. All three meanings are central for our discussion: (1) Positions in the diatonic step interval pattern, (2) generic scale degrees and (3) note names. The simple reason for this confusing clash of syllable-meanings is their singableness in comparison to the musical clumsiness of the dry latin note names A, B, C, D, E, F, G or the numbers 1, 2, 3, 4, 5, 6, 7 in whatever language. The traditions differ in the evaluation of the relative importance of these different meanings and consequently every tradition chooses the syllables for their favorite one. In the context of this music-theoretical outline we stick to the oldest of these traditions and use them to denote the step positions. But SolFa Mode-Go-Round allows the user to switch between the three meanings, both in display and audible solmisation. See the user manual for details of the interface.

  2. When Octave stands for a concrete note interval, it happens that the above abstract meaning takes more concrete shape. In addition to the abstract step interval pattern one obtains an (ascending) sequence of eight notes, whose step intervals exemplify the abstract pattern. Potentially every note may serve as the lower boundary of a species of the octave. The boundary notes are also called final or tonic. But these terms require some care and relate to the wider scope discussion. In order to denote concrete species of the octave one prepends the note name of the lower boundary note to the pseudo-classical mode name. Examples are C3-Ionian, C4-Ionian, D4-Dorian, C4-Dorian, C4#-Dorian. We should allow ourselves to occasionally omit octave registers and to write C-Ionian, D-Dorian, etc. But for theoretical reasons it is insightful keep sensitive towards octave registers.

Our interest focuses upon the combinatorics of the concrete modes. We will distinguish two transformations d (upward diatonic transposition) and f (flatward alteration) which can be applied to any concrete mode, and whose concatenations provide a connection between any pair of concrete modes.

• 1. Upward diatonic transposition removes the lower boundary note from the note sequence of a mode. The second note becomes the new lower boundary. In compensation one appends the note to the sequence which is an octave higher than the new lower boundary. Likewise, one removes the first step interval from the associated step interval pattern and appends in two the end.

The subsequent video shows the implementation of this idea in the design of the Playing Interface. Each configuration is associated with a concrete species of the octave (see the Music-Theoretical Outline). Its name and number can be inspected in the mode selection menu. The gray shadows are only shown in the outer ring of the playing area. A rotating regular 12-gon represents the centers of the 12 segments. The selected vertices are marked with a light beam, connecting the center of the disk with the vertex in question.

• 1. The iterated application of d yields concrete instances of all seven species of the octave in terms of "white note" modes within the ambit of a double octave. The 7-fold application of d yields an complete octave transposition of the entire mode: d⁷(C4-Ionian) = (C5-Ionian).

  2. Flatward alteration lowers one particular note from the note sequence of a mode by an augmented prime. This is the difference interval between tone and semitone. To that end the note in question must be surrounded by a tone from below and by a semitone from above. In the step interval pattern one exchanges the surrounding step intervals of the altered note. The lower tone becomes a semitone and upper semitone becomes a tone. There are two such possibilities, but only one of them is the right one. The new step interval pattern again needs to be a diatonic species of the octave. The iterated application of f yields concrete instances of all seven species of the octave in terms of the seven common-tonic modes, or tropes, within the ambit of a single octave.

The overall shape of the seven light beams visualizes the step interval pattern, consisting of 5 whole tones and 2 semitones. Each light beams always reaches precisely one of the seven elementary segments between in shadows. These are denoted in counter clockwise order by tone syllables fa, so, re, la, mi, ti in accordance with the actual step interval pattern (compare the corresponding graph which is cited from the music-theoretical outline).

• 1. If the altered note happens to be the lower boundary tone (Locrian) then also the upper boundary tone has to be altered. The flatward alteration of a Locrian mode is a Lydian mode. The 7-fold application of f yields an complete flatward-alteration of the entire mode, i.e. of every single note: f⁷(C4-Ionian) = (Cb4-Ionian).

The two transformations d and f of may be freely concatenated and thereby the allow us to reach any species of the octave from any other. Subsection Ma-2 (in the right column) describes a trick to reduce the combinatorial plenitude and explores its application to the navigation through the playing states of the SolFa Mode-Go-Round.

Mu-2: Scale Degrees versus Step Positions

The descriptions of the transformations d and f differ considerably from each other, and so do the two illustrative figures with the seven white-note modes and the seven common-tonic modes. Nevertheless there is a deeper analogy – actually, complementarity – between them.

In (upward) diatonic transposition the step positions remain unchanged, while the scale degres are shifted cyclically downward by one degree each: the second degree in C-Ionian becomes the first degree in D-Dorian, the third degree in C-Ionian becomes the second degree in D-Dorian etc.; and the octave displacement of the first degree in C-Ionian becomes the seventh degree in D-Dorian.

Ma-2: Navigatian through the 84 playing States

Diatonic transpositions and alterations commute with each other. The concatenations df and fd describe different pathways to the same results and thus they represent the same transformation. Under df (f followed by d) C-Ionian is transformed into D-Aeolian via C-Mixolydian, while under fd (d followed by f) C-Ionian is transformed into D-Aeolian via D-Dorian. The full repertoire of transformations, generated by f and d, can therefore by parametrized by the expressions d^mfⁿ denoting an n-fold flattening followed by an m-fold diatonic transposition. Here, m and n, can of course also be negative integers. It is convenient, though, to introduce a new symbol s = f ^-1 for the sharpward (= negative flatward) alteration.

For any two concrete species of the octave X and Y there is a unique transformation of this form d^mfⁿ, sending X to Y. For example, we may transform C#4-Phrygian into Db4-Phrygian by virtue of the transformation d¹f¹². The 12-fold flattening sends C#4-Phrygian to Cb4-Dorian which is then diatonically transposed to Db4-Phrygian.

The SolFa Mode-Go-Round reduces the free combinatorics by virtue of the following two relations:

• 1. Octave Identification: d⁷ = Identity.

  2. Enharmonic Indentification: d = f ^-12 = s¹²

NB: This reduction is not harmless from a theoretical point of view and it has a significant impact on the design of the modal navigation in the instrument. Nevertheless it does not confine the user to a standard all-encompassing 12-tone system. Pragmatically one may circumvent this constriction while gaining additional theoretical insights.

Substituting d within the first relation d⁷ = Identity for s¹² by virtue of the second relation d = s¹² yields the relation (s¹²)⁷ = s⁸⁴ = Identity. In other words, in the reduced picture the sharpward alteration s generates a long cycle of 84 modal transformations. A similar reduction has to be done to the modes themselves.

The screenshot displays the mode selection menu. In order to apply the sharpward alteration s to the actually chosen mode, one may press the large #-button. The selected alteration amount should be equal to 1 then. If the selected alteration amount is 12, pressing the #-button has the effect of performing a s¹²-transformationat once. When choosing the alteration amount of 12, the user is reminded about the relation 2 in the display: "12 is (enharmonically equivalent) to a diatonic transposition".

In (flatward) alteration the situation is quite complementary. The scale degrees remain unchanged, while the step positions are shifted cyclically sharpward by one position on the generic cycle of fifths: the fa-position in C-Lydian becomes the do-position in C-Ionian, the so-position in C-Lydian becomes the re-position in C-Ionian etc.; The actual flattening occurs at the ti-position in C-Lydian, which becomes the fa-position in C-Ionian.

The figure below summarizes this observation in a more abstract form. Diatonic transposition corresponds to a cyclic permutation of the scale degrees. Alteration corresponds to a cyclic permutation of the step positions:

There are three other musically relevant transformations, namely

• 1. s⁷ (with alteration amount 7), which is enharmonically equivalent to a semitone-transposition.

  2. s⁴⁸ (with alteration amount 48), which is enharmonically equivalent to a diatonic fifth-transposition.

  3. s⁴⁹ (with alteration amount 49), which is enharmonically equivalent to a perfect fifth-transposition.

See the user manual for details of the interface, and see the article of Julian Hook (2008) for further theoretical details and inspect the following tutorials for practical instruction:

Tutorial 0d: Diatonic Transposition

This complementarity can be studied in much more detail. For a reference see:

• • Clampitt, David and Thomas Noll (2011): "Modes, the Height-Width Duality, and Handschin’s Tone Character" Music Theory Online, Volume 17, Number 1, March 2011.