This study is a detailed extrapolation of harmonic possibilities within the half-whole scale, often called the diminished scale by jazz musicians, and the second mode of limited transposition given by Messiaen in his Technique of my musical language (1944). The symmetry of the scale gives rise to interesting harmonic logic which is "square", in the sense that everything that occurs from one point in the scale also occurs in direct transposition at three other points. I look at harmonisation, voice-leading and intervallic possibilities.

For more technical background, see Models of Octatonic and Whole-Tone Interaction: George Crumb and his predecessors by Richard Bass.

This is a work in progress, and so I will add more over time. If you want to discuss this project with me, you can get in touch here.

• Scalar architecture
• Chordal possibilities
• Parallel movement
• Harmonisation of a single note
• Voiceleading
• Intervallic possibilities

Scalar architecture

The diminished scale is created by alternating 2 and 1 semitone intervals.

The scale is comprised of two diminished arpeggios. Odd degrees 1, 3, 5, and 7 form the primary diminished; even degrees 2, 4, 6, and 8 the secondary diminished.

I have used C as my root-note; however one could equally well use any other primary scale-tone for this purpose. In other words, because of the symmetry of the scale, there are four possible roots. The entire scale exists in 3 transpositions. When I use the phrase diminished scale, I always refer to the version with the half-step first.

As anybody reading this study will no doubt be aware, there are three dimished arpeggios. Say that "type 1" refers to the diminished arpeggio starting on C, "type 2" to that beginning on C#, and "type 3" to that beginning on D. Then these three discrete arpeggios combine to form diminished scales as follows:

Chordal possibilities

Taking the primary degrees as roots, the following conventional chord types are possible:

From the secondary degrees there are some ambiguous structures. In the notation, the names are these from the given secondary root note, but it is also possible to describe them in terms of other roots, especially in inversion.

A few other non-diminished possibilities:

• Implied whole-tone clusters using degrees 1, 4, 5, 8, or 2, 3, 6, 7
• Nested perfect 5th intervals from primary degrees, separated by a semitone (i.e. C, F#, G, C#)
• Nested minor 7th intervals from primary degrees (i.e. C, A, Bb, G)
• Nested major 7th intervals from secondary degrees(i.e. Db, Bb, C, A)

Parallel movement

First: 3-note chords. Descending from tonic 1 major the following triads occur:

Rules in a nutshell:

• Descending, virtual root moves down maj 6th from major, down tritone from minor
• Ascending, virtual root moves up a tritone from major, up maj 6th from minor

Here is a diagram I made of the basic possibilities from primary scale tones. Each major and minor chord occurs once. Click to view larger version.

Due to the symmetry of the scale, moving stepwise in parallel motion always alternates between two forms.

Here are the main 4-note forms with their evil twins:

Here are these chords, with root note G, written out in full:

Harmonisation of a single note

Now I will look at the various functions available to primary and secondary tones as a part of various chords. I begin with triadic harmony with a primary tone in the melody.

Triadic harmony with a secondary tone in the melody:

It seems easier to use a table to demonstrate the available tones in the dominant seventh, and whether they are primary or secondary.

Voiceleading

By moving only certain voices it is possible to create different progressions of non-diminished chords.

Triadic movement with one mobile voice:

Triadic movement with two mobile voices, where majors and minors are naturally separate:

When moving two voices, it is also possible to punctuate the same sequence of majors with incomplete dominant shapes (they lack the fifth), while maintaining the two-voice movement rule.

This section is being expanded

Intervallic possibilities

All possible intervals are present in the diminished scale, but only those comprised of minor thirds are found from both primary and secondary tones.

The following chart applies to intervals ascending. Available intervals descending from a given tone-class are those available ascending from the opposite class, that is, a major seventh is only available ascending from a secondary tone, but only descending from a primary tone. Intervals contained within a diminished seventh chord are available from any tone, ascending or descending: these are combinations of minor thirds, for example the tritone and the major sixth. Obviously larger intervals can be built using an additional octave. Those interval types which, after their application, have not changed the function of the tone from primary to secondary or the reverse, are shown in italics.

All the intervals can be arranged in order of size and played consecutively without requiring tones from outside the mode.

Structural unity can be derived from an ordering of intervallic sequence in a longer linear system.

Such interval-ordering can be treated strictly, as in a literal sequencing of their relative sizes, as in the following example:

Did you find this interesting? Do you have any input? Let me know.