Tonality, Harmonic Progressions, and Voice Leading with the Bohlen-Pierce Scale (DRAFT) Dustin Schallert January 5, 2012 Within the Bohlen-Pierce scale, relationships can be found among triadic harmonies that closely parallel the harmonic structures associated with 12ED2 tonal Western classical and popular music. By identifying and implementing these harmonic structures within Bohlen-Pierce compositions, it is possible to create a piece which establishes a tonic, moves through related harmonies or modulates to a new key, and returns to the tonic with a compelling sense of resolution and closure. In other words, there are chords within several Bohlen-Pierce modes that closely mimic the function of “tonic”, “subdominant”, and “dominant” chords of 12ED2 music – both in their aural impact and in their structural implications for the relations of keys and modes. The mode which will be analyzed here is a hybrid of the Lambda and Pierce modes of the BP scale. The hybrid mode, in which the second degree of the mode can be either one or two steps above the tonic, emerges as the collection of pitches which form the most closely related harmonies to the tonic chord. These modes are also ideal for supporting tonal compositions because they omit the three steps (5, 8, and 11) that form dissonant wolf intervals against the tonic, which if included would work against affirming the tonic within the context of a harmonic or melodic progression. The following letter-based notation will be used in this article. Highlighted below is the C Lambda/Pierce hybrid mode. Name C Db D E F Gb G H Step # 0 1 2 3 4 5 6 7 Scale Degree I II* II* III IV V VI *The function of the split scale degree II will be explained below

Jb 8

J 9 VII

A 10 VIII

Bb 11

B 12 IX

A comprehensive and useful system of harmonic relations should be able to support any pitch of the diatonic scale. In the most basic and familiar of harmonic progressions found in 12ED2 popular music, three chords of a single type (4:5:6 “Major”) can be used to harmonize the full diatonic major scale. Since the Bohlen-Pierce scale is derived from the 3:5:7 (“Wide”) triad, and since this is the most consonant triad found within the scale, it will serve as the chord used for building a harmonic framework. Building triads from notes available in C Lambda gives the triads [C G A], [E J C], [G B E], [H C F], [J D G], and [A E H]. This collection is sufficient to harmonize each note of the diatonic scale, but it doesn’t give any insight regarding how these chords are related. So, another way to relate triads to each other is by comparing the most closely related keys (those which have the most pitches in common) built on the tonics of each triad. The below table compares the number of notes that differ between each pair of keys. Lambda modes are compared to simplify the issue of the second scale degree. For example, E Lambda differs by one note compared to C Lambda because it uses F-sharp instead of Fnatural.

C Db D E F Gb G H Jb J A Bb B

C 0 4 5 1 3 6 2 2 6 3 1 5 4

Db

D

E

0 4 5 1 3 6 2 2 6 3 1 5

0 4 5 1 3 6 2 2 6 3 1

0 4 5 1 3 6 2 2 6 3

F

Gb

G

H

Jb

0 4 0 5 4 0 1 5 4 0 3 1 0 5 4 6 3 1 5 4 2 6 3 1 5 2 2 6 3 1 6 2 3 2 6 Table of key relationships

J

A

Bb

B

0 4 5 1

0 4 5

0 4

0

The most distantly related pairs of keys—such as C and Jb—also happen to have roots related by the most dissonant BP intervals (in this case, the 8th “wolf octave”). When these keys are arranged in a circle that puts the most closely related keys adjacent to each other, a structure forms that is somewhat analogous to the 12ED2 “circle of fifths”. The interval by which closely related keys are found in BP is the tenth (or its inversion, the third), so it may be called a “circle of tenths”.

BP Circle of Tenths

The colored lines in this illustration connect the three most closely related keys in either direction of C. If the triads of each of these modes are compared, we find that each shares one common tone with the [C G A] triad. For example, the note A is shared by [C G A] and [A E H]. The note G is shared by [C G A] and [J D G]. Thus, for any given 3:5:7 triad in the full chromatic BP scale, there exactly 6 other triads with which it shares a common tone. This is a useful property for creating smooth voice leading and connections between chord changes in BP. If this is circle is interpreted as a strict analog to the 12ED2 circle of fifths, then A ought to function as the dominant of C. (The 12ED2 triad [C E G] has for a dominant the triad built on its top note, giving [G B D]. Following this construction in BP gives [C G A] > [A E H]). What is the aural impact of this progression? Of the available two-chord progressions, the author believes it exhibits the strongest quality of resolution suitable for the function of dominant. The triads found in a strict C Lambda mode were C, E, G, J, A, and H. The circle of tenths suggests that to achieve symmetry, the F chord should also be included, but it contains a Db which is not diatonic to Lambda. D cannot be completely replaced by Db, as D is needed for the J triad. So, the author suggests that both Db and D (i.e. the II and bII in any key) be retained as usable notes in a diatonic, tonal setting. What considerations determine where to use each, since each is present in only one of the seven diatonic chords? Consonance against the root of the chord may be a useful factor for deciding this. The below diagram compares the intervals of Db and D above the roots of each chord:

Db D

F 10 11

H A C E 7 4 1 11 8 5 2 12 Intervals of the second scale degree against chord roots

G 8 9

J 5 6

Over the three chords leading in one direction from C along the circle of tenths (A, H, and F), Db is a consonant note against the bass, while D is dissonant against it. Conversely, over the three chords leading in the opposite direction from C (E, G, and J), Db is dissonant while D is consonant. Against the tonic itself, either note may be suitable, but depending on the surrounding harmonic and melodic context, one or the other (or neither) may be preferable. The presence of a scale degree which is raised or lowered depending on context is analogous to the treatment of 12ED2 minor keys, in which the sixth and seventh scale degrees are adjusted depending on the harmony. The above regarding Db and D is a particular example, but a useful general melodic practice would be to avoid sustained tones that are dissonant against the bass or other prominent notes in the harmony (i.e. 5, 8, or 11 steps). When occurring in the context of passing tones from one consonant interval to another, these dissonant intervals are not problematic and may even be useful for color or flavor. Having established a working set of chords and the properties of a hybrid scale mode, the final consideration for tonal BP writing is how to form longer chord progressions with the use of subdominants and the like. The author suggests that any chord which shares a common tone with the desired dominant may function as a subdominant. For example, H > A > C is an effective progression

with common tones in each chord change, and identical root motion to each change. Other progressions cadencing in C may include E > A > C, F > A > C, G > A > C, or longer progressions involving common tones. A distinct effect is created by chromatically adjacent parallel chords such as G > H – this may be useful for transitions and breaks, but may have a disruptive effect in the middle of a phrase. Jumps between very distantly related chords such as J > F will also create an abrupt juncture useful for transitions to contrasting sections.