Building Vocabulary with Seventh Arpeggios
I revisited a past post on building vocabulary from scratch today and decided to extend it a bit. Here's what I came up with.
The idea in the post was to partition the 12 available notes into three categories:
- Chord tones
- Consonant tones
- Dissonant tones
The categories are supposed to be disjoint and exhaustive, meaning that each of the 12 notes is placed in exactly one of them. It's particularly tempting to think of four-note chords (sevenths) and make the categories all of equal size. This produces, in my jargon, a disjoint cover of the chromatic scale in which all the components are the same size.
Naturally, I wondered how many ways there are to do this. In particular I wondered how many there are if the chord tones belong to one of the four common seventh chords and either the dissonant or consonant set do as well (at some transposition).
I take the common seventh chords to be:
| Major 7 | 0, 4, 7, 11 |
| Dominant 7 | 0, 4, 7, 10 |
| Minor 7 | 0, 3, 7, 10 |
| Minor 7 b5 | 0, 3, 6, 10 |
(The numbers are pitch class numbers, which are the easiest thing to work with in this context). The question is entirely answered by the following: for each of these, how many ways are there to choose a second one of them without overlapping notes?
Here's how to read the runes. The first heading says we're dealing with a Maj7 chord, e.g. CMaj7. This, as you probably know, contains the pitch classes {0, 4, 7, 11}. The first line of the table contains the two sets of consonant and dissonant notes (you decide which is which, or whether these are applicable terms at all!). The first entry says bVII Maj 7, so we play the Maj 7 arpeggio built on the bVII of the key, i.e. BbMaj7. With C=0 this arpeggio is the PC set {2, 5, 9, 10}. What's left is the set {b2, b3, b5, b6}. So the whole picture is:
| C Maj 7 | 0 | 4 | 7 | 11 | ||||||||
| Bb Maj 7 | 2 | 5 | 9 | 10 | ||||||||
| {b2, b3, b5, b6} | 1 | 3 | 6 | 8 |
Maj 7
| bVII Maj 7 | {b2, b3, b5, b6} |
| bV Maj 7 | {2, b3, b6, 6} |
| II Maj 7 | {b3, 4, b6, b7} |
| bIII Min 7b5 | {2, 4, b6, b7} |
| bVII Min 7 | {2, b3, b5, 6} |
| bIII Min 7 | {2, 4, b6, 6} |
| bVII Dom 7 | {b2, b3, b5, 6} |
Dom 7
| II Maj 7 | {b3, 4, b6, 7} |
| VII Min 7b5 | {b2, b3, b5, b6} |
| bVI Min 7b5 | {b2, b3, 4, 6} |
| IV Min 7b5 | {b2, 2, b5, 6} |
| bIII Min 7b5 | {2, 4, b6, 7} |
| VII Min 7 | {b2, b3, 4, b6} |
| bVI Min 7 | {b2, 2, 4, 6} |
| VII Dom 7 | {b2, 2, 4, b6} |
| bII Dom 7 | {2, b3, b5, 6} |
Min 7
| VI Maj 7 | {2, 4, b5, 7} |
| II Maj 7 | {3, 4, b6, 7} |
| VII Min 7b5 | {b2, 3, b5, b6} |
| bVI Min 7b5 | {b2, 3, 4, 6} |
| VII Min 7 | {b2, 3, 4, b6} |
| bV Min 7 | {2, 4, b6, 7} |
| bII Min 7 | {2, 4, b5, 6} |
| III Dom 7 | {b2, 4, b5, 6} |
| bII Dom 7 | {2, 3, b5, 6} |
Min 7b5
| VI Maj 7 | {2, 4, 5, 7} |
| VII Min 7b5 | {b2, 3, 5, b6} |
| bII Min 7b5 | {2, 4, b6, 6} |
| III Min 7 | {b2, 4, b6, 6} |
| bII Min 7 | {2, 4, 5, 6} |
| VI Dom 7 | {2, 4, b6, 7} |
| V Dom 7 | {b2, 3, b6, 6} |
| III Dom 7 | {b2, 4, 5, 6} |
| bII Dom 7 | {2, 3, 5, 6} |
