This essay was drawn largely from “The Forms of Tonality” by Paul Erlich available here. It’s great information, and I think it’s interesting to take it in a slightly different direction than Paul did. Also related is my essay A Musing On Modes, which has some sound samples not included here.
To start, consider the classic 5-limit just intonation tuning of the two commonly used scale modes, Major and Minor, and look at how the classic triads in them break down. Triads that contain the interval 9/8 to 4/3 are marked as dissonant. Erlich shows a graph of these scales in Figure 4.
Minor Scale:
1/1 9/8 6/5 4/3 3/2 8/5 9/5 – A B C D E F G
Minor Triads
i: 1/1 6/5 3/2 – A C E
iv: 4/3 5/3 1/1 – F A C
v: 3/2 9/5 9/8 – G B D
Major Triads:
III: 6/5 3/2 9/5 – C E G
VI: 8/5 1/1 6/5 – E G B
Diminished Triad (dissonant)
ii: 9/8 4/3 8/5 – B D F
Major Triad (dissonant)
vii: 9/5 9/8 4/3 – G B D
Major Scale:
1/1 9/8 5/4 4/3 3/2 5/3 15/8 – C D E F G A B
Major Triads:
I: 1/1 5/4 3/2 – C E G
IV: 4/3 5/3 1/1 – F A C
V: 3/2 15/8 9/8 – G B D
Minor Triads
iii: 5/4 3/2 15/8 – E G B
vi: 5/3 1/1 5/4 – A C E
Diminished Triad (dissonant)
vii: 15/8 9/8 4/3 – B D F
Minor Triad (dissonant)
ii: 9/8 4/3 5/3 – D F A
Compare this result to the graphs shown in Erlich’s paper, Figure 6. Major and minor triads are shown as triangles rotated 180 degrees from each other; and diminished, augmented, and suspended triads are shown as the 3 directions of straight lines. Other scales than major or minor can be formed with different shapes. There’s nothing particularly special about these two modes except perhaps that they are more ‘compact’ to describe in graphs than other modes, and (in a way) nice complements of each other.
Erlich’s paper includes 3 scales that include 7-limit intervals (Figure 7), which have the same kind of ‘compactness’ and complement-ness to each other. I think it’s quite interesting to consider these scales. It’s reasonable to think that adding another harmonic should drastically increase the complexity of potential harmony, and these scales do contain more notes – but the complexity isn’t increased by ridiculous amounts. These scales are still approach-able. Note that I call note names as letters like the 5-limit scales above, but here A-J are all contained in one octave instead of A-G.
Andrew Meronek 