If you are not familar with how I can use fractions and ratios to represent pitches, I suggest you first read Musings on Otonality and Utonality.
Sometimes, I like to use just intonation as a tool to think of how music is constructed in ways that break convention. It’s fun, and it can help us think about music at a deeper level. So, here’s a way to break the convention of how modes can be relative, in the sense that two modes can share all the same notes.
Our major and minor scales typically can be referred to as a set of pitches formed from the 3 primary triads of our desired mode. A major scale includes all the pitches from the major I, IV, and V chords. Put another way, form 3 major triads in which one is a central “reference” or tonic, and the other two get mapped to the bottom or top of that tonic as it is spelled in “root” position. The IV will share it’s top note with the bottom note of the I, and the V will share it’s bottom note with the top note of the I. The resulting set of pitches in relation to the “root” of the I chord is, in 5-limit:
1/1 9/8 5/4 4/3 3/2 5/3 15/8 2/1
If we do the same thing with 3 minor triads, we get the minor scale:
1/1 9/8 6/5 4/3 3/2 8/5 9/5 2/1
We can create other forms of the major and minor scales; for example 3-limit Pythagorean.
First, these two 5-limit major and minor scales are not equivalent, and are not exact modes of each other – unlike the scales in 12-note equal temperament (12ET) which are. If I rewrite this major scale as starting from its sixth degree I get:
1/1 9/8 6/5 27/20 3/2 8/5 9/5 2/1
And the minor scale as starting from its third degree:
1/1 10/9 5/4 4/3 3/2 5/3 15/8 2/1
To implement good tuning, I find it more useful to think of major and minor scales as being constructed from my first pair above with the primary-triad building blocks, instead of being relative modes of each other. I can create different scales resembling modes by mixing up which primary triads I use. For example, a Dorian mode could be minor I and V chords, and a major IV chord:
1/1 9/8 6/5 4/3 3/2 5/3 9/5 2/1
And, with this approach, the problem of pitch drifting I wrote about in the article Unintentional Pitch Drifting can generally be dealt with in the same way in these different modes. Its a bit different with Lydian and Locrian modes, but the altered note can still be related to a primary triad, if I allow more types of primary triads. For example, in C Lydian, the IV chord can be thought of as F# diminished instead of F major, still using the same A and C, and placing the F# an additional minor third below the A:
1/1 9/8 5/4 25/18 3/2 5/3 15/8 2/1
Of course, stacking two minor thirds in this way and using the 25/18 produces a raised fourth degree that is 34 cents away from the 12ET sharp 4. I think it sounds great, but it is noticeably different. For something a bit closer to 12ET, a 7-limit interval can be used for F# instead, going 7/6 below the A instead of 6/5:
1/1 9/8 5/4 10/7 3/2 5/3 15/8 2/1
If we assume that the major scale and the lydian scale are relative to each other, then let’s map the major scale according to the 4th scale degree:
1/1 9/8 5/4 45/32 3/2 27/16 15/8 2/1
And just to drive the point home, how these three diminished iv chords sound, in sequence:
Obviously, this produces something pretty different from the two other Lydian scales, and in this mode, the IV is completely 3-limit. This makes sense if we consider that, going back to our original major scale, the IV of IV has notes that fall on the 7th, 2nd, and 4th scale degrees. So, it’s pretty obvious in this case that a major scale and a lydian scale, constructed using these tools, are not relatives of each other, that they do not actually share all the same notes.
Being a trombonist, I have seen tuning problems pop up in live performance (of pitch-adjustable, non-keyboard instruments like in wind ensembles) after mode changes a lot. I think that at least part of these tuning problems have to do with musicians believing that different modes, when they share a written key, share tuning tendencies; when in fact some notes will change tuning even if the key does not. The key thing to keep in mind are the primary triads: take a moment and see if those primary triads change as the result of a mode change, and if they do, forget about the previous section’s tuning and take the new mode as its own environment.
For the curious, I used Musescore2 to create these scale samples, due to it’s easy-to-apply tuning adjustments on notes, with the bagpipe sound specifically because it did not have vibrato. Vibrato tends to hide intonation issues.
A great tool to experiment with these scales is Scala. Note that once you have the scale created, the KEY command is what you want to realize scales with different roots, and that Scala (like many good computer tools) starts counting scale degrees at 0, not 1. So, to see the relative lydian mode of the 5-limit major scale (also called the Ptolemy scale), use command KEY 3.