I have a natural curiosity for mathematical things, but I’ve made the decision to put as much focus as I can on music. So, naturally, I’ve been studying the math of music.
From the perspective of a musician, it may seem frivolous to study music mathematically. After all, does all this left-brain stuff really help you express yourself? I think it does. By investigating musical structures systematically, we can build our intuition, our understanding of how the pieces fit together, and then more ideas are available to us. It also allows us to hear others music in a more detailed way, with more language available for our minds to describe, and thus understand, patterns.
Today I wanted to share some of the investigations I’ve been making into modes and scales, from a mathematical perspective rather than a sonic perspective. Some of this stuff is well-known to musicians, and other bits of it are rather novel, as far as I know.
Here I’ll quickly review modes. Play a C major scale, all the white keys starting on C. Now play another scale starting on A with all the white keys. This, as you may know, is the minor scale. It has all the same notes as C major, but because it starts in a different place it sounds different. This is how modes are generated — we use the same notes but start in a different place.
We always refer to the key of a mode by the note it starts on. So the two scales we just played are “C major” and “A minor” — they have the same notes, but they started in different places, and we refer to them by their starting position. In “C major”, C is the key, and “major” is the mode.
The modes that have the same notes as a major scale are called “natural modes” (I’m defining this right now, I don’t think this is standard jargon). There are 7 of them (because there are 7 different places to start!). Play all the white keys starting on each of these notes and you will get the different modes:
- Start on C: major (aka “ionian” if you want to be pretentious)
- Start on D: dorian
- Start on E: phrygian
- Start on F: lydian
- Start on G: mixolydian
- Start on A: minor (aka “aeolian” for the grandiloquent)
- Start on B: locrian
It is a bit hard to hear how these sound when you play them next to each other — our mind aligns to the key of C major and it just sounds like we are playing runs in C major. To hear how these sound, and also to exercise your music math skills, transpose all these modes so they start on C. So look at the pattern of whole and half steps, e.g. minor goes whole-half-whole-whole-half-whole-whole, and then do that same pattern starting on C. It looks like this:
Do this for all the modes above, starting on C. Doing it this way you will get a feel for how each of the modes sounds.
(Another way to think of this if you know your major scales already, is that for dorian you play a Bb major scale starting on C, for phrygian you play a Ab major scale starting on C, and so on.)
For expert mode, do it starting on all the keys (I do this as part of my daily practice). Obviously don’t burn yourself out though, learn a little at a time.
It turns out there’s a best order to do this in: it goes lydian, major, mixolydian, dorian, minor, phrygian, locrian. Doing it in this order will show you something about the relationship between the different modes (I’m not going to give it away!).
After we get going here, we’ll see that there are actually 21 different 7-note modes. It took me a long time just to learn the names of the 7 natural modes, and the other modes we will study often don’t even have agreed-upon names. So first I’m going to give a scheme for naming modes, which also helps you think about them.
We will focus our attention on “4 note scales”. In the natural modes, there are only four of these that ever appear. Their names come from the most common natural mode that they are the first four notes of.
- Major (maj)
- Minor (min)
- Phrygian (phr)
- Lydian (lyd)
It’s of course about the pattern of whole and half steps, not about the specific key we’re starting on. Every natural mode can be made by putting two of these guys together, with an appropriate step in between. For example, the minor mode:
Thus I might refer to the minor mode as “minor phrygian” or “min-phr“.
Notice how there is a whole step between the two components. You put in whatever step between the two fragments that will make the last note the same as the first one. It’s usually a whole step, but it will be a half step when lydian is one of the components, because there’s an extra half step between its first and last notes. For example:
This mode (which is not a natural mode, by the way!) has a half step between its two fragments.
(Theoretically the lydian-lydian mode should have a 0th step between its two components. Instead we just fuse the two identical notes into one, and you get the six-note whole tone scale.)
With this vocabulary, we can refer to the 7 modes in a way that makes them easier to understand:
- Major: maj-maj
- Dorian: min-min
- Phrygian: phr-phr
- Lydian: lyd-maj
- Mixolydian: maj-min
- Minor: min-phr
- Locrian: phr-lyd
Play each of these modes, concentrating on their fragment anatomy.
This anatomy suggests more modes than the natural ones, for example the min-lyd mode we saw above, which doesn’t appear as one of the natural modes. What’s up with that?
Melodic Minor Modes
In standard music theory, the min-lyd mode we saw above is one of the modes of the melodic minor scale, and it does not fit into the scheme of natural modes (i.e. it won’t be all the white keys starting from any key). The melodic minor scale is a major scale with a flatted third:
And it sounds really cool (very classical sounding when played as an ascending scale, and becomes much more jazzy when you start building riffs out of it). The melodic minor scale has 7 of its own modes, just like the major scale. You can find these by playing the melodic minor scale above starting on each of its consecutive degrees, just like we did for the natural modes. The colloquial names I’ll give for these are from Mark Levine’s excellent book “The Jazz Piano Book”:
- Start on C: min-maj (“melodic minor’)
- Start on D: phr-min
- Start on Eb: lyd-dim* (“lydian augmented”)
- Start on F: lyd-min (“lydian dominant”)
- Start on G: maj-phr
- Start on A: min-lyd (“half-diminished”)
- Start on B: dim*-lyd (“altered”)
Notice that there’s a new fragment that we haven’t seen before in a few of these scales, the diminished (dim) fragment:
- Diminished (dim)
Like the lydian fragment, it does not have a perfect fourth between its first and last notes. Whenever there is a diminished fragment as part of a mode, the two components should be separated by an extra half step. Because we’re only looking at scales with whole and half steps right now, and usually fragments are separated by a whole step, that means that the diminished fragment is always paired with a lydian fragment to cancel it out — otherwise we’d end up with a minor 3rd somewhere in our scale (which happens, e.g. in the harmonic minor scale, we’re just not considering that at the moment).
Transpose each of these modes into C (or wherever you like) to get a feel for how they sound.
If you make any mode randomly out of whole and half steps, chances are it’ll be either a natural mode or a melodic minor mode. They account for two-thirds of all the possible modes. There is one more type of mode that we haven’t covered, and is very uncommon in western music (which makes me itch for opportunities to use it!). I’m not sure what to call it, right now I’m calling them “exotic” modes alongside the “natural” and “melodic” modes. It looks like this:
These modes are strange because they have two half steps in a row (separated by an octave in the the above version, but notice it has B,C,and Db). This means many of its chords have major seconds as intervals, and it’s quite weird and foreign sounding.
But I am interested in it because together with natural and melodic modes, we have now covered all the seven note scales. No matter what scale you make, as long as it has seven notes and no interval greater than a whole step, it will be a mode of one of these three families. Why?
Let’s consider what it takes to make 7 intervals cover 12 half steps. If every interval were a whole step, the scale would span 14 half steps, which is 2 too many. So we need to shrink it by two half steps: change exactly two of the whole steps into half steps. The different families come from the different places we can do that.
Put your focus on the two half steps in the scale. They can either be separated by 2 whole steps, 1 whole step, or none. They can’t be separated by 3, e.g.:
Because you can just rotate it around
And now we can see that they’re actually just separated by two.
So when you have your scale, it should just have two half steps in it. If they are separated by two whole steps, it’s a natural mode; if they’re separated by one, it’s a melodic mode; and if they’re separated by none, then it’s an exotic mode.
There you have it, some interesting mathematical music stuff. I have more — there are lots of neat symmetries between all these modes, but I’m tired of writing for today. Remember to follow me if you’re into this kind of thing (lower right). I get a little burst of approval happiness every time I get a new follower, and it makes me want to write more! The same goes for sharing, of course. ;-)