Just wanted to let y’all know that I’m making videos sharing the life tools I’ve learned, which have been so valuable and supported me over the years. I think it’s fun to watch, it’s got my piano music in it, and you might even get something out of it.
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. ;-)
I think I’m going to switch my blog to medium. My first post there: Infinitives of an Oppressor, a reflection on some of my inner workings wrt social justice.
Here is some new music, a piano/vocal improvisation from tonight.
The voice is a gift
and he gives it not to some but to all
to YOU all
the small and the large
the weak and the in charge
We all have that thing
we were born to say
you may not be able to access it today
let me hear you say someday
I will stand tall
I will take charge
Outward my inner prayer I will pray
I used to be ashamed of my voice
Couldn’t make the choice
to be heard
I had hurt
I had heart
but I couldn’t speak the words
Couldn’t sing the songs
that I knew all along
Every day a new invention
A hidden intention
Well this is my intervention
I make the choice to use my voice
I stand tall
I take charge
And outward this inner prayer I pray
I pray to come alive today
There ain’t no other way
My brothers can think what they may
Cause I know they also got something to say
Let my courage show you the way
And let yours be inspiration to another
This is the situation there’s no other
It is WE who light the world on fire
Who inspire the weak and the tired
To be strong and admired
To change ourselves from liars
mask wearers, fear bearers
Heart hiders, insiders
To fearless riders
carrying our word
be it simple or absurd
the truth is the word
OUR truth is the word
I make the choice to use my voice
I stand tall
I take charge
And outward my inner prayers I pray
On a day long ago was learned
a terrible lesson, my friend.
In the memory burned
Too swiftly to comprehend
And dry a great river turned.
What happened that echoing day?
To this little boy, full of passion
So much to give and to say
A life for beauty amassing…
Then for watching it fade away.
“I’m sorry,” he began to cry
Before giving his gift.
It became too much to try,
A burden too heavy to lift,
And he almost told it goodbye.
Someday, it is said, he will find
Holding the hand of a friend
The memory locked away in his mind
And the courage to give once again
With shame left far, far behind.
What would you have me do?
For your sake I’d hidden myself from you.
I knew inside so I built a wall,
Attached a mirror, so whatever you wished you would see as true.
Might you I ought to’ve cared for,
But I resolved not to pretend anymore.
A false heart I’d not caring call,
And when I did finally speak, it stopped feeling so awfully sore.
This sharpening pain I have made,
I hope is better than how I’ve betrayed.
All I really want is to bring down this wall.
I asked for love, truth asked of me, and I obeyed.
When a particular mutation causes some cells to start dividing faster than the others — a self-replicating explosion that is characteristic of life — we call them cancer and they kill their host. With their host they also die. The latent bacteria and fungi kept in delicate balance by the sophisticated systems of the body take over, each exploding in its own way, recycling and reintegrating the materials that composed this human machine into the larger world. What was the human’s individual consciousness reintegrates into the Earth in whatever way that happens.
At some point a mutation in the memetic code altered humanity. We began wearing clothes, building houses, and with this developed a concept that we are separate from each other and the rest of nature — Self, Ego. We have expanded much faster than our surroundings, and the amount of the Earth’s resources we consume to continue doing so outweighs the value we give back into our host.
The Singularity is based on the observation that the rate of technological innovation has been accelerating for a long, long time. This mystical belief system projects what might happen as this acceleration continues. What happens as the structure of self-replication and transformation of nature continues to its limit? By analogy, what happens when the small mutation that causes cells to divide faster continues to mutate for its own propagation? In some accounts, a structure of energy will be created that expands outward at the speed of light, integrating the consciousness of our local nature into the universe — Stepping into the Light.
The Singularity is a concept of death. Environmentalists see our unhealthy habits and want to get our act together so we can continue living a healthy life; technologists see the unstoppable advancing tumor and are trying to accept our fate and embrace death — not the end of all things but indeed a major transformation in being resulting in the loss of all our attachments. It is a peculiar contradiction, in a divine-truth sort of way, that many technologists believe that we will invent a way never to die. Each of us transforming ourselves through the advance of technology is participating in the larger process of death that the Earth faces. Environmentalists hope that our cancerous expanse can reconstruct a functioning body on top of ourselves, keeping the host just alive enough to preserve our Selves — restoring the Earth to the past in which humans are few and in parity with other animals is not what we have in mind. With hope and desperation we awe in Mother Nature’s embrace.
We belong to the Earth, and she is dying.