There are 12 notes in an octave, separated by half-tones. Say you choose 7 of these notes to be in your scale. Then there are 792 possible scales. But a lot of them are pretty goofy.
So suppose the space between consecutive notes in your scale is never bigger than a whole tone. Then there are only 21 allowed scales!
These scales must have 2 consecutive notes a half-tone apart, and 5 a whole-tone apart. For example the major scale has spaces like this:
w w h w w w h
This is why there's a group of 2 black keys on the piano and then a group of 3. By cycling this pattern around we get a total of 7 scales, called the modes of the major scale.
But the melodic minor scale is not among these, because it has spaces like this:
w h w w w w h
If we designed the piano with this scale in mind, it would have a group of 1 black key and then a group of 4. By cycling this pattern around we get 7 more scales, called the modes of the melodic minor scale.
There are also 7 scales where the two half-tone steps are right next to each other, like this:
w h h w w w w
But these are more awkward, and less commonly used.
The 7 modes of the major scale have standard names, listed here. Ionian is just another name for the major scale itself.
The 7 modes of the melodic minor scale have lots of different names, and in some cases I'm not even using the most common names! For example, nobody ever talks about "Ionian ♭3" - everyone calls it melodic minor, or 'melodic minor ascending' to be precise. But I'm using names that emphasize how you can get these modes by slightly tweaking the modes of the major scale. You get melodic minor by flatting the third note in Ionian.
@johncarlosbaez Seeing them like this, I get a silly urge to interpret them as matrices and start multiplying them by something. I have no idea what that something would be though ... probably a rather goofier entity than those remaining scales
@TorbjornBjorkman - you can multiply one by the transpose of the other, treating w and h as 0 and 1 mod 2. But I have no reason to suspect that this is a wise thing to do. 🙃
@johncarlosbaez What does "awkward" mean here? Is it an important criterion? IMO "w w w w w w" would be the least awkward, and the reason why it is not used is presumably not that it has 6 instead of 7 but that it is lacking harmony.
@zenorogue - w w w w w w has been widely used since Debussy, it's called the 'whole-tone scale' and it has a very interesting 'natural' feel due to its high symmetry, which also kills off many of the standard tools of so-called 'functional harmony', like exploiting our desire for the 7th to go up a half-tone to the octave (a fundamental trick in Western music).
Here I'm limiting myself to 7-tone scales just because I feel like it - and because these include a huge pile of scales used in jazz.
'Awkward' is hard to define in general, since it's more about what we're used to than anything set in stone. The main reason I didn't include scales like this
h h w w w w w
w h h w w w w
w w h h w w w
is that I've never seen anyone discuss them.
But also, going up two half-tones one right after the other sounds less like a 'scale' to me than what you'd get by taking the whole-tone scale and sticking an extra note between two notes of that scale - which you might do occasionally in a whole-tone piece to liven it up, but not all the time consistently.
But I'd like to know if someone talks about those scales! I have no moral objection to them.
Weird #scales aren't usually in my #MusicTheory wheelhouse (they're way more popular among the hardcore jazzers and theorists) but by accident I happen to know an example of the scales you're asking about!
They're modes of so-called "Locrian Major" (which wikipedia says is sometimes called the "Arabian Scale", but I would avoid that terminology since I've also seen people use it to mean "Double Harmonic Major"). Here C Locrian Major is (up to enharmonics):
C D E F F# G# A# C
Though it's probably "more correct" to write
C D E F Gb Ab Bb C
to view it as an altered locrian.
They also come up as modes of the "Leading Whole Tone" scale, which (as you might guess) is a whole tone scale with the leading tone thrown in:
C D E F# G# A# B C
Back before I realized that this kind of modal experimentation wasn't really for me, I tried to write a piano piece in locrian major, and I remember it being really hard for me to handle how closely related it was to the leading whole tone scale -- it made my sense of tonic really hard to maintain.
I would volunteer to share it, but I've shared almost none of my music online, and I'm not sure a piece from 6 years ago that I thought was bad when I wrote it is the first impression I want to have :P
Regardless, hopefully you find this interesting! If nothing else it's a keyword to google ^_^
Thanks very much! I hadn't bumped into "Locrian major" before, or the "leading whole tone scale". So now the 7 scales I did not discuss are starting to come into view!
Btw, I find scale names like "Arabian scale" and (worse) "Hindu scale' and "Oriental scale" irritating, not so much out of general-purpose opposition to what Said called orientalism, as because there are probably lots of Arabian and definitely shitloads of "Hindu" or "Oriental" scales. It's sort of like calling major the "European" scale (which would probably be more accurate).
I'd be pretty surprised if you'd bumped into locrian major, haha. Like I said, I found it back when I was going out of my way to be "different". I think the "leading whole tone scale" has much more application (at least in my music) and is the kind of name you could reasonably guess from the scale.
As for all these orientalist names, they're kind of a historic artifact, and I completely agree that we should move away from them. There's a huge variety of reasons to do so, three of which are already in the thread, haha.
@hallasurvivor - on the other hand I feel utterly unmotivated to rename the Phrygian etc. modes.
The only reason I'd have bumped into Locrian major is that I'm reading stuff about weird modes, and watching videos about them. Most are a lot less amusing/irritating than this one, but this lists all 7 of my "missing modes":
@johncarlosbaez@zenorogue An example worth considering is the “be-bop scale” (more a family of scales). These are 8 note scales created by inserting a passing tone into a familiar scale with the goal of making the chord tones (1-3-5-7) occur on the down beats when played up and down the scale with swung 8th notes. In your notation, an example is wwhwwhhh which has the adjacent half steps for the addition of a flat 7. You can alternatively add a flat 6 to get wwhwhhwh. Both are common.
@R4_Unit - yes, it's definitely relevant and cool! I've made some charts of all 7-tone subsets of the chromatic scale that have two half-steps in a row. Here's one listing their most common names. So, you're pointing out that these can be extended to 8-tone bebop scales.
@hallasurvivor was the one who alerted me to the names of some of the scales here.
@Dyoung@rivfader - nice! But scales where all notes are 3 or 4 semitones apart are pretty unusual, since those count as huge intervals between notes. For example double harmonic major is considered fairly far-out, and it has just two 3-semitone steps.
@johncarlosbaez before I studied Physics I was at University for a year in music. In theory class I could rattle off the notes in a scale or chord, but when they asked me to sing an interval... Oh boy. It was then I knew I needed to change programs:)
@Dyoung - yeah, sounds like me doing experiments in physics lab! That's when I knew I was going to major in math.
I'm pretty crappy at singing a given interval, but I was immensely proud of myself when I first managed to sing a scale in the Lydian mode. Since I play the piano more than sing, it helps to visualize a keyboard for some reason.
@johncarlosbaez Now you combine this with all the possible chord progressions, and you and up with nice ways of implying tonal centers that might or might not be the actual tonal center (see e.g. in Satin Doll)
@johncarlosbaez e.g. in Cm harmonic or melodic, 5th chord/arpeggio is G7= GBDF, a dominant 7th chord with a major third B which resolves to the key center C a half-step above in the dominant cadence V7 --> i
@jaztrophysicist - Thanks! Harmonic major indeed breaks out of that criterion, giving 7 more modes, and harmonic major does too, giving you 7 more. I just saw a Beato video where he explains (after an annoyingly long chat with a friend) how each mode of harmonic minor, inverted, gives a mode of harmonic major. (The title of his video mentions a 'paradox', but there's no paradox.)
But my ability to absorb the math of modes greatly runs ahead of my ability to play anything. 😢
Of course, the moment you leave the western tradition, the assumptions, "There are 12 notes in an octave" and "There are 7 notes in a scale" go out the window, and you get to explore things like microtonal or pentatonic scales found in many different cultures.
@BradRubenstein - Yes! But instead of trying to maximize the territory I explore, I'm trying to slowly expand it while really getting to know it. It's keeping me plenty busy to learn the particular flavor of these 14 modes - and it's even harder to learn how to improvise in these 14 modes, which are all quite widely used in jazz.
Add comment