Rotationally Symmetrical Scales

I think yesterday's post was very bad in terms of the artistry of a blog post.  What was it about?

I've got a theory that there are three musical scales which are epitomes of emotional resonance, and that the majority of music we create and consume relies on bastard approximations of these three scales.

There's a number of assumptions you have to make to get to this position.  I'll list them:
1. A note with a frequency twice or half that of another note is somehow equivalent (the octave).
2. The set of possible notes is the set of notes produced when you extrapolate the circle of fifths through the octaves (this gives twelve semitones in an octave).
3. Divisions of three semitones (giant steps) are ugly (we must therefore pick six or more notes from the 12 semitones to use).
4. Wholetone scales with six notes in carry little emotional value except as a contrast to another scale.
5. The fewer notes in a scale, the greater the emotional affect (7 is therefore optimum).

This much is common to all western music, but it's good to state your assumptions because then you know where you are.  There are flutes from the palaeolithic (the time of cave paintings) whose design debatably agrees with all of these points.  Human beings have always used these ideas, and you use them every time you listen to music.  But they're not just a product of emotions and evolution.  If they're not objectively true on the level of proof and logic then they're at least sensible mathematical choices.

It happens that there are three meaningful mathematically possible combinations of two semitone and five wholetone intervals, each with seven flavours depending upon which note you start on (these are not trivial for reasons of temperament and instrumentation), each in turn with seven modes depending on which interval you begin at.  There are one hundred and forty seven possible diatonic scales without giant steps.

My proposition is this: scales which are rotationally symmetrical about the octave are the purest.  A quick glance at a piano or a fretboard will show that to maintain rotational symmetry and the rule of no giant steps, the fifth and the fourth necessarily have to be in the scale.   That's a good start as these are the most commonly used intervals in all music where their absence is not a feature.  The diminished second is not possible because to preserve the symmetry you would need to have the fourth, the tritone and the fifth, which is more note selections than there are available.  So we must have the ordinary second, and then you have to have the sixth to maintain the symmetry.  This leaves only the question of the third and the seventh.  There are thus two rotationally symmetrical scales available without giant steps.

The first of these scales is the normal everyday major scale (called the "ionian" scale when you're talking about modes).  This makes sense, as western (and many non-western) musical theories assume this as the basis for music without looking any deeper into it.  If it's that easy to accept it as true, it may well be so.

The second scale is the dorian.  It is my belief that western theory dropped the ball when it adopted the aeolian as its minor mode.  You can see that something's not right when you look at all that classical fudging over harmonic and melodic minors (classical theorists straight-facedly insist that you should play different notes going up compared to coming down a minor scale, even in equal temperament).  When you need to stick that many plasters on a system, it's broken.  The only thing that has sustained the supremacy of the aeolian mode over the years has been too much education.  Wherever musicians have been left to figure it out on their own, they've usually reverted to the dorian as their minor mode, often without realising they've done it.  As examples, I present Scottish, Irish and English folk musics, bluegrass, the 60s, the 70s... even 90s dance music.  Wherever musicians lack a man in a wig to tell them not to, they have consistently flattened the seventh.

Interestingly, if you accept the presence of giant steps in the scale there is only one additional scale that preserves the symmetry.  I do not have a name for this scale but it belongs to the family of "locrian majors" (which are neither locrian nor major).  The scale is giantstep-semitone-semitone-tone-giantstep-semitone-semitone.  This scale has a certain bluesy feel about it, as though the quarter-tone african scale that was the father of the blues is somehow related to this.  It certainly looks that way on a fretboard, but I don't know enough to comment further.

However, there is some theoretical dissonance with this third scale.  We needed to assume that giant steps were bad in order to chose to form seven-note scales.  How can we then turn around and say that giant steps are OK at a later stage?  If giant steps are OK, why not form five-note scales?  In practice, that's exactly what those old bluesmen did.

One last comment: insisting that the diminished second be rotationally reflected in the tritone is a little pedantic as this reflection falls outside the home octave.  If you ignore this consideration you end up with the lydian mode.  So you could make an argument that there is something pure and special about the lydian mode, too.  This does seem like a viable idea; there is a unique quality which is neither major nor minor that is probably strongest in the lydian.  I wish some 50s jazz leader had looked into it.

If you accept both compromises (giant steps and diminished second) there are at least another two scales with rotational symmetry, one of which is the "double harmonic" scale from the Pulp Fiction theme.  The other has two semitone intervals at the beginning and sounds like it's derived from the chromatic (that is, is sounds pretty characterless).

Coming soon: experiments in thirds-propagated temperaments.