Background: Major & Minor Intervals

Every major interval (major second, third, sixth & seventh) has a corresponding minor interval that is a half step smaller. The major intervals can generally be described as having a “brighter” or “happier” sound; the minor intervals as having a “darker” or “sadder” sound.

Interestingly, the inversion of every major interval is a minor interval and vice versa. To see this, remember that if an interval is made up of n half steps then its inversion comprises 12-n half steps. If we look at intervals in terms of the number of half steps they comprise, we see that the inversions are as follows:

Background: Neutral Intervals

So Major and Minor intervals are always inversions of one another.

If we divide the octave into quarter tones (giving us 24 pitches) rather than half steps (12 pitches), there is another note in between every major and minor scale degree. The interval it forms with the root is a quarter tone larger than the minor interval and a quarter tone smaller than the major interval, and it is called the neutral interval. A neutral second is a minor second plus a quarter tone (or a major second minus a quarter tone). A neutral third is a minor third plus a quarter tone, etc.

We can notate quarter tones using the half-flat ( B ) and half-sharp ( µ ) symbols. Here are the intervals notated, along with audio of the neutral intervals compared to their major and minor counterparts:

Neutral Second

Neutral Third

Neutral Sixth

Neutral Seventh

Neutral Inversions

We can see that the inversions of all neutral intervals are also neutral intervals: An octave is made up of 24 quarter tones, so two intervals are inversions of each other if the number of quarter tones they contain together add up to 24. Indeed, 3 + 21 = 24 and 7 + 17 = 24. So a neutral seventh is an inverted neutral second, and a neutral sixth is an inverted neutral third.

Neutral Fourth & Fifth

In a previous blog post, I argued the advantages we’d see if we adopted the (admittedly very nonstandard) terminology of the major & minor fourth & fifth. Well, under that nomenclature, the definitions of neutral intervals extend naturally to fourth and fifths as well:

Just to reiterate: this terminology I’m using for fourth and fifths is not standard at all. Fourths and fifths, as we all know, are usually referred to as perfect, augmented and diminished, not minor or major. Though the terms “major fourth” and “minor fifth” are sometimes used in microtonal music to refer to what I have just called the “neutral fourth” and the “neutral fifth”.

That said, I’ll be using the “minor fourth” and “major fifth” terminology instead of perfect intervals for the rest of this blog piece because, as we’ll see, it just makes it so much easier to express generalities succinctly. I apologize if that causes confusion. (I’d welcome your comments on the notion of major/minor fourths & fifths—if you’re interested, please read that original post and comment there.)

Given this system, it is natural to define neutral fourths and fifths as we have done above: The neutral fourth and fifth each fall midway between their minor and major counterparts, just as for all the other intervals. And the neutral fourth and fifth form an inversion pair, just like all the other neutral intervals.

Background: Major & Minor Modes

When most people think of major and minor scales, just a few options come to mind… such as the major scale (white notes of the piano, starting on C) and the natural minor scale (white notes starting on A). But there is a whole spectrum of scales ranging from “very” minor to “very” major.

The Modes, Bright & Dark

When we look at the notes of a major scale—say, the white notes if we’re in C major—we get different different scales depending on which note we start on. The seven possible starting notes give us seven scales, which are called the “modes” of the major scale. The darkest sounding one of them, which we could call the “most minor,” is the Locrian scale, which starts on B. The brightest one of them, which we could call the “most major,” is the Lydian scale, which starts on F.

To better compare these scales, let’s transpose them both to C:

Notice that aside from the root, C, each of the notes in the Locrian scale is a half step lower than its corresponding note in the Lydian scale. That accounts for the darkness of the Locrian scale. The Lydian scale is made up entirely of major notes. The Locrian scale replaces each with its minor counterpart, a half step lower.

If we look at all the modes, transposed into the key of C, Locrian and Lydian are the two endpoints of a continuum from bright to dark. We can generate all the modes by darkening (lowering), one by one, each note of the Lydian scale until we reach the Locrian scale:

We can see the process graphically:

Dorian in the Middle

In this scenario, Dorian is the mode midway between bright and dark. We can arrive at it from a Lydian scale by lowering three notes, or from a Locrian scale by raising three notes:

Dorian’s Symmetry

Whereas the Lydian scale is made up of the root plus six major notes, and Locrian is made up of the root plus six minor notes, Dorian is made up of the root plus three major notes (the second, fifth and sixth) and and equal number of minor notes (the third, fourth and seventh). You can see this on the circle of fifths, where major notes are clockwise of the root and minor notes are counterclockwise:

The symmetry of the Dorian scale can be expressed in many ways. Dorian is “closed under inversion” which is to say, for every note in the scale, the note’s inversion is also in the scale. The major and minor notes in the scale all form pairs of inversions:

Each pair is chromatically symmetrical in that the two notes of each pair are equidistant from the octave’s midpoint, F#.

Each pair is also symmetrical on the circle of fifths, since the two notes of each pair are equidistant from the root, C.

Lastly, we see can see the symmetry in the interval sequence that makes up the scale. It reads the same forward or backward—which is to say, ascending the scale or descending:

The “Neutral” Scale

Consider for a minute the following scale:

It is made up of the root, plus (by the terminology we’re using) the six neutral scale degrees. Since each neutral degree falls midway between the corresponding major and minor degrees, this scale has the interesting property that the six non-root notes are a quarter tone higher than Locrian and a quarter tone lower than Lydian:

For fun, if we wanted we could call the scale the “Lydocrian” scale — but I’ll just go with “Neutral”. 🙂

Let’s listen to the three scales:

The Neutral Scale’s Symmetry

The Neutral scale is symmetrical in much the same way as Dorian. It too is closed under inversion, since it contains the six neutral intervals, and their inversions are those same six neutral intervals:

Each pair is chromatically symmetrical in that they are equidistant from the octave’s midpoint, F#.

And we see can see the symmetry in the interval sequence that makes up the scale. It reads the same forward or backward—which is to say, ascending the scale or descending. Here are the number of quarter tones separating successive scale degrees:

These quarter tones add up to 24, as we know they must for the scale to span an octave. And all but the first and last intervals are the same as for both Locrian and Lydian. Locrian begins with a half step (2 quarter tones) between the root and 2nd, and ends with a whole step (4 quarter tones) between its seventh and root. Lydian begins with a whole step and ends with a half step. All their other intervals are identical. The neutral scale simple uses 3 & 3 quarter tones to begin & end the scale instead of 2 & 4 or 4 & 2… which accounts for its symmetry.

An Example Phrase

Let’s pick a phrase and listen to it in Lydian, Locrian and Neutral.

Here is a phrase that uses the Lydian scale:

When we lower every note in the phrase except the root by a half step, we get a phrase that uses Locrian instead:

And now, here is a version “in the middle” which uses the Neutral scale. That is, except for the root, every note in the phrase is a quarter tone below the Lydian version and a quarter tone above the Locrian version:

Interesting, huh?

For comparison’s sake, let’s listen to the same phrase constructed out of the Dorian scale. This will be the same as the Lydian version but with all 3rds, 4ths and 7ths lowered a half step. Or, equivalently, the same as the Locrian version but with all 2nds, 5ths and 6ths raised a half step:

Dorian versus Neutral

The Dorian scale and the Neutral scale we’ve presented here both represent midpoints in the brightness/darkness continuum.

Dorian’s notes are equally divided between bright and dark: it has three major notes and three minor notes.

The Neutral scale has no such division of labor; all its notes are sort of half-major. Whereas Dorian makes an assertive statement, the Neutral scale sounds betwixt & between… neither here nor there.

Of course it’s bound to sound somewhat weird: it’s made up entirely of notes that are off the tonal grid that we, as Westerners, have been listening to all our lives. But the reason for its weirdness goes beyond that. The frequency ratios of the notes in the standard chromatic scale very closely approximate ratios of small integers, which makes their waveforms align beautifully and harmoniously. Not so the notes of the Neutral scale. And there seems to be no reasonable way to integrate the notes into the circle of fifths, which is so fundamental to how Western harmony operates.*

Parting Thoughts

I have no background in microtonal music. I figured that what I’ve called the Neutral scale here would be one of the standard quarter tone scales because it’s conceptually so simple. I was surprised that I couldn’t find any reference to it. Certainly not by that name—and not by another name either, though it is much harder to search for that. If you’re familiar with it, please leave a comment!

I was very curious about what the scale would sound like. I have to say, I haven’t found much musical use for it. Not yet, anyhow. But, then, I don’t play a quarter tone instrument so I’m limited to trying things out on the computer, which I find pretty limiting. Still, it makes for an interesting study, and I’ve enjoyed pushing my ears out of their comfort zone. I hope you have too.

I have more thoughts about integrating neutral interval elements into jazz, using a variation of the neutral scale. But this post has gotten long, so I’ll save that for a second installment.

In the meantime… comments welcome!

Footnote:
*  Actually, there are two logical ways to make the equivalent of a circle of fifths out of the quarter tone chromatic scale. The chromatic scale maps to the circle of fifths (and vice versa) by this function on pitch classes: [n]12→[7×n]12. We can create a mapping from quarter tone chromatic scale to give us the equivalent of a circle of fifths using one of two mappings.

The mapping [n]24→[7×n]24 gives us this ordering:

The mapping [n]24→[19×n]24 gives us this ordering:

As we can see, they both keep the notes of the standard circle of fifths in their usual position: C, G, D, etc. simply have additional notes placed between them at the odd-numbered positions on a clock with 24 instead of 12 positions. So in that sense it meshes with our existing system. But how does movement in 5- and 7-quarter tone intervals relate to all we know about harmonic resolution in the circle of fifths? This is just one of the things we’d have to sort out to make sense of the expanded circle in practice…