How Many Scales Are There, Really?

One thing my students often ask about is “scales” and which ones to learn. In an effort to clarify this, here are some permutational ways to look at this question. Keep in mind, I’m not really talking about which ones are most useful, or how to practice and apply them. What we will do here is do a little surveying, to see what’s out there. First, let’s establish some boundaries:

1. We will start with seven-note forms, since they get the most “mainstream” use. Our music notation system is clearly oriented towards that definition, with seven letters in our alphabet, and seven Roman numerals for analysis.
2. The term “mode” and “scale” will be used interchangeably. We will think of these forms as a sort of key signature, even if they don’t replicate any of the standard key signatures. Best to think of a mode as a collection of tones.
3. While useful, pentatonic and other reduced forms can usually be thought of as modes with missing tones. A pentatonic scale can be expanded to seven notes in more than one way, and a mode can be reduced to several different pentatonics, so the seven-note form seems to be a better definition of a tonal center.
4. There are symmetrical modes made by interval patterns. These can be very useful, but they are less prone to establish a strong tonal center. Examples of these are chromatic, whole-tone, diminished and augmented scales. We will look at them after exploring the more “neutral” sounding seven-note modes.

Methods

There are a number of ways to organize these forms; I have chosen an interval-based approach, using numbers of half-steps as the variable. It is possible to use terms like “minor 2nd” or to use scale degrees like “#4,” but they make the permutations messy and difficult to define. Once done with our work, it’s easy to convert back to these terms.

1. We will look for modes made of seven tones, made up of whole and half-steps only, and also those modes with one augmented 2nd (sounds like a minor 3rd.) Modes with two augmented steps cannot be formed without a pair of consecutive half-steps. (Not if we insist on seven tones. The augmented scales and other forms have multiple augmented steps.)
2. Modes that have two consecutive half-steps are difficult to use as a tonal center. The tone in the middle of the half-steps will typically sound like a chromatic passing tone, rather than part of the harmony. If used in the harmony, there will end up being two “competing” sounds that don’t typically co-exist under the same chord definition.
3. Modes that are rotations of one another are not considered as distinct. Rotation means to start in the middle of the scale and “go around” to the beginning; the intervals are still in the same order, just with a different root. Our interval system makes finding these easy. By this definition, Major, Minor, Dorian etc. are all lumped under the “parent” mode of a Major scale. (This doesn’t mean we shouldn’t practice all those forms separately, but for “collection” purposes, they can be put together.)

The Basic Keys

Let’s begin by building all the seven-note modes that are only made from whole (2) and half (1) steps. The intervals must add up to a total of twelve. (otherwise our scale won’t repeat at the octave) To build these, we will need five whole-steps and two half-steps; This is the only combination of seven “2s” and “1s” that adds up to twelve. (I checked)

1. 2222211 – Two half steps in a row, sounds like a whole-tone scale with a passing tone.
2. 2222121 – Rotated to 2122221, becomes Melodic Minor.
3. 2221221 – Rotated to 2212221, becomes Major.
4. 2212221 – Major
5. 2122221 – Melodic Minor
6. 1222221 – Same as number 1 above.

As you can see here, the only combinations that fit our requirements turn out to be Major and Melodic Minor. All the others are either rotations (modes) of these or break our rule of consecutive half-steps.

Counteth Thou to Three

To add a “3” or augmented second to the mode, we need six intervals of 1 and 2, adding up to nine to finish up the scale. It turns out that 222111 is the only possibility, three twos and and three ones. The only combinations avoiding two consecutive “ones” are:

212121
121212

Adding a “3” to either of these creates a symmetrical diminished scale missing one tone:

2121213  —> 212121(21)
1212123  —> 121212(12)

Placing the 3 between any pair of these tones creates a rotation of the above patterns. The diminished scale sound is a strong one, and this missing tone will usually get “filled in” by our ear.

Combinations with a single pair of consecutive ones:
112122
112212

These two yield two scales when the 3 is placed between the ones:

1312122  rotates to 2122131, Harmonic Minor
1312212  rotates to 2212131 “Harmonic Major” (Major with b6)

What about placing a 3 between a pair of 2s?

2+3+2=7, requiring four more numbers to create a 7-note scale. Those four numbers need to add up to 5 to reach the octave. The only result is 1112. This makes such a scale impossible without consecutive “ones,” unless we reduce the number of overall tones.

The Result

All the combinations of 7-note modes made of either whole and half-steps only, or with one augmented step, as in harmonic minor, yield only four “parent” modes:

Major: 2212221
Melodic Minor: 2122221
Harmonic Minor: 2122131
Harmonic Major:2212131

Each of these has a subset of seven modes. Without considering the symmetrical scales mentioned above, or the relative (pun intended) usefulness of the scales, we have a total of 28 modes or tonal centers.

Next time, we will look at the more “exotic” world of interval-based symmetrical scales, and even more unusual forms that don’t repeat at the octave. (gasp!)

Questions

1. Have you ever been overwhelmed by the number of scales and modes to learn?
2. Do you have a “favorite” mode?
3. Have you ever “discovered” a scale that you later found out was already documented?