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Pythagorean Ratios and Scales

From the Tetrachord to the Scale

In the previous section we saw how a scale fragment can be created by "filling in the tetrachord" with a series of half steps and whole steps, and then combining these tetrachords into a variety of modes (scales). We are going to take a step back now and look at how this "filling in" is conceived, using Pythagorean principles. In so doing, we will learn the ratios between all the steps of the scale.

Pentatonic Scales from Pythagorean Ratios

As we learned from the chapter on ancient music theory, the scales that emerge from the note we today call D are the most elemental. We see this essential "Dness" of our music in many ways, most notably in the fact that our stringed instruments are still centered around D. Violins and mandolins are tuned GDAE, violas and cellos are tuned CGDA. The bottom four notes of the guitar, corresponding with the notes of the bass guitar and orchestral bass are EADG. The presence of D (not C!) in all of these tunings point to the historical ancestry of the ultimate Dness of our music. We will be exploring this relationship between Dness and Cness (as I call it) shortly. For now, let's stay in the more ancient world of Dness.

The first principle is that we take a central tone—let's call it the tone center—and build (and tune!) perfect fifths in precise 3:2 intonation above and below the tone center. Let's start with just the fundamental pentatonic scale, which for our purposes begins as:

From this first principle, a variety of pentatonic scales can be realized, depending on which tone is taken to be the resting tone. Thus, using the central, the high and the low tones as the actual tonic or resting tone, we can create these five note patterns (with the resting tone being the low tone), using octave reduction/expansion to bring the other notes within an octave:

  1. The first version takes the central tone as the tonic or resting note. We could think of this as the Dorian pentatonic, due to its relationship to the Dorian mode.
  2. The second one uses the lowest tone and builds up from the bottom, pulling all the notes down into the octave. This is the most "tonal" version of the pentatonic and is the scale we are most used to in the West (Amazing Grace, O Freedom, lots of other "spirituals," huge numbers of English and Continental folk songs).
  3. Number 3, with A as the resting point, keeps the original D as the central tone and pulls the other pitches closer in with octave reduction, resulting in a scale beginning on A. It is the second most common pentatonic in the West and we usually refer to it is as the minor pentatonic. Since we will be more modally based in our class, we'll call it the aeolian pentatonic.
  4. The last version is the reverse of number two, as it builds down from the top and could be considered the reciprocal version of the pentatonic, which we call also call the Phrygian pentatonic.

These contrasts encapsulate some fundamental principles and polarities that are worth pointing out:

If we now look at the ratios of these notes, we can observe the following. These ratios will hold, regardless of the type of pentatonic we are creating, because the intervals are all based on 3:2 relationships. This is beauty and simplicity of 3-limit systems, which work great for melodic music but create problems with harmonic music, as we will see.

The Ratios of the Pentatonic in the 3-limit system

The ratios are:

Please work this out for yourself if it doesn't immediately make sense to you.

Make sure you can clearly sing and hear these in your imagination, and that you can internalize the sound and feeling of them. Try very hard to hear these as a relationship of pure perfect fifths (in other words, don't just play them at the piano but internalize them as pure harmonic ratios - sing them above a central tone).

Building the Dorian Mode

The full Dorian scale (mode) can be conceived of as being created by an additional pairing of fifths above and below the pentatonic set. This give us:

Taking the central tone as tonic (method # 1, above), it is easy to see the Dorian mode emerge, using octave reduction/expansion to pull the notes within the octave:

It is important to note the beautiful symmetry of the resultant Dorian mode. The Dorian mode has the same order of half steps and whole steps whether one ascends or descends. No doubt this has much to do with its rather calm and noble character. This relates back to our very first page, when we discussed the symmetry of the bass clef. We can now see another reason why D dorian and "Dness" is more ancient, since the bass clef (F clef) is one of the two most ancient clefs, dating back to Gregorian chant. This clef is centered around the symmetry of the Dorian mode ("The Symmetry of Dness").

Thus the scale is TSTTTST whether ascending of descending.

To our ears, it is that characterstic "minor third with a major sixth and minor seventh" that is the Dorian modes distinguishing characteristic. The minor third is there whether you go from D up to F or from D down to B. Similarly, the major sixth is there whether you go from D up to B or D down to F. Minor seventh? D to C or D down to E. And of course the perfect fourth and fifth are inversions of each other. Check this our for yourself: start at the central D note, find an interval in one direction and find the interval again in the other direction. Minor third balances with minor third, major with major, and so forth.

The Remaining Ratios

As we move into determining the other ratios, a little mathematical reminder is needed:

We also need to introduce the concept of octave reduction. We did this in a musical way in discovering the pentatonic scales above. We found fifths above and below a central tone, then brought them up or down an octave or two to bring them within the octave range of a scale.

Octave Reduction, explained

A simple process for finding the ratio that corresponds to the harmonic relationship of a scale tone to its tonic is as follows:

Let's clarify this one last time with musical notation:

Back to the Dorian Mode

What is also important to notice when creating the Dorian mode from the Pentatonic scale is that we have added in the phenomenon of the semitone, which is of course what distinquishes the septatonic (7-note) scale from the pentatonic. The ratios of D to E, G, A, and C were derived above. Now we need to add the ratios of D to B and D to F, and then find the intervals between them.

What remains is to find the ratios that exist between each step of the scale. These ratios are found by dividing the various fractions and reducing down as needed.

When this is accomplished, the following result is obtained. All whole tones work out to be the ratio of 9:8, while all semitones work out to be the ratio of 256/243 (!):

1

 

2

3

4

5

6

7

8

1/1

9/8

32/27

4/3

3/2

27/16

16/9

2/1

D

E

F

G

A

B

C

D

9/8

256/243

9/8

9/8

9/8

254/543

9/8

The 9:8 ratios are obtained by dividing the various ratios and then reducing down the fraction. For example, the interval between the F and G can be shown to be 9:8 by dividing the two ratios (27X4/32X3 = 108/96) and dividing both by 12, which gives 9/8. A similar process occur between the A and the B, using 6 as the reduction factor. Do this yourself!

The semitone between E and F and between B and C is derived by the same process. Thus 8 X 32 / 9 X 27 = 256/243. Notice how much more complex is the the ratio of the semitone! As this is already within the octave, it can't be reduced down to a simpler fraction.

I know that these numbers may seem a bit un-musical. No musician ever says "Hey, man, I liked how you played that series of 9/8 ratios over that chord tuned to the ratio of 4:5:6!" But the concept behind this is what I want you to carry with you.

You are now grasping a fundamental principle of music theory, known to every trained musician for millennia. Tuning septatonic scales by Pythagorean, 3-limit intonation results in scales with identical whole tones and semitones (9/8 and 256/243, respectively). But, and it's a very big but:

As neat and tidy as this system seems, a fundamental problem occurs. In this system, two semitones do not add up to a full tone! Thus we can see that this is not an equally tempered scale, because the ratio 256/243 is not exactly a half of a tone. If you do the math and add two Pythagorean half steps (multiply the semitone ratio by itself), you get the wonderful ratio of 65536/59049! This does not reduce down to a 9/8 ratio, is it slightly smaller. (It's easier here to express it in decimal notation:

This discrepancy between ratios is but the first of many important ones we will encounter in understanding intonation. It is one of the Pythagorean commas, which will be covered in the section after this.

Meanwhile, we need a new page to cover one last aspect of tuning ratios: the comparison of the Pythagorean and Just systems of intonation. This is also known as comparing the 3-limit and the 5-limit systems.

Next: 3C Just Intonation