Pentatonic Scales, Revisited
Tuning with 3:2 and 4:3
We turn for the third time to the pentatonic scale. Thus far, we have looked at the following concepts regarding the pentatonic:
- Possibly related to an innate propensity for the descending minor third pattern, with a whole tone above. Combining two of these patterns on different pitch levels would give us a variety of pentatonic scale patterns.
- Pentatonic scales can also be conceived, more theoretically, as being based around a central tone, with fifths in perfect 3:2 relationships derived above and below the central tone. Once these fifths are found above and below, a variety of pentatonic patterns can be discovered, depending upon which tone is considered to be the central or tonic note.
- The pentatonic scales derived from finding two fifths above and two fifths below a central tone have one unifying characteristic—they lack the tritone that we find in septatonic (7-note) scales.
- These patterns also lack a semitone between any of the pitches in ths scale.
Thus we have so far discovered pentatonic patterns based on 3-limit system, using only perfect fifths (3:2). What if we were to expand our system to include 5-limit intonation?—what new tones or patterns might this bring us?
As we saw at the end of the last chapter, a problem occurs with 3-limit, pure Pythagorean intonation when we confront the tuning of the major third:
- When we add two Pythagorean thirds, we take the tone between the arithmetic and harmonic mean and discover the ratio of 9:8. We then "fill in" the tetrachord with two whole tones, giving us two whole tones above the tonic note, with a remainder that we call the Pythagorean semitone. In Cness, we can give these the notes C D E.
- When we add these two semitones, we find the ratio of 81/64 between the C and the E. (9/8 X 9/8).
- However, we know from "listening in" to the harmonic series that the series itself tunes the third to a ratio of 80/64 (5:4 expanded to give a shared denominator with the Pythagorean third (16/16)).
Thus, we can conclude that it is "better," or more resonant to tune our major thirds with 5:4 intonation. What would we find if we did so using pentatonic scales?
Well, instead of finding the notes of the pentatonic scale using all 3:2 perfect fifths, we can find them by limiting the "spine" of fifths to C and G, find the Pythagorean whole tone above each note, and then working out the major third above the C as being a pure 5:4 ratio above the E, rather than a distant fifth more than two octaves ablve. Thus we have tuned our intervals according Just intonation:
Extending this idea further, we could create another scale by finding the third above the G as well as the one above the C. This would give us a pentatonic scale with a raised seventh degree.
As a good example of these two pentatonics, and by way of introduction to the world of melody and Indian classical music, let's listen to two examples of these scales, as performed within the Indian classical tradition.
Two Ragas: Bhupali and Hansadhavani ("Flying Swan")
Introduction to Ragas
Here are two North Indian raga performances, based on the two pentatonic scales above.
Ragas are a wonderful way to first understand melody, because the entire Indian classical tradition is based solely on the employment of complex rules of melody in service to an expressive intent. Since all raga is conceived and performed over a droning tonic and fifth, the variety and interest in raga music is solely based on the manner in which the raga contains a specific subclass of notes, defined within the two tetrachords, and the way in which those notes are approached and performed.
We can only briefly define raga here and in the next page. In essence, a raga is a collection of notes that form a scale. These can be as few as the five note ragas described here, or as complex as seven note ragas that contain different variants on the basic seven tones, depending on whether one is moving in an upwards or downwards direction. Different ragas have different emotional contents, and often they are meant to be performed only at certain times of the day or even only in certain seasons of the year. The mood of a raga is a complex combination of tonal movement, emotional expression and cultural assumptions. To us, we can simply appreciate them for their beauty of sound and simplicity of expression.
First, listen to these two melodies, each based on the two ragas listed above. The first, played on the Indian Bansuri (bamboo flute), contains the sixth (the A) andis more quiescent in nature. The second, sung in a traditional Indian classical style, contains the seventh, with its tendency to reach up to to the tonic, and has more of a quailty of longing and urgency. It is often used to sing love poetry about the yearning of the soul when it is away from the beloved, whether that beloved is a human relationship or, as if often true in Indian classical (Sufi-influenced) music, a yearning for the soul's re-connection to the divine.
Both raga performances are on this same player. You can toggle between them.
Next: 4B Ragas and Chants