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An interval is simple the relative space between two notes. Some intervals sound purer to us than others. The purer an interval sounds, the more consonant it is. The less pure, or harsher, it sounds, the more dissonant. What makes an interval sound pure? There are two main things involved here. First, is the relative nature of the frequencies.
Take the simplest case: a unison. A unison is two notes that have the same frequency. If your guitar is well tuned, you can play a unison by playing the open high E string, and the second string at the fifth fret. They sound the same note, and if they are in tune, they are perfectly in tune.
The next example of a consonance is an octave. The note an octave above another has twice the frequency. This means that the higher will coincide with lower once every two cycles.
The next harmonic is the fifth. A perfect fifth has a frequency relationship with the bass note of 3:2. This whole number relationship is very satisfying. It creates resonance and balance, since the overlapping of the waves of the two notes will happen on a regular, predictable basis.
A dissonant note will have a much more complicated ratio. For example, the tritone (which is the b5, or blue note), has a frequency ration of 45/32. This is the most complicated, and therefore dissonant of the diatonic intervals. It is certainly not the most dissonant interval possible. Microtonal music contains examples of dissonance which make diatonic dissonance a positive joy to the ears by comparison.
For an example of extreme dissonance, check out Penderecki’s Threnody to the Victims of Hiroshima
The context is the other part that determines how dissonant or consonant an interval is. That usually means its relationship to the key. For example, in the key of C the major third from C to E is a very consonant interval. But if we are firmly in the key of B, that same interval could sound extremely dissonant. What matters is not just the relationship of the two notes to each other, but also their relationship to “home.”
When people create scales, there are typically two goals in mind. One is to divide octaves up into usable divisions that do not have leaps that are too large. And the other is to make those divisions contain as many consonances as possible. As a result of these goals, lots of cultures have arrived at the pentatonic scales.
From Brian’s lessons, we know that there are two common pentatonic scales: major and minor.
The major pentatonic scale is Root, Maj2, Maj3, 5th, Maj6th. In terms of half steps (or fret jumps) the pattern is 2, 2, 3, 2, 3.
The minor pentatonic scale is Root, Min3, 4th, 5th, Min7. In half steps, or frets, thats 3, 2, 2, 3, 2.
What about their frequency ratios to the tonic note?
For the major pentatonic, the ratios are Root 1:1, Maj2 9:8, Maj3 5:4, 5th 3:2, Maj6th 5:3, Octave 2:1
For the minor pentatonic, the rations are Root 1:1, Min3 6:5, 4th 4:3, 5th 3:2, Min7th 9:5, Octave 2:1.
Notice that these are all very consonant intervals. The ratios involve relatively simple fractions. If you factor any of the numerators or denominators, you are dealing with factors of only the prime numbers 2, 3 and 5. It’s for this simple reason, one straight from physics and human physiology/psychology, that pentatonic scales are so common.
The last thing I want to mention, but won’t get into, is that all of these ratios have been perverted a bit in Western music by our compromise of Equal Temperament. The ratios I mention are pure, but they are also rarely heard in any of our music. To make every key be approximately like every other key, we distort their intervals away from pure intervals. Everything we play is actually slightly out of tune.
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