This essay is kind of a brain dump for me, so here’s the first of my geeky tangents. This section may be safely ignored by those only wishing to learn to improvise (just skip to The Black Keys).
If you think of pitches as vibrating strings, an octave vibrates twice as fast as its root (ratio 2/1). A perfect fifth vibrates three times for every two times the root does (3/2). If we tune by fifths, the major 9th (fifth of the fifth) vibrates 9/4 the rate of the root (3/2*3/2), so the major 2nd (an octave down from the ninth) vibrates at 9/8 the rate of the root.
People seem to naturally like consonance, which is basically multiple vibrations with frequencies related by small whole number ratios. This gives another reason the pentatonic sounds good: musically it’s always the five most closely related notes by fifths, so produces lots of consonance.
The system of tuning by fifths is called Pythagorean Tuning. If we go all the way around the circle with perfect fifths, (3/2)^12 * ½^6 = 2.0273, we end up a bit sharp for the octave. (The ½^6 moves the pitch to the octave we’re interested in. Going up six major seconds (9/8)^6 gives the same result.) But it’s close, and this near coincidence is the reason we have twelve tones. We end up sharp by +23.5 cents, a cent being a hundredth of a semitone. The 23.5 cents is called the Pythagorean Comma.
There are other ways to tune than just using fifths. Tuning according to ratios of small whole numbers is known as Just Intonation. The just fifth (3/2) and just fourth (4/3) are the same as in Pythagorean tuning, as is the just major second, a ratio of 9/8. A just major third has the ratio 5/4 and a just minor third 6/5.
We see that consonance in fifths results in some other intervals being more or less out of tune. In western music we usually think of the major third as the next most consonant interval after the perfect fifth (and its inversion, the perfect fourth), even though it’s four steps away on the circle of fifths. 3/2*3/2*3/2*3/2*¼=1.265625 (the ¼ gets us back in the starting octave), which is pretty close to the consonant 5/4=1.250000.
But pretty close isn’t really that close. The system of tuning by fifths results in a pretty out of tune major third (it’s sharp by +22 cents) when compared to the just 5/4. That is pretty dissonant to the ear.
Western music generally solves this by decreeing all the semitones must be equally spaced (i.e., it’s the same ratio between every pair of adjacent notes) and 12 semitones has to make exactly an octave, a ratio of 2/1. If we call the ratio s, we have s^12=2, i.e. s is the twelfth root of two. So, s=2^(1/12)=1.05946 for the semitone. This makes a fifth 2^(7/12)=1.4983, very close to 1.5 (3/2) but a little flat (-2 cents, a twelfth of the Pythagorean comma). The major third is now 2^(4/12)=1.2599, +14 cents sharp compared to the just 1.25 (5/4). It’s closer than what we just got when we went four steps around perfectly tuned fifths. (The major third is relied on heavily today, but one imagines the pretty out-of-tune Pythagorean major third was more avoided by ancient musicians.) The minor third is now 2^(3/12)=1.1892, flat (off -16 cents) compared to the just 1.2000 (6/5).
You can see how in our system of equal temperament we sacrifice some consonance to make all the keys equally in tune, that is to say, equally out of tune.
Tangent on the tangent: Are there any other circles of the twelve tones? Why is there no circle of thirds? Only by adding 1, 5, 7 or 11 semitones can you make a repeating list of all twelve pitches. These numbers are relatively prime to 12 (no common factors). They correspond to a circle of semitones (c c# d d# e f …), fourths (c f Bb Eb …), fifths (c g d a …) and descending semitones (c b a# a g# …) respectively. When you use an addend that’s not relatively prime to twelve (like 4), your list repeats before you get all 12 notes in (c e g# c).
Next: The Black Keys
Next: The Black Keys
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