Music Theory: Harmonics, the Circle of Fifth & Equal Temperament
Harmonic series is at the heart of music harmony. When any object shakes, it creates a fundamental wave frequency vibration, and an infinite series of overtones whose relative frequencies & wavelengths follow the harmonic series:
1/2, 1/3, 1/4,…
As mentioned, music, like story-telling, is a series of conflict resolutions using musical notes
These notes form frequency ratios & combinations (harmony & chords), motions (melodies) and frameworks (keys)
We’ll look at how the musical notes are constructed via the “circle of 5ths” (Co5)
It’s called 5th bc it’s the 5th note in a scale of, typically, 7 notes, from traditionally 12 (still 12 today) extant notes in an octave.
Remember, the “Major scale” “mode” means 7 notes starting from 1 like this,
where 1 is a “half-step”, 2 is a “whole-step”:
2 2 1 2 2 2 1
Starting on note 6, 2 below the end, changes the intervals pattern and forms the 6th “mode” called the relative “minor scale”. It sounds dark and sad. The other modes have other names. Mode 2 is Dorian, Santana uses it
Co5 is used to arrive at (12) musical notes using the simplest harmonic series of tones — the “5th” interval, which comprises 2 notes at a frequency ratio of 3:2.
2:1 string lengths creates an octave interval difference. The next simplest harmonic interval is a perfect-fifth made by 3:2 string lengths.
The simpler the frequency ratio of 2 notes, the less dissonant/conflicted and more consonant it sounds.
We get 12 notes by moving up in 5ths, 12 times, over 7 octaves.
This forms the “circle of 5ths” (Co5).
A to E is a 5th (5 letters).
The letters wrap back around from A-G (7 notes in a scale).
A2 means the note A in octave 2. Each octave has 2x the freq.
A2 ~100 hz base (110 hz)
E2 ~150 6/4 (add 50%)
B3 ~225 9/4 (add 50%)
B2 ~112.5 9/8
The Equal Temperament (ET) music scale on piano breaks the purity of the “circle of 5th” notes construction in the pure “Pythagorean temperament” whereby 1 octave is 2:1 freq and the 5th scale note (7 of 12 half-steps in the octave) is the next simplest 3:2 harmonic ratio.
A minor drawback with ET is
2⁷ = 128, ie, 7 octaves make freq go up 128x
BUT:
1.5 ^ 12 = 129.75, ie, 12 5ths makes freq go up 129.75x
It’s a little off!
ET condenses everything to fit perfectly into the octaves (otherwise it would all sound off on piano)
There aren’t perfect solutions for piano
The results are less pleasing frequency combos (chords), but it’s barely perceptible to all but 1 in 10,000 people with perfect pitch
However, Pythagorean and other non ET temperaments have “wolf intervals” where things sound terrible
ET is a happy-medium
As seen above:
A to B is a 12.5% freq delta increase (from ~100 to ~112.5 hz)
Let’s see how close this 12.5% is to what ET would use to make a perfect octave of 2x freq
12 1/2 steps to octave
~2 = 1.0595¹²
= 5.95% freq increase each 1/2 step
6 whole steps to octave 2x
~2 = 1.1225⁶
A to B is 1 whole-step
ET is 0.25% off wrt to the original A
12.5–12.25%
ET is 2.04% off wrt to the delta
12.5/12.25 = 1.02040816
The 1/2 step freq delta (5.85%) is less than 1/2 of the whole step freq delta (12.5%) bc octaves are 2x so it all works by geometric freq % increase
Vocals, violin & other instruments can do what piano can’t — play perfect 3rds, 5th and other simple harmonic ratios in the Pythagorean temperament pure Co5. Harmonics can affect resonant frequencies
Here’s an amazing 1st lecture by Leonard Bernstein at Harvard Music in a 6 part series on the mind & music: from linguistics, mono-genesis, phonology, syntax, semantics, ambiguity, crises & poetry
Read on in my other blog, which covers the bigger picture of how the results of this blog are used to construct actual music
Music should be heard
Find great videos about music theory at bottom of my other blog post here: