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It does cause subtile problems in situations where musicians do tune their fifths to a 3:2 ratio, such as a choir in a church.) The Solution: Equal Temperament The inescapable bad news is that our notion of an octave-the most basic interval, found throughout nature and representing a 2:1 ratio of frequencies-and our notion of a perfect fifth-the second most basic interval, also found throughout nature and representing a 3:2 ratio of frequencies-are incompatible with the very notion of the circle of fifths-the notion that moving up a fifth twelve times should move us up seven octaves from the original note.īut the good news is that they’re close enough that we can “fudge it” and never even worry about the difference. What you’re hearing is the 23.45 cent difference that’s accumulated over the course of the journey around the circle of fifths. x/music/blog/IntonationAroundCircleOfFifths.mp3 Moving around the Circle of Fifths: Intonation Diverges

And notice how out of tune it sounds with the original C: Listen to the sequence and the final C that it arrives at. (Sometimes we lower it an octave so it stays in the same general range.) We keep the original C playing underneath. In each case the next note is a fifth higher, 1.5 times the frequency of the previous one. Here’s an example of a starting note (middle C), followed by G, D, A… going around the circle of fifths and finally arriving back at C. The two pitches differ by 23.45 cents, or approximately a quarter of a half-step. That’s close to 7136 Hz but not the same. After all, moving up an octave just doubles a note’s frequency, right?īy that reasoning the frequency of that high A is, which is to say 7040 Hz. Since this A is seven octaves above the low A (55 Hz) we started with, we could have found its frequency by just doubling 55 seven times. The frequency of this high A, 7 octaves above the original one, is 55 Hz multiplied by 1.5 twelve times, which is to say or 7136 Hz. If we continue to generate all the notes by going up in fifths twelve times, we arrive again at A:Ī → E → B → F# → Db → Ab → Eb → Bb → F → C → G → D → A

We can get the frequency of B by going up another fifth from E: it has a frequence of 123.75 Hz because 123.75 = 82.5 × 1.5 = 55 × 1.5 × 1.5. (The unit of frequency, hertz, abbreviated Hz, means vibrations per second.) When we multiply 55 by 1.5, what we get is the frequency of the note (E) a fifth higher-namely, 82.5 Hz. The A string on a bass has a frequency of 55 Hz, or 55 vibrations per second. Almost all of the #music we listen to relies on a crazy #coincidence of #math. The problem is… (1) and (2) can’t both be correct. The frequencies of two notes an octave apart have a ratio of 2:1.

If you’ll stick with me for a minute I’ll show you what I mean. Did you know that much of what we understand as music rests on a big mathematical coincidence? Namely, that two to the power is approximately equal to 1.5.