How to calculate the extra semitone in one circle of fifths

This is an extension to my previous question when I tried to do the math to prove the following statement:

going up 12 perfect fifths takes one up 7 octaves plus one-fourth of a semitone extra

Going up 12 perfect fifths is equivalent to one cycle in the circle of fifth (or precisely the circle of C1-G1-D2-A2-E3-B4-F#4-C#5-G#5-D#6-A#7-E#7-C8). Since each fifth has a frequency ratio of 3/2, and with 12 perfect fifths in the circle, the end frequency should be (3/2)^12 or 129.75, roughly 1.75 more than that of the 7th octave in equal temperament, which is 2^7 = 128.

If my math is done correctly, the multiplication factor of 1.75 should be equivalent to one-fourth of a semitone extra, but it isn't! I tried comparing the 1.75 extra factor to a semitone in equal temperament and that in Pythagorean scale, and neither matches that statement.

• Comparing to a Equal Temperament semitone
The ratio between neighboring semitones is always 2^(1/12) or 1.059. One-fourth of that is 1.059/4 = 0.265. Well...not even close to 1.75. One-fourth of that is 2^[1/(4*12)] = 1.0145.

• Comparing to a Pythagorean semitone
If I choose a semitone between C and its closest/neighboring C#, their ratio should be (3/2)^7 / 2^4 = 2187 / 2048 = 1.068. Somewhat closer to the value of 1.75, but still not quite there.

How is this one-fourth of a semitone extra derived?

• "If my math is done correctly, the multiplication factor of 1.75 ": it's not a multiplication : 129.75 - 128= 1.75.
– Tom
Jul 4 '21 at 17:43
• Also : "One-fourth of that is 1.059/4 = 0.265" no. That would give lower frequencies. One fourth of that is 2^(1/(4*12)). (I don't have a calculator right now to do the NA), but that has to be over one.
– Tom
Jul 4 '21 at 17:52
• @Tom if I start with frequency x, by the end of the circle I have frequency x * 129.75 or x * (128 + 1.75). The 1.75 * x is the extra (or overshoot) from the octave 128 * x. That 1.75 is the multiplication factor of that extra overshoot.
– KMC
Jul 4 '21 at 17:57
• @Tom you are right on my mistake calculating the one-fourth. Let me edit it
– KMC
Jul 4 '21 at 18:00
• Yes, but that's not a multiplication factor: if I multiple 128 by 1.75 I don't get 129.75. To get that I need to multiply by 129.75/128. And that's a completely different number.
– Tom
Jul 4 '21 at 18:01

The problem is that intervals are ratios, so using subtraction to calculate the "extra" interval (frequency) is the wrong operation.

The "extra" interval is ~ 1.014.

(3/2)12 / 27
= 129.746337890625 / 128
= 1.0136432647705078125

One quarter of a semitone is

In equal temperament ~ 1.015

(21/12)1/4
= 21/48
= 1.0145453349

In Pythagorean tuning ~ 1.013

(256 / 243)1/4
= 1.053497942386831275721/4
= 1.0131142475

Thus, the differences are just a rounding error.

• Thanks for taking time to write a nice answer and not a bunch of comments ;).
– Tom
Jul 4 '21 at 19:19
• Nicely done, Aaron! Jul 4 '21 at 22:09

Sound pitch increases with logarithm of frequency, which makes calculations somewhat non-intuitive. For this reason pitch is often given in the units of cents, which are additive.

Number of cents of interval between frequencies f and f₀ is calculated as

x = 1200 · log₂(f/f₀).

For example, an octave, or f/f₀ = 2 has 1200·log₂(2) = 1200·1 = 1200 cents.

An equal temperament semitone of f/f₀ = 2⁽¹/¹²⁾ has 1200·log₂(2⁽¹/¹²⁾) = 1200·1/12 = 100 cents.

A perfect fifth defined as f/f₀ = 3/2 has 1200·log₂(3/2) ≈ 701.96 cents.

Twelve such fifths has simply 12 times more cents, 1200·log₂[(3/2)¹²] = 12·1200·log₂(3/2) ≈ 12·701.96 ≈ 8423.46, while 7 octaves has 7·1200 = 8400 cents. Difference between 8423.46 and 8400 is 23.46 cents, which is close to quarter of a semitone, 25 cents.