Why does it take 700 cents to get to a perfect 5th, which is 3/2 (one and a half) from the root?

I think I have the wrong perspective.

If you take an octave and express it in cents, you get 1200.

Half of that is 600.

A perfect 5th is 1.5 (3/2) of the root.

But 600 is not a perfect fifth, 700 is.

Even more so, if you look at the decimal ratios in this table https://en.wikipedia.org/wiki/Equal_temperament#Comparison_to_just_intonation, you will see that it takes "longer" to get to the 5th, and then from the 5th to the 8th is very "fast".

I have a feeling this has to do with the logarithmic nature of pitch, I feel like I am close to understanding why this is, but wondering if someone can help me out.

• The ratios are multiplied, not added. An interval of two fifths is 3/2 x 3/2, or 1.5 x 1.5 = 2.25 times the root frequency, not the 2.0 of an octave relation. – Russell Borogove Dec 5 '15 at 16:03
• Actually, a perfect fifth is 701.955 cents. But I don't believe anyone would ever be able to hear a couple of cents' difference. – Dawood ibn Kareem Dec 5 '15 at 21:48
• Yeah, logarithms aside, the answer is "because two fifths aren't an octave." They're about an octave plus a whole tone. – hobbs Dec 6 '15 at 8:12
• @DawoodibnKareem harmonically, the difference manifests itself as beating. An equal-tempered above A4=440 Hz is E5=659.26 Hz, so it will beat with a frequency of 0.74 Hz, which corresponds to a period of about 1.34 seconds. The higher the pitches, the faster the beating, so the beating between A5 and E6 is is twice as fast. Given the right tone quality, anyone ought to be able to hear that. – phoog Jul 18 at 15:57

You are exactly correct that it is the logarithmic nature of pitch that causes this effect.

In cases like this, I find that a picture is helpful. Here I've labeled equally spaced octaves (1200 cents) along the x-axis (representing pitch). I've then labeled the corresponding frequencies on the y-axis as multiples of some arbitrary base frequency f. Note that each octave doubles the previous frequency, so I've drawn the exponential curve that connects them. The vertical dashed lines represent the pitch that is halfway (600 cents) in between two octaves. The horizontal dashed lines represent perfect fifths, which are 1.5 times the previous frequency. Note how these lines do not meet the curve at the same point. Mathematically, the frequency of a note that is 600 cents higher than a note of frequency f would be given by the formula 2^(600/1200) * f = sqrt(2) * f ~ 1.414 * f.

On the other hand, 700 cents is not quite a perfect fifth either. It gives you a ratio of 2^(700/1200) * f ~ 1.498 * f. This is almost imperceptibly close to a true perfect fifth, which is exactly 1.5 * f. Close enough that the difference is negligible.

• This is perfect. Thanks for takin the time to draw the picture. This leads me into questions about the nature of pitch, and the why/how of its measurement by cents. According to wiki, Pitch is each person's subjective perception of a sound, which cannot be directly measured. Though, it appears to me that over the centuries we have agreed to position pitch based on octave and fifth ratios, 1:2, 3:2, because they are consonant to most of us. But numbers tend to get confusing, I guess I should draw more pictures ; ) – Vigrond Dec 5 '15 at 15:58

While 1.5 lies perfectly in the arithmetic middle of 1 and 2, the arithmetic middle is not relevant for music. The intervallistic middle of two frequencies is their geometric mean.

Go two octaves up, and your frequency goes from 1 to 4 times its value. But one octave up is not 2.5 times the frequency, but rather 2 times.

So if you have two notes with frequencies f1 and f2, the middle of their interval has the frequency (f1 f2)^0.5, the square root of their product.

Consequently, "half an octave up", 600 cents, corresponds to a multiplication of the frequency with the square root of 2.

That ratio applies to the frequency which is an absolute measurement, not cents which more of a relative distance measurement between notes. They are different in nature.

Just a simple example, the perfect fifth above A4 (440 Hz) is E5 (660 Hz Just intonation/ 659.26 Equal temperament). This is where it makes sense to describe the interval in a ratio. A4 itself is 7 semitones away from E5 and with each semitone being 100 cents is 700 cents. See how in one we are looking at absolute values and one we are looking at relative distances? They each have separate uses and meant for different purposes.

A perfect fifth is just that. It's a fifth from the root, but that's not exactly the halfway point. That's saved for the TRITONE, which actually sounds an odd interval to some - used to be called 'the Devil's interval'. The tritone is equidistant from the root either way, so must be halfway. You're right that the P5 is not in the middle.

700% is 7 moves from the root. The 3/2 notion is a little weird it refers to the occurrence of the 2nd overtone which is 3x the vibration of the root.. via laws of physics.. but the 1/2 step interval = 100%/root x twelfth root of 2 is a human invention.. i think, like comparing golf balls to mushrooms

• Could you clarify this? I'm having trouble following what you're saying. It might be the use of "700%" that's confusing me, along with the word "moves." – Richard Mar 14 '18 at 16:21