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I heard that after the sound of the octave the most pleasant interval to people is the perfect fifth.

If we take a middle C (C4) with frequency of 261.63 Hz
If we take one octave higher that'd be 2*261.63 Hz (C5) = 523.26 Hz

Now looking at wikipedia I see the perfect fifth of the key of C is G, at 391.995 Hz. How did they get to that number?

I thought the nicest sound would be if we took a half of the way between C4 and C5. Which is 261.63 * 1.5 = 392.445. Shouldn't that be the nicest sound to people? (so why is the perfect fifth 391.995 and not 392.445).

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    Lovers of the slendro scale (roughly 5-tone equal temperament, at least in Java) do NOT agree that the perfect 5th is "the nicest interval". Careful when talking about interval preferences with fans of gamelan (ethnic music of Indonesia, uses the slendro scale).
    – Dekkadeci
    Commented Apr 20, 2017 at 12:19
  • I would counter that an octave or even a unison is "nicest" if a "nicest interval" is even possible.
    – gingerbreadboy
    Commented Apr 12, 2023 at 16:15

4 Answers 4

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Too long for a comment.

The existing answer does a good job of explaining that it's because of equal temperament, but as to why we use equal temperament, an equal temperament fifth is 1.4983... which sounds almost exactly like 1.5 but it's cleverer.

1.49830708...^12 = 128 exactly. 2^7=128

i.e. if you stack 12 tempered fifths on top of each other, you will be at a note 128 times higher than the start, or exactly 7 octaves up.

If you use 12 just fifths you get 1.5^12=129.746338. 128/129.746338 = 1.01364326... or 531441/524288 exactly. This difference is not small, it's about a quarter of a semitone.

On old keyboards you would have 11 perfect fifths, and one fifth called the "wolf fifth" that was out of tune but got you to where you needed to be. On modern keyboards we take this difference and spread it out between all 12 notes, so each fifth is made just a tiny bit narrower, but after 12 of them, you end up exactly where you started.

Why 12? Because it's the first low number where the numbers work out nice and close to "just" intervals, like 3:2 (1.5) 4:3 (1.333...) 5:4 (1.25) 5:3 (1.666) etc...

The next one that works well is 19 but it's not really better. After that the next one that's really good is 31, and like how many keys do you want man. 7 is OK ish, but not good enough for us apparently.

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    I'm unaware of any keyboards that were actually tuned with 11 perfect fifths and one wolf fifth narrowed by the Pythagorean comma.
    – Dave
    Commented Apr 20, 2017 at 12:35
  • nitty picky: it's actually exactly a quarter of the diatonic semitone (that is, the semi tone between natural b and c) and exactly a fifth of the chromatic semi tone (the one that comes from sharps and flats). That difference is actually called a comma, and there are 9 of them in a tone.
    – Félix Gagnon-Grenier
    Commented Apr 20, 2017 at 17:13
  • @Dave that might be correct. I don't know, but my timeline could be off with regards to when Pythagorean temperament fell out of fashion, and when the modern 12 key keyboard came into common use. However, for the purposes of this answer, I think it's a justifiable simplification (I guess I could have said "lute" not keyboard and made everyone happy, but the image of a piano keyboard certainly helps to visualise it conceptually).
    – Some_Guy
    Commented Apr 22, 2017 at 16:29
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edit: The other answers do an excellent job of describing the difference between pythagorean and tempered tuning systems and the related maths, so this answer is to add additional information regarding the other part of the question as well as a followup answer the original poster added. I'm assuming that "nicest" in this case means "consonant". edit

Historically the intervals based on ratios can be traced back to Pythagoras. To quote a book on the subject:

After researching what notes sounded pleasant together Pythagoras worked out the frequency ratios (or string length ratios with equal tension) and found that they had a particular mathematical relationship.

So to address the followup question from the OP:

I think I made a mistake in my question...

the perfect fifth is a result of the nicest ratio of LENGTH of strings that Pythagoras the mathematician found. So octave was 1/2, and perfect fifth was 2/3 of the length of the string. He found that after the octave the perfect fifth was most consonant sounding. Because after splitting the string into 2 equal pieces, he then split it into 3 equal pieces and that's how he found the perfect fifth ratio. Why is 2/3 nicer than 1/3 is beyond me.

edit The mathematics and perception of tone and intervals was researched by Hermann Von Helmholtz, and his work on the subject "Sensations of Tone" still stands as an excellent resource and information. edit

Why a perfect 5th is considered the most consonant interval other than the octave has to do with how the waveforms of the pitches interact with each other.

Assuming a sine wave (no harmonics, pure tone) for each pitch, the combination of two pitches will create more or less complex patterns depending on the interval. It has to do with the way waves combine.

An imperfect example would be waves on a pond. If you throw two rocks into a pond and the waves line up, you get waves flowing together, some becoming larger, others fitting in between each other. If the waves don't line up, you get square peaks in an interference pattern.

Here is a picture of some of the intervals with their waves combined: waveforms of music intervals

The site the image is from has a good description of density degree.

The perfect 5th has the simplest form with the fewest peaks and valleys, making a smooth sounding tone. More peaks and valleys in a tone we will hear as a dissonance or "grinding" sound. The only thing smoother than the 5th would be an octave, or 2:1 ratio.

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    It seems like you only read the title, not the question
    – BlueRaja - Danny Pflughoeft
    Commented Apr 20, 2017 at 3:42
  • 3
    The title is a question too, isn't it?
    – BobRodes
    Commented Apr 20, 2017 at 4:11
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    The question in the title is pure woo-woo. Asking why an interval is the most consonant would be fine, but asking why it is the nicest is as subjective and nonsensical as asking "why is puce the nicest colour". On the other hand, the question itself seems to be about not understanding the difference between just intonation and tempered intervals (which is also a good question).
    – user19146
    Commented Apr 20, 2017 at 9:35
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    @alephzero Someone who already has the exact language to precisely define the phenomenon they are perceiving probably wouldn't need to ask the question in the first place. Everyone knows what OP means, and it's a very perceptive question that opens up a world of interesting music theory. While "niceness" is subjective, approximating consonant as pleasant, harmonius or nice when it comes to intervals is not as arbitrary as calling puce the "nicest" colour. Calling a perfect fifth "nicer than" say 440hz played against 910 hz is like saying honey smells nicer than rotten meat.
    – Some_Guy
    Commented Apr 20, 2017 at 10:52
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    @BlueRaja-DannyPflughoeft agreed. This answer completely misses the point of the question! The OP basically says "why don't we use a 1:1.5 (3:2) fifth". This answer proceeds to give some pretty spectroscopes of just intonation intervals (like 3:2) and goes, "that's why just intonation intervals are in tune". Which a) completely misses the point that the reason those spectroscopes look so smooth is because of the fact that there is a simple integer relationship between the 2 frequencies and b) doesn't address the OP's question; if these intervals are so great, why aren't we using them???
    – Some_Guy
    Commented Apr 22, 2017 at 16:25
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The exact frequency of an interval is depending on what temperament you are in. Specifically you are looking at notes in equal temperament which is based on the harmonic series, but is slightly altered to allow playing in different keys and make modulating easier for instruments with fixed pitches.

To really understand the difference, let's look at equal temperament compared to just intonation. Let's look at a chart comparing the frequencies of each interval in cents:

enter image description here Source

As you can see, equal temperament has the steps in cents be equal while just intonation does not. It should be pointed out that the difference between the perfect 5th in equal temperament and just intonation is small compared to some other interval differences.

Both of these are valid temperament that you can hear music in or perform in. There are pros and cons of each of these and I'm not going to get into it here especially which is better. For more information about the differences between these two temperaments, see the linked question source.

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    If you insert the numbers for equal temperament, with a fairly large accuracy, you get the tempered fifth at f=391.987, so that would actually come pretty close to the error stated in the question. Also, in regards to the OP's question, I would claim the just fifth would, without context, usually be heard as more pleasant than the tempered one.
    – Some Math Student
    Commented Apr 19, 2017 at 22:59
  • so shouldn't I be able to take the difference of "-1.96" and apply it to my number? But 392.445 - 1.96 = 390.485 not 391.995... but maybe I'm not understanding something correctly..
    – user34288
    Commented Apr 19, 2017 at 23:55
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    @foreyez the difference is in cents (hundredth of a semitone) not Hz. We don't use Hz for musicalintervals, because they have a different size in Hz depending on the starting note e.g A1 = 55hz A2 = 110hz A3 = 220Hz A4 = 440Hz . All those differences are 1200 cents, 12 semitones, or 1 octave apart, but a different number of Hz. Mathematically speaking, a cent is a logarithmic unit not a linear one, which means it expresses a ratio not an absolute value. So cents tell you the ratio between 2 frequencies not the number of hertz between them. Tip: for easier examples, use an A not a C
    – Some_Guy
    Commented Apr 20, 2017 at 0:15
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    This isn't arbitrary, human perception of pitch is logarithmic not linear. So 50 hz and 60hz sound MUCH further apart than 600hz and 650hz but the same distance apart as 500hz and 600hz. As an example, lets take A3; 220 Hz. An octave (2:1) is 440Hz. A just fifth (3:2 ratio) from this would be 330Hz. A 12 TET interval is calculated like f * 2^(interval in semitones/12) or f * 2^(interval in cents/1200) (so an octave is f*2^(12/12)or f*2^(1200/1200) i.e. just a doubling). So 7 semitones up from 22hz = 220 * 2^(7/12) = 329.63 aka really really close to 330hz.
    – Some_Guy
    Commented Apr 20, 2017 at 0:30
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    Might be worth noting that in arbitrarily pitched instruments (especially fretless strings and voice), it's very common to bend pitches up or down to match the surrounding chord--the old string player's complaint that A# is not Bb.
    – chrylis -cautiouslyoptimistic-
    Commented Apr 20, 2017 at 6:36
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I think I made a mistake in my question...

the perfect fifth is a result of the nicest ratio of LENGTH of strings that Pythagoras the mathematician found. So octave was 1/2, and perfect fifth was 2/3 of the length of the string. He found that after the octave the perfect fifth was most consonant sounding. Because after splitting the string into 2 equal pieces, he then split it into 3 equal pieces and that's how he found the perfect fifth ratio. Why is 2/3 nicer than 1/3 is beyond me... still need to look into that.

enter image description here

But yeah, it seems like I was thinking of ratios of frequencies and not ratios of lengths of strings (which lead to frequencies when plucked).

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  • Ratios of lengths of strings correspond to ratios of frequencies, using the rule that you get a frequency X times as high using a string 1/X times as long. So a 1/2 length string gives you 2 times the frequency; a 2/3 length string gives you 3/2 times the frequency.
    – David K
    Commented Apr 20, 2017 at 3:14
  • You can see it if you graph out the ratio frequencies. 1:3 of a base frequency clumps the events together with a space in between. 2:3 creates an even pattern between the events.
    – Alphonso Balvenie
    Commented Apr 20, 2017 at 3:18
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    Plucking the string at 1/3rd will give the exact same result as plucking it at 2/3rds. To see this, rotate the string 180°
    – BlueRaja - Danny Pflughoeft
    Commented Apr 20, 2017 at 3:41
  • BlueRaja, the picture isn't completely right.. I meant, you hold the string at the point of 2/3, and you pluck the portion of the the string to your left. If you hold the 1/3, and you'd pluck the left of it. you'd get a much higher frequency because it's a shorter segment
    – user34288
    Commented Apr 20, 2017 at 3:53
  • @foreyes the relationship between frequency and wavelength is inverse; halving the length is doubles the frequency etc. This leaves you with exactly the same problem just expressed oppositely; and while it is helpful to think about it in both ways to get an understanding, neither way is "right". Take a guitar A string of length L=60 cm at f=110Hz. L/2 gives 2f (an octave up) L=30cm f=220hz . L/3 (a third of the length) gives 3f (3 times the frequency) so L=20cm f=330Hz (this is an octave and a fifth up from 110Hz). (2/3)L gives (3/2)f ; L=40cm gives f=165Hz (a perfect 5th up).
    – Some_Guy
    Commented Apr 20, 2017 at 9:05

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