While working on my software synthesizer, I've noticed a disturbing anomaly in the traditional western twelve-tone musical scale.

I've been making note calculations by multiplying each successive note by pow(2, 1.0/12), a.k.a. the twelfth root of two. This has the desired effect of multiplying the frequency by two every twelve half-tones, a.k.a. every octave.

After seven of these calculations, a.k.a. a perfect fifth, the calculated frequency is very, very close to 1.5 of the original. pow(pow(2, 1.0/12), 7) is about 1.4983.

I know it's not a rounding error, because I've checked with an unlimited-precision calculator to my satisfaction. I know it's not an error in my algorithm, because my calculated data matches that of several published frequency tables, which show the same anomaly.

I've heard that the perfect fifth interval is supposed to be a difference of exactly 1.5, and it makes sense to me that the fewer cycles two waves require to match up, the better they would complement each other. Octaves would take two cycles of the higher pitched wave and one of the lower; a "true" perfect fifth would take three and two; more dissonant intervals would take many more.

What I'm saying is that something is very wrong here. I don't know what it is. Not having a background in "conventional" music theory, I don't know who to talk to about this scandal or even what questions to ask, so I'm putting it here in hopes that someone more knowledgable will be able to enlighten me.

Allow me to point out that 3/2 ^ 12 is not exactly equal to 2^7. What this means is that twelve fifths is not quite equal to seven octaves (In fact, it's a little higher). In other words, the problem is not some basic misunderstanding of the physics of music, nor is there an error in music theory. Essentially, the problem exists because that's just how things are. While seven and twelve are nice numbers, this time things just happen to not work out.

This little problem that you have put your finger on has bedeviled musical traditions for some millenia.
Read all about it in why it's fundamentally impossible to tune a piano perfectly and equal temperament.

Also notice that it is somewhat dangerous to call the temperament you were using traditional western, since from the Middle Ages to now we have gone through at least three temperaments, namely just intonation that was substituted around 1500 by the meantone temperament, now largely superseded by the equal temperament.

The phenomenon owlbreath has discovered is known in music theory as the "Pythagorean comma."

As the above posts indicate, the issue is a lot more complicated than it first appears. If you dig deeper, the history behind tuning is a mess, the maths is a mess, and the theory of consonance - again a mess - and not fully understood.

I'll have to disagree with the view that equal temperament (12-ET from now on) is inherently out of tune. Yes, 1.5 might look better than the irrational 1.4983..., and 1.25 'looks' better than 1.25992 for the Major third, but many important numbers in maths are irrational - like pi and e.

To my ear, 1.25992... sounds 'sweeter' than the Just Intonation equivalent - 1.25. Don't get me wrong, pure ratios are ideal for creating unique timbres (the basis behind the harmonic series), but I am suspicious whether this ideal should be carried over to melody and chords.

So if 12-ET isn't based on Just Intonation, then where does the number 12 come from? Put simply; Why are there 12 notes in the chromatic scale? My pet theory is that you can perfectly surround a sphere with 12 equally sized spheres. Coincidence maybe? You decide...

By the way, the reason why it's impossible to tune a piano properly isn't because there's anything fundamentally wrong with 12-ET. It's because the lower notes on the piano form slightly inharmonic overtones. These overtones will then clash with the higher notes - making the piano itself imperfect. Because of this, it's necessary to compromise the tuning by 'stretching' the octaves slightly. However, it's important to note that this type of compromise is not to be confused with the type of 'compromise' that is usually associated when comparing Just Intonation and Equal Temperament. Yes, confusing I know.
Incidentally, it goes without saying that synthesized sound doesn't suffer from this 'piano problem', as all harmonic partials can be tailored to exact specified frequencies.

Finally, there's a third type of 'compromise'. It's well known that two tones played together will produce extra 'unwanted' frequencies such as summation tones, difference tones, interference beats, not to mention the phenomenon known as virtual pitch (which is based on the harmonic series). Anyway, upon closer study, it turns out that both Just Intonation and Equal Temperament (and every other tuning) 'suffer' from these added 'flavourings'. Thankfully, these extra tones are very quiet and are barely (if at all) noticable in music.

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