Hacker News Reader

Comment by smlacy | Hacker News Reader

Another possible explanation, which I'm surprised the author didn't go through is the "Circle of Fifths" which basically says:

Since Fifths sound so great, why not just keep doing that? When we get to the next octave, then come back down. If we get to a place that's "pretty darn close" to another note, then stop. The Python explanation looks like:

  f = 440
  for i in range(13):
      print(i,f)
      f = f * 3/2
      if f > 880: f=f/2.0

  0 440
  1 660.0
  2 495.0
  3 742.5
  4 556.875
  5 835.3125
  6 626.484375
  7 469.86328125
  8 704.794921875
  9 528.59619140625
  10 792.894287109375
  11 594.6707153320312
  12 446.00303649902344

Note that after exactly 12 steps, we're back at 446 which is "pretty close" to 440. So, we take this set of notes, sort them, and just jigger it a little bit to get the 12 notes we know today.

View on HN

It's worth mentioning that stacking fifths this way creates something pretty close to an octave, but it's still noticeably different from an octave. The difference between an octave and 12 fifths is called "Pythagorean comma", it's about 23.46 cents and it'll be obvious to all humans who don't have a speech/hearing impediment, even if you were never musically trained. (It's believed humans are sensitive to small intervals like this because it's required to process spoken human language). Traditionally, it's considered anything more than a synctonic comma (i.e. 21.51 cents) will feel different to even untrained ears. (but of course, this is just the theory, in reality there is some small variance between humans, background, culture etc).

https://en.wikipedia.org/wiki/Pythagorean_comma

This is pretty significant to mention, because even though 12 fifths are "very close" to an octave [1], they're far apart enough that no one will feel an octave. In music, near misses like this are very significant since they cause the feeling of harmonic "dissonance". Since 12 fifths is a very dissonant interval (since it's so close to an octave but still noticeably out-of-tune) Western music developed techniques (such as well-temperament, equal temperament etc) to make sure this "error" is blend in. We achieve this by changing other notes ("tempering") ever so slightly so that critical intervals like fifths (or in other cases thirds etc) are stable. Other cultures, such as classical Indian music, have their own way dealing with Pythagorean comma! Since music is a universal phenomena found in all cultures, but it doesn't manifest the same way in all cultures (e.g. not all cultures give the same kind of emphasis to pitch or harmony Western music gives) various cultures developed their own different and interesting ways to work around this "error".

[1] To be precise, we're referring to the difference between 12 fifths and 7 octaves. Since an octave is so consonant, sounds N octave(s) apart feel "equal" albeit with different timbre.

Somewhere there is a perfect universe where 12 fifths form an octave.

You don't need a new universe. You just need a species that hears sounds a little differently. We won't like listening to their music, but it'll be really great for them.

This may or may not be true. In fact, it seems unlikely to me your claim is true.

In nature, sounds produce harmonics i.e. when two objects collide they usually create waves of frequency f, 2f, 3f, 4f... in various (usually exponentially decreasing) weights. It's very rare to find pure sounds (i.e. only f frequency) in nature. The interval between f and 2f is an octave apart (1:2 ratio); the interval between 2f and 3f is a perfect fifth (2:3 ratio). So, when you actually hear a sound, you actually hear an octave and a fifth too, and how dominant this octave and fifth changes the "timbre" of the sound. This way, you know the source of the sound independent of the frequency. For example, both a violin and a piano can produce the note A4 at 440Hz, but anyone can easily determine if it's a piano or violin. The reason is, when a piano produces A4, it sounds not only just 440Hz but also 880Hz and 1320Hz etc... too and the relative volume of 880Hz and 1320Hz will be different than that of violin. Your brain automatically interprets these volume weights as "timbre" and the fundamental frequency 440Hz as "pitch".

Consequently, in order for your brain to be able to process the timbre of a sound it needs to find octaves and fifths between each fundamental note it hears. This means there might be something universal about octave and fifth (and other decreasingly consonant intervals such as major third etc...). Maybe we "understand" music because our brain is hard-wired to search for octaves and fifths in all sounds, in order to analyze timbre and in order to process spoken language. If this hypothesis is true, maybe an alien species could have octave/fifth/major third based music too! (if they have music at all, of course)

There is also the shape of the individual waveforms to take into account a piano has a more or less sinusoidal wave and a violin is more of sawtooth (due to the stickslip of the bow moving across the string(s)).

You're saying the same thing - the combined sawtooth wave is just the sum of all the sinus harmonics.

Is it not true that the shape of the waveform (sinusoidal, saw-like etc) is created by the relative weights of each harmonic? E.g. if you take any random sound wave, Fourier-transform it, you'll find the weight of each harmonic. Or are you saying there is a separate quality to sound waves that can cause their shape to be different even if each harmonic has the same relative weight with respect to the fundamental?

Yes, this quantity is the relative phase of the harmonics, although the human ear is generally considered to be insensitive to phase.

The human ear is insensitive, but the human ears are not. Or, more accurately, the human brain is not.

> This means there might be something universal about octave and fifth

There is! At least for the kinds of instruments that are conventionally used in Western music. The harmonic series arises naturally from the physical properties of a string or wind instrument (e.g. violins, guitars, pianos, flutes, brass, organs, etc). As a very rough description of the physical phenomena, the tones we hear arise from a full spectrum, atonal excitation (like a pluck or a reed flapping) bouncing back and forth along the length of string or tube, which is basically a one-dimensional "waveguide". Frequencies that are aligned with the harmonic series naturally reinforce themselves, in the same way that putting energy at the top of the arc of a playground swing has more of an effect than in the middle.

Notably, musical instruments that are not strings or tubes, or more general sound-producing bodies, have more complicated patterns of sound waves dispersing through them, and don't typically follow the harmonic series. Pitched percussion, drum heads, or bells have more complicated harmonic spectra than the standard harmonic series (as they are generally thought of as 2D or 3D waveguides where cancellation/reinforcement patterns are less straightforward), as do less musically conventional sounds like knocking two rocks together or striking an arbitrary surface.

Eh, I think you'd need a new universe, as it's a pretty basic principle of math:

  3^12 ~= 2^19

You can take two long strings of equal length (A & B), and pluck them, and they'll make the same sound. Then you can take scissors cut string A in half, and it'll sound different. (This is an octave.)

Then you can cut string B into thirds, and it too will sound different.

If you pluck both of your new strings at the same time, you'll find they sound quite nice together (the difference between these two is called a "fifth").

And after 12 rounds of cutting string B into thirds, and 19 rounds of cutting string A in half, you'll happen to have a string from each group that are almost identical in length and pitch.

But it won't line up exactly! They'll be about 1.4% different in length, which roughly works out to a quarter-semitone difference in pitch (i.e. 1/4th the distance from one piano key to the next).

But who says the creatures need to perceive sound in such a way that the harmonic series has sensory significance? To be honest I’ve never seen a compelling evolutionary explanation for why “hearing the harmonic series” developed in the first place. It obviously seems useful to be able to perceive sounds generated by (roughly) harmonic oscillators, since those occur naturally for various reasons, but why octave equivalence?

I don't think you need to imagine "creatures" - there are humans on earth whose culture's music doesn't have the concept of octaves, let alone fifths. All the real action in their music is in its rhythmic complexity.

But my comment was in reply to this statement specifically:

> Somewhere there is a perfect universe where 12 fifths form an octave.

For this to be true, I think you'd indeed need a new universe where 3^12 = 2^x, where X is a whole integer.

I'm not talking about explicit notions of octaves and octave equivalence, although those do exist in very many musical traditions and appear to be extremely widespread in musical traditions where tones have names (if not ubiquitous—I'm not aware of any exceptions). I was referring to the claims that octave equivalence is in some sense hard-wired in the human brain, or sometimes claimed to be all or many mammalian brains. I'm not qualified to evaluate these claims or even whether how well-accepted they are among experts, but such claims do seem to pop up all over the place when discussing music perception.

I'm much less qualified: I took a "World Music 101" class in college that described some Native American tribes whose songs had drums, and cries that fell in pitch from high to low to call down the spirits from above, but their music had no concept of octaves.

And I agree with you. While I can imagine a culture that hadn't "discovered" the octave, it's difficult to imagine people who can't even perceive octaves when presented with them, and quite easy to imagine other creatures that cannot perceive them.

The basic reason for octave equivalence is not from human subjectivity. It's from the fact that something that repeats "x" times is also something that repeats "x/2" times, eg a sequence such as:

  ABCABCABCABCABCABC

Can be thought of as "ABC" repeated 6 times or "ABCABC" repeated 3 times. If you replace the letters with numbers, you can treat the numbers as samples of a sound wave.

Just to be clear: the magical number here is 2 (not 12 or 8 or 7) since "octave equivalence" refers to the fact that you can multiply a frequency by 2 and get the same note.

Unless the sound is a perfect sine wave, there isn't a particular frequency associated with it due to these alternative interpretations.

Look at it from a frequency perspective instead of genetic.

When you overlay two waveforms that are related x:1, the zero points of the waveforms align. The wave resets at the same instant. If instead the waveforms are related not exactly, then you get a change in where the reset point is that drifts and causes the wave to exhibit beating (changes in volume).

If you have a 2:1 (octave) relationship between 2 waveforms, then you won't hear the beating as the beat frequency exactly overlaps the frequency of the lower frequency.

If you have say two notes that are not quite in the octave ratio, then you hear beating at the difference in frequency. E.g. say 440Hz and 888Hz (instead of 880Hz), you have beating occurring at 448Hz, which you'll hear as an 8Hz (448Hz-440Hz) wobble in the sound volume of the combined wave.

There are some assumptions about phase in there.

Yeah, you're right.

> If you have a 2:1 (octave) relationship between 2 waveforms, then you won't hear the beating as the beat frequency exactly overlaps the frequency of the lower frequency.

If we take a look into a human ear we'll see there an apparatus which physically splits sound into a spectrum of frequencies (a long narrowing tube, sound comes from one end and it creates resonances at different places) and a lot of receptors which placed in such a way that allows them to specialize on different frequencies.

I know nothing that would point to an innate ability of this apparatus to feel octave as something special compared with a mix of two random frequencies. So the peculiarity we hear pops up on later stages of sound processing. But why it pops up?

Moreover there is an evidence (I have no link, sorry) that the peculiarity of an octave is a cultural thing, not a genetic one. European music teaches us to feel octave as something special. There are tribes not exposed to European music who doesn't feel consonance and dissonance like we do.

My hypothesis that a mind picks correlation between a frequency and a double frequency. I mean it is not because of some funny math comparing two sine waves, it is because sounds essentially are not single sines. Two sounds forming an octave are both sums of many sine waves, and there is a huge overlap between sets of frequencies, so they sound similar. And then, when mind trains on this data, it becomes conditioned on a similarity of double frequencies, so it starts think of two sines with frequency ratio of 2:1 as of similar. The similarity is just a correlation.

And people who didn't tried to make a music with strings cannot grasp the idea, because natural sounds mostly much more complex than just sum of a several sines with frequencies that are multiples of some base frequency.

What? No, you said it, the ear splits the sound into frequencies and physically puts sensors into places where they resonate. Octave resonates in the same place. (but higher octave resonates in one additional place)

No, octave resonates in two different places. If we talk about octave as of sum of two sines.

I was sloppy. First i meant the octave as two separate tones (one after another), then I meant only the higher pitched one.

The higher tone also excites the sensors for the lower tone.

Wait, does it? I might be spouting bullshit, sorry.

> Wait, does it?

Thinking about it, I'm not so sure it doesn't. I'm not a physicist to know it for sure.

It depends on what type of oscillator we are talking about, but strings, chambers or beam oscillators for example will resonate at double frequency of their main frequency.

The octave rule definitely has some physical/biological founding, it’s not purely cultural. Although culture also plays a role in its preponderance.

Every note is a sum of harmonic sine waves, usually not just a single one. The first harmonic of 440hz is 880Hz. So plucking a harp string for A4 will also produce the A5 and A6...

That would have to be a universe in which the fundamental theorem of arithemetic was false. Otherwise, the only way to cross an interval that is an integer number of octaves is to take steps that are also octaves.

What the parent proposed doesn't require different math.

Just a species that hears out current slightly offset divisions in 12-tet as perfect, as opposed to only hearing integral ratios as perfect.

Viewing sound waves that don't synchronize with each other as being better matched than sound waves that do synchronize is less plausible than violating the fundamental theorem of arithmetic. There's no element of coincidence in whether two frequencies harmonize.

That hypothetical species wouldn't recognize two notes an octave apart as being similar, so there would be no reason to imagine a circle of fifths in the first place.

>Viewing sound waves that don't synchronize with each other as being better matched than sound waves that do synchronize is less plausible than violating the fundamental theorem of arithmetic

Actually it's perfectly plausible. People couldn't imagine others enjoying hearing a tritone -- and nobody in 1800 would imagine we'd enjoy listening to punk, hip hop, or Death Metal, and yet, millions do. We can surely consider a race that doesn't require intervals to absolutely synchronize.

>That hypothetical species wouldn't recognize two notes an octave apart as being similar, so there would be no reason to imagine a circle of fifths in the first place.

Note how I said that this imagined alien race would consider the "divisions in 12-tet as perfect". Note how those do include a perfect octave, that we already recognize as such. The alien race wouldn't change that, they'd just need to also consider perfect the slightly off ratios in 12-tet.

> Actually it's perfectly plausible.

No, for this to be plausible, you would need to have some theory of why the two notes matched with each other. There is no such theory; they have been chosen to be as unmatched as possible.

> nobody in 1800 would imagine we'd enjoy listening to punk, hip hop, or Death Metal

This is false.

>> That hypothetical species wouldn't recognize two notes an octave apart as being similar

> Note how I said that this imagined alien race would consider the "divisions in 12-tet as perfect". Note how those do include a perfect octave, that we already recognize as such. The alien race wouldn't change that

Sure. In that case, we can also imagine an alien race that perceives all and only the light that fails to reach its eyes.

Then again, perhaps being able to form a sentence describing something doesn't guarantee that the situation described is possible.

OK, I see that the discussion is futile.

This is both pedantic, condescending, and unwilling to entertain an idea.

In any case, just for anybody interested:

>No, for this to be plausible, you would need to have some theory of why the two notes matched with each other. There is no such theory; they have been chosen to be as unmatched as possible.

Nothing has in 12-tet has been "chosen to be as unmatched as possible". Instead, we've chosen it to match the ratios as close as we can, given the compromises we had (simpler instruments, larger range, etc.).

The 12-tet values still has strict mathematical properties, it's not just some random off collection of pitches - and it could be that, which is perceived as "perfection" by the imaginary alien race (that it is equally spaced values on a logarithmic scale).

> The 12-tet values still has strict mathematical properties, it's not just some random off collection of pitches - and it could be that, which is perceived as "perfection" by the imaginary alien race (that it is equally spaced values on a logarithmic scale).

This is an analysis you can't even apply to two notes. Every pair of notes is "equally spaced along a logarithmic scale", for the same reason that every pair of geographic locations is "equally spaced" along a linear scale, and of course also equally spaced along a scale with sinusoidal spacing. You have a single data point, and it's equal to itself. The claim has no meaning unless you're simultaneously evaluating more than two things.

In reality, of course, we perceive sounds as being related to each other when they have frequencies that are related to each other. This removes the need to evaluate them against imaginary background standards.

And two notes drawn from a 12-tet scale cannot have related frequencies unless they are separated by an integer number of octaves. The period of the combination of a note with its own fifth is double the period of the base note. The combination of a note with its 12-tet fifth is aperiodic.

> Nothing has in 12-tet has been "chosen to be as unmatched as possible".

Think before you speak.

Distinguishing different pitches implies distinguishing dissonant intervals from consonant intervals: there is no way they can sound equally "perfect".

What is definitely possible, instead, is liking dissonant intervals more than consonant intervals, for example because they sound "fatter".

Yes, that's correct.

Interesting tidbit: the ratio of a circle to its diameter isn't necessarily always Pi, but Pi will always be Pi as long as the foundation of mathematics holds.

I've heard there is a secret chord, that prophets play to please their lords.

But if you care for music less than algebra, just know it goes like this; the fourth, the fifth: the minor falls, the major lifts.

Well played

But, you don't really care for music; do ya?

That would be a non-Euclidian (ie. curved) universe?

Maybe it is possible to construct a non-Euclidean universe for sound, by modifying properties of the propagation medium as a function of space or time?

I'm not comfortable with this first refering to a natural human tendency and then a western harmony, which is pretty much accquired, as if it were a natural consequence.

A simpler way to go about this is using the chromatic scale, drawing multiples of C0 upto C8 so that C7 to C8 spans an octave, and then fixing F according to a table of equal temperament.

>I'm not comfortable with this first refering to a natural human tendency and then a western harmony, which is pretty much accquired, as if it were a natural consequence.

Well, western harmony is based on a set of natural human tendencies formalized.

There are other ethnic music practices, also based on natural human tendencies.

The parts that are acquired are built on top. But most/all music practices (western or otherwise) start with natural human tendencies, as their foundations.

You're both describing two similar consequences of the same mathematical fact: 2^(7/12) is close to 3/2:

* The reason that in their 12-note graph the red line very nearly overlaps with the seventh green line is that (2^(1/12))^7 is very close to 3/2.

* The reason that twelve fifths nearly make an octave -- (3/2)^12 is ~2^7 -- is that if you use 2^(7/12) to approximate 3/2 then it's (2^(7/12))^12 which is exactly 2^7.

Since you're applying the approximation twelve times instead of once, that also explains why we've gone from being off by 0.11% to 1.4%.

Fifths don't sound so great after a while. This tuning leads to the dissonant "wolf interval" - https://en.wikipedia.org/wiki/Wolf_interval - so it was largely replaced by the well temperament - https://en.wikipedia.org/wiki/Well_temperament - used by Bach in The Well-Tempered Klavier.

are the post and your comment not mathematically equivalent statements?

Its the use of the musical concept of Fifths that's the key here: We just derive the notes from what sounds good, not what makes sense mathematically? I'm just using Python to mirror the author's analysis -- you can derive "12 notes" pretty much just by using your ear and listening to the Fifths.

A perfect fifth is just 150% (3/2) the frequency of the root, just as an octave is 200% (2/1). "What sounds good" is in a certain sense based on the harmonic series, and in that sense it is equivalent to what makes sense mathematically.

A problem is that if you stack perfect fifths to get 12 notes then they won't sound in tune with each other across different keys. It's this issue which forms the crux of the linked post.

I think it's integral to the theorie seeing the fifth as 3 times, the octave as 4 times and the prime as 2 times an arbitrarily low root key. Trivially, the 2^n multiples form octave intervals but between C12 and C24 there's your twelve tones on a logarithmic scale. Naturally, it doesn't transpose in integer intervals if stepping down.

No, the article is essentially pointing out that the 12th root of two to the seventh power is really close to 1.5, whereas the comment you're replying to is saying that if you raise 1.5 to the 12th power, you get really close to a power of 2.

The former is more interesting when it comes to how music works psychoacoustically: the interval of a perfect fifth is fundamental to almost all music. Whereas the "circle of fifths" is more of a convenience that makes it easier to think about keys. Few songs would ever traverse the whole circle and come back to where it started, and if you stick with strict just intonation there is no circle of fifths anyways. (Maybe you could better call it a "spiral of fifths" or something.)

Eh eh as other people say, you have discovered the Pythagorean scale, that is known to drift away slowly from the correct frequencies, that's is why for a long time people didn't use chords that overlapped octaves, because they sounded weird and they called those evil chords

In other words, 1.5^12 = 129.746… ≈ 2^7.

(For those wanting to hear these frequencies:)

aplay -d 2 -r $freq

> Note that after exactly 12 steps, we're back at 446 which is "pretty close" to 440.

If you’re not careful you’ll summon the ~elders~ people who are convinced that frequency scale is wrong and that there’s a more ideal (to human ears) frequency step for the same 12 note scale. They might even be right, but goodness… prepare yourself for it to get weirder than finding out whether someone really believes it’s legal to be barefoot in all public settings.

This is the reason. You can keep going but it's just pointless. Things sound good enough at this point

I love math explanations in Python, thank you.

Reply on news.ycombinator.com