I was skeptical of his prior dissonance study. I feel there is a bit of accommodation by the participants to maximize the surprise of the researcher. Getting goods for singing funny.
The paradigm I would have liked to see is where two tones are held constant and the participant moves a third tone to the location that sounds the best or the worst.
It is remarkable work. But I don't believe it. Octaves are just too basic a phenomena to be viewed as a cultural invention.
> Octaves are just too basic a phenomena to be viewed as a cultural invention.
Octaves are consonant or dissonant depending on the timbre of the sound. For sounds produced by harmonic instruments, like a vibrating string or air vibrating inside a long tube, the sound is a superposition of waves whose frequencies are integer multiples of a fundamental frequency. Then, playing two sounds an octave apart will match exactly all these frequencies and it will sound nice. But there are other instruments (not privileged in the western music tradition) whose timbre is not composed of integer multiples of a fundamental frequency; and in these instruments octaves sound very dissonant.
You can argue that the octave is a "basic phenomenon" inasmuch a vibrating string is basic. Yet, from the point of view of a person who uses a synthesizer, the octave has nothing special with respect to other intervals.
What are these instruments? I guess their behavior has to be described by some interesting partial derivatives equations that give rise to non-integer multiplies harmonics.
Bells, gongs, drums, and other instruments that can be a modelled with a 2D surface that vibrates in an assortment of simultaneous modes are semi-harmonic.
Pipes and strings are better modelled by a 1D resonator, which is more likely to allow integer overtones - although they can still have inharmonic elements due to stiffness and - in the case of orchestral strings - rotation caused by the scraping bow.
None of which changes the fact that octaves are primary in any instrument which produces a range of pitches with a clear and reasonably sustained fundamental.
In fact researchers rely on the concept of pitch chroma/pitch class to distinguish between absolute frequency. Humans reliably hear the octave/not octave distinction, as do some animals. Obviously the animals aren't musically trained.
But it's not enough to say that "octave perceptions are learned." In this experiment it's more likely that octave use is learned in musical contexts.
There's a huge difference between saying that someone can't hear octave relationships at all, and that they can hear them perfectly clearly, they just don't find them culturally relevant - perhaps because their music theory is built on a grid of absolute pitches, and not on a repeating pattern of pitches.
Also worthwhile to point out that brainwave coupling is based on "octaves" (i.e., roughly speaking -- gamma is double frequency of beta, which is double alpha, which is double theta, which is double delta....)
Just from experimenting with synthesizers, I’ve noticed that the perceived pitch of a tone is defined more by the relative frequencies of the harmonics, rather than their absolute frequencies. So, you can take a harmonic series like 100Hz, 200Hz, 300Hz... and add a constant to each of them and get something like 123Hz, 223Hz, 323Hz... and it will still sound more like 100Hz than 123Hz.
EDIT: Actually adding 23hz is probably way too much for the result not to just sound dissonant.
I think that is different – that’s simply the case where all the harmonics are integer multiples of the fundamental, but the fundamental itself is missing. Every cycle of the waveform will be identical, so the period will unambiguously be the frequency of the missing fundamental.
In my case none of the harmonics are an integer multiple of the perceived overall pitch. And while the waveform will be aperiodic, it shows clear signs (on visual inspection) of similarity at the perceived frequency.
There's a double bass player trick for when you don't have a low C extension to play the low note up an octave plus the note a fourth below. This creates an illusion of the low note being present. It works best with a bow since it's hard to get the simultaneous sounding of the notes perfect pizzicato.
You probably know this, but due to the 12 note tuning, most intervals on a piano are slightly off from perfect intervals.
EG, on a string instrument or with voice it would be natural play notes a major third away such as C and E with a frequency ratio of exactly 5 to 4. There's no way to get every natural ratio exactly right and have 12 notes in an octave, so most of the intervals on a piano are tuned to approximations of an exact frequency ratio.
So pianos have at least 2 things working against having perfectly consonant chords.
Which is why I once sang in a choir for a couple of years. Nothing beats perfect harmony.
Similarly, I was once convinced everyone in our jazz-rock band was out of tune after I had experimented with alternative tunings at home the whole day. Yeah, equal temperament is really quite a bit out of tune, but somehow we seem to manage just fine.
When you're singing in perfect harmony, are you trying to be in tune with the people singing simultaneously to you or with the previous note sung? Presumably there are times when you can't do both at once.
For example if you were singing a major second above someone else, and then you had to jump a perfect fifth and they had to jump a major sixth to end up singing in unison. At least one of those intervals is going to be out of tune.
Oh yes, sometimes you actually notice this, often when approaching a certain note from above and below. Depending on the harmonic function, the major seconds are not always the same size and you end up singing out of tune.
So, one always tries to understand the harmonic function of the choir as a whole in relationship with the structure of the piece. For example, if tension is needed, one can sing even more towards a dissonant diminished fifth. Close harmony is all about that.
So in a choral context, you would typically aim to be in tune with the people singing simultaneously. The root note of each chord is pitched according to equal temperament.
This has its downsides as well. Consider the chord progression Em -> A7: both chords contain the note G, which in the first chord acts as a minor third, and in the second chord as a minor seventh. The minor third should be pitched 16% of a semitone (cents) above equal temperament, and if we're doing really just intonation and pitching minor sevenths as harmonic minor sevenths, the minor seventh should be pitched 31 cents below equal temperament. So consider a voicing where one voice should hold a G across both chords: this means that even though it's singing the "same" note, the voice should drop 47 cents when the chord changes - almost a full quarter tone!
This video from the excellent Voces8 ensemble has an example of how this might sound, at 55:50 (between the first two notes):
I highly recommend watching the whole thing, it goes into a lot of detail about the practical implications of singing in just intonation. For the problem described above, one solution is to avoid the issue by tuning minor sevenths using equal temperament in tricky cases like this one.
Really nice! Just got a chance to watch it. Such a nice explanation and he gives very nice examples of how intonation is a puzzle and a bit of an art, even for choirs.
Maybe not so much for a whole band, but for the classical guitar, people seem to tune the instrument slightly different for each piece (in a given key) and sometimes in the middle of a performance! Anecdotally, moving from something in e to d sounds terrible without adjustment.
And you could tune a piano that way. But it would only be in tune for the particular key that you tuned it to. So if it was tuned to C (as is normal for a piano) it would be (more noticeably) out of tune for any other key. Further, since other instruments are normally tuned for other temperaments, you'd be out of tune relative to them, even in your chose key.
That's one of the advantages of digital pianos. Many models have a setting where you can tune it for the key of your song. Then the basic chords on that key sound perfectly in tune, and the chords of nearby keys sound slightly dissonant, thus creating a beautiful tension.
I believe the human voice is the most popular instrument, world wide by a long shot followed by clapping. You are probably right that most drums are inharmonic, but I wonder what percentage of percussive instruments have a strongly identifiable fundamental anyway (or some tone if not the fundamental). At any rate, perfect intervals sound really consonant in singing, so I'd expect a bunch of cultures to independently stumble into that knowledge because singing is so popular. I didn't read the actual study, but I'd lean towards distrusting a single study or two, and in this case, wonder if this group of people is into polytonal singing.
> You can argue that the octave is a "basic phenomenon" inasmuch a vibrating string is basic. Yet, from the point of view of a person who uses a synthesizer, the octave has nothing special with respect to other intervals.
I am skeptical of this. I maintain that 1:2 is special. I would love to find more evidence or resources about this.
There's the book by Dave Benson [0] (available online), mostly about the mathematical modeling of instruments, that has a nice ethnomusicological compendium of instruments with weird timbres.
And then there's the infamous book by Sethares [1] that is all about the dependency of harmony on timbre.
1:2 is special in that it gives the maximum possible consonance when using harmonic timbres, but there are other good scales built around different ratios. See:
You might want to be wary of your bias. From your statements, it seems as though you're somewhat dismissive of this study and are searching for studies/sources that support your point of view. If you find yourself dismissing the person you're replying to and other comments in this thread, even though they raise interesting points about instruments, you might want to think about whether you're being open-minded.
Maybe. If you knew me, I think you'd be hard pressed to label me as someone who isn't open minded. But these are not arbitrary positions. I study harmony intensely, integrating neuroscience, music theory and classical philosophy. Did you know that the Pythagorean test of small integer relationships in consonant bronze chimes (described by Plato and Aristoxenus) is considered to be the very first quantitative, hypothesis-driven experiment in western history? And yet, we still don't know the basis for consonance and dissonance!
I will only say that my "bias" got me what I wanted-- evidence about the phenomena! (And thank you for that)
I think that the OP is maybe making too much drama about it. Skepticism (even extreme skepticism) is always welcome in a scientific/technical discussion!
EDIT: Regarding the basis for consonance/dissonance, the mathematical part of it is straightforward. When superposing pure waves of close frequencies you obtain beating (a slow frequency modulation of the amplitude of your sound), and beating does not appear when you superpose pure waves of very different frequencies, regardless of the interval, integer or not. Thus, the only dissonant intervals of pure sinusoidal waves are those that are very close to the unison. If you compound this with the fact that western instruments have harmonic spectra, you see why some intervals are consonant and dissonant: the dissonant intervals are those that have some partials that are close, but not exactly, unison.
> Then, playing two sounds an octave apart will match exactly all these frequencies and it will sound nice.
I don't think this level of consonance is necessarily something one has to consider as sounding "nice". It could just as well be thought of as sounding "hollow" or "uninteresting" or "weak", compared to more dissonant harmonies.
Do you plan to publish this data? I'm very interested.
I have found that the "worst" audible beating has frequencies between 6hz and 20hz. Higher than 20hz you do not perceive it, and lower than about 6hz it becomes an agreeable "tremolo" instead of as annoying beating. Thus, it would seem that when the frequencies of two notes differ between 6 and 20hz you get the worst dissonance possible. When you are in the middle of the scale, this is more or less about a semitone.
A drum is a good example, though we don't often think of it as a melodic instrument. The resonant modes of a drum don't look anything like someting as nice as those of strings, with their integer multiple frequencies. Makes you wonder, what music might have been like if our ear was more "drum-based" than "string-based" (however that might work).
Music already "is" something different! not all music is western. One of the most strikingly different kinds of music (besides synthesized stuff) is gamelan music, which is performed on a set of non-harmonic bells, which sound consonantly at non-integer intervals.
Thanks! I've been listening to gamelan music for the past couple of hours now, and have thoroughly enjoyed the experience. From my limited exposure, I get the impression that the music leans on repetition of sound patterns to give the listener something to hold onto musically, creating more of an atmosphere of sound than a melody you might hum along to. Is that fair to say? Perhaps I just need more exposure before things "click".
I might compare this non-harmonic genre with the artist Sevish, whose work I've enjoyed a great deal. He too tries to step out of the realm of 12-tone equal temperament. Some of his work is quite reminiscent in structure to this gamelan music (e.g. Desert Island Rain). However, he also manages to build melodies and something you could probably call chord progressions (though I wouldn't know), in a new and very foreign musical world (his entire album Harmony Hacker is amazing). Besides the music being amazing, getting used to this new landscape is enjoyable in and of itself.
My point about the drums, though, was more aimed at what scales, chord progressions and melodies might be developed by a species who had as harmonic basis the drum's spectrum of resonant frequencies. We, by comparison, have the integer multiple of some base frequency, the canonical "harmonic sequence", whereas for them it would be quite different:
https://en.wikipedia.org/wiki/Vibrations_of_a_circular_membr...
So there's this thing called a wave, and waves with wavelengths that form a simple ratio interact to form periodic waveforms that are observably stable.
Pulses too, actually.
> waves with wavelengths that form a simple ratio interact to form periodic waveforms
That is certainly not the case. The superposition of waves is linear, and waves of separate frequencies do not really "interact" besides being linearly combined. The fact that a waveform is periodic is not perceptible in our ears. We hear the frequencies separately and the ratio between them does not matter.
This is not a theoretical concept, but a physical observation that anybody can check. You can readily try it with a synthesizer. Create two waveforms f(x)=sin(x)+sin(2x) and g(x)=sin(x)+sin((2+e)x) where "e" is a small irrational number. The wave f is periodic and the wave g isn't. They sound just the same: you cannot tell which one is periodic just by hearing them.
You are right, this is not a discussion worth having when you can do the experience. If you have a digital synthesizer, just try the experiment that I describe and listen whether you can tell the periodic from the non-periodic waveform. You say that they are perfectly distinguishable, but I just did the experiment (even drawing the lissajous figures) and there's no real difference except the perceived pitch.
I didn't say they were perfectly distinguishable. You have reasons simple ratios don't result in physical waveform attributes relevant to music. I posit you should just really think about that for like five seconds. Otherwise I don't know what to tell you. Look up Pythagoras or maybe what a wave is or something.
I agree with most skeptical commenters. I once read about a study of singing in traditional cultures. Men and women would sing the same melody, except one octave apart due to differences in physiology. Reportedly, they did not perceive their singing to be separated by any interval; they perceived that they were singing in unison.
... a result that almost reinforces the findings in the paper reported in the article: octave (equivalence) perception is learned/culturally influenced rather than being hard-wired.
Also if the result would have been:Tsimane can hear octaves as everybody else, then there would be no publication. So there are rewards not to question ones own study too carefully.
Octave harmony in instruments and singing are established physical phenomenon. Higher harmonics of two voices line up and resulting frequency spectrum is more clustered and concentrated. I think the study disproves that all brains consider this to be preferable, and instead ascribes it to cultural/learned phenomenon. I agree it's not compelling theory, but hard to argue against if they can get more proof.
You'd be surprised at the number of "basic phenomena" that turned out to have cultural origins. Look up the language of the Piraha people, who famously does not contain any concept related to numbers or counting (because they have no use for it and introducing it would disrupt the trades they have), among many other things.
Also, cursorily reading a study that contradicts one's preconceptions and going like "nah, my gut feeling doesn't agree with it" is so quintessentially Hacker News ;-)
They could reproduce the intervals without problem, they just transposed them by something else than a multiple of octaves. You make an interesting point in the last sentence though.
My horse in this race is that I know somebody who was raised on western music and has the same problem. I don't know if it's amusia per se or some related pathology, but I definitely think it's not just acculturation.
We spent several hours arguing whether music theory was a valid field of study or complete bunk until we noticed he didn't perceive melodic harmony at all. It was fun, if exhausting. That person also nailed an amusia test, which makes me suspect there's a separate condition.
>The researchers acknowledged in the study that the results might reflect differences in how people sing, and not in how people perceive pitch. But they argued that the totality of collected data, including some more direct measures of octave perception, put the weight of evidence on the side of a perceptual explanation. [..]
Moreover, if the Tsimané’s performance in the tests has more to do with singing ability than their perceptual experience, then it would mean that all people have absolute pitch
Since I only have access to the abstract: How did they argue this in detail? Because that is something that immediately sprung at me. In the process of learning to sing, reproduction of relative intervals is far easier (and much earlier) learned than correct absolute positioning °) which actually takes some effort to train.
And the reasoning in the second paragraph eludes me completely. If I haven't completely misunderstood something, this has nothing to do with absolute pitch, and everything with tonal memory, which - again - won't work if it isn't trained:
https://en.wikipedia.org/wiki/Tonal_memory
°) EDIT: Apology, the choice of words is actually incorrect. It's still relative, but with reference to the notes heard before, in contrast to the reproduction of intervals where you have your own previous sung note as (control) reference. Simply put: it's much harder to learn to hit the first note (and thus the general pitch) of the sequence correctly than the intervals in the sequence.
Quite a bold title. My understanding (while limited) is that the human ear senses sound by means of tiny hairs in the inner ear that vibrate in resonance with incoming sound waves. I've always believed that the reason certain intervals are more harmonious to us than others was because there would be some overlap in hairs that resonated for the different harmonious pitches. The same way guitar strings will resonate with other strings that are tuned an octave, or perfect fifth, or fourth higher, these ear-hairs ought to behave by the same physical laws and principles.
As such, I'm a bit skeptical of the method and conclusion of this research. It seems like physics to me, not psychology. Ability to reproduce pitches and intervals does not strike me as entirely related to what they were trying to find out.
How about instead playing different soundbites, of consonant and dissonant pitches, and asking for their opinion in some way? Maybe even allowing the participant to find a relationship by some sliding instrument themself? That would seem more aimed at studying the Tsimané's ability at "perceiving octaves."
Maybe my interpretation of the header and conclusion is wrong. Perhaps the idea is that _some training_ is needed in order to perceive octaves, the same way _some training_ is needed in order to learn how to walk. That I might be able to wrap my head around.
Music is definitely an acquired taste. My family was not very good at nor into music. I didn't understand the point of instrumental music until my mid teens. Before that I only listened for the lyrics.
On a different note, people often think of the octave as the most fundamental interval, but the most fundamental is the very same tone twice. Even here people like a little dissonance. Two tones at nearly but not exactly the same pitch will produce a pleasantly shimmering chorus effect.
Anyway music is subjective and there's is no sound that is better than everything else in all respects and contexts.
I have never heard "out of tune unison" described as an interval, except in a joke about viola players. Sure, there's a difference between 440 and 442 but that's not considered an interval in western music theory.
I wonder if this is more a problem with the article than the study. I haven't read the study. It feels oddly described. Like, are they saying that when a Tsimané sings a song (solo, without instruments) on Tuesday, and then later sings the song on Friday, it's guaranteed to always start on the same pitch, without hearing some sort of reference note from an instrument? If so, that's basically perfect pitch.
We all have comfortable vocal ranges, and I know generally where to start the Star Spangled Banner to hit both the high notes and the low notes, but since I have relative pitch and not perfect pitch, my starting point is still always going to vary within a minor third or so.
Incidentally, I noticed in my college ear training courses that I (I have excellent relative pitch) would routinely score higher than the folks with perfect pitch. There's something about perfect pitch that can be really distracting to musicians for certain exercises, as it's not quite as flexible as relative pitch. So maybe that's related to the Tsimané.
Also, there are aspects of different notes that very much are noticeable in physical reality and don't need to be culturally learned. In an old singing group of mine, we would have fun with a game where we would stand really close to each other, face to face, and sing unison, and then one of us would start to go slightly sharp or flat while the other would try to stay steady. There's an ugly "beating" sensation when this happens that goes away when you come back to unison. It's not just theory, you hear/feel it. That's present to a lesser extent with octaves, and then fifths, through the harmonic sequence.
Those subtleties are of course covered up when you're using separate instruments or timbres, but using different timbres doesn't disprove the underlying presence.
I find this hard to believe from a mathematical point of view - octaves are as far as you can get from arbitrary which is I guess what gives the headline it’s shock value.
There are multiple definitions of the word octave. Some have to do with frequencies and some have to do with musical scales. Furthermore, no common instrument emits a pure tone, there are always overtones and other frequency components (aka timbre). If the 'note' on the instrument is based on the fundamental frequency, there's no guarantee that all of the modes in the timbre are going to re-align at double the frequency of the fundamental.
So, basically, we're all talking past each other (again).
Having a harmonic at twice the frequency is not arbitrary. Lots of things that vibrate will have a fundamental frequency and harmonics at integer ratios. The name "octave" implying 8 is a bit more complicated, but if you just treat it as a name for a moment, the "octave" is the ratio 2:1. Another common ratio is 3:2, and again, if you ignore the meaning of the name, it's called the "fifth".
So why 8 notes per octave? Well, it's really 12 notes per octave. Maybe think of it as the white and black keys on a piano. 8 white keys gets you back to where you were, but you skipped some black keys along the way. So why 12 notes per octave? Well, that's because (3/2)^12 (1.5 to the 12th power) is almost a power of 2. So if you step up by fifths 12 times, you very nearly land 7 octaves up. Each of the notes you stepped on along the way becomes one of the 12.
Heh, there are also "fourths" at a ratio of 4:3. So music has a "fourth" plus a "fifth" equals an "octave"! It's kind of silly :-)
Anyways, that's the quick version. If you go further down this road, there are "wolf fifths", various tunings with subtle (but perceptible) differences, and you can even find 19 and 31 tone scales. It kind of goes on and on.
There are 12 semitones in an octave because western scales are based on the mathematical coincidence that 1.5^12 ~= 2^7
Because of consonance/dissonance, on most of our instruments, some of these 12 notes sound better in combination than others.
Imagine you ran a clustering analysis to group the ones that sound better together and ended up with two clusters, a cluster of 5 and a cluster of 7.
The cluster of 7 gives us the name "octave" as musicians double-count the first note when it is repeated at 2x the frequency at the top end of the scale.
There is no particular reason, that part is certainly cultural. There aren't even seven, they are twelve equal slices, called semitones. There are infinite slices an octave can be divided into, we just picked one kind of division completely arbitrarly ~2000 years ago because we needed to build organs that sound good together and those people then thought those divisions were enough to express all kinds of emotions. Of course that process took a long time to finalize to what we call now western music.
The ear doesn't care too much about which frequencies, but that they are the same ones. That said, an octave is the doubling or halving of those frequencies which we identify as them being the same note. So, I find it very hard to believe what the article is claiming, also because one reason we can listen to music in low quality speakers is that our brain fills in the missing fundamental[0] so the doubling/halving part seems to be integral to our perception. I don't see how such a psychoacoustic effect be trained or be culturally based.
But on the other hand, tuning into a particular division is very hard to get rid of, westerners can't hear scales that have more than 12 semitones, arabic, indian and eastern music in general is like that and to a westerner ear these sound mostly dissonant and you need to spend a lot of time listening to start appreciating the expression.
Mostly because primitive flutes and horns historically had those seven notes (plus octave) [1], or they sounded good when sung or played in sequence or together. It "just made sense" to stumble upon them, because of the harmonic series [2].
Some intervals were more obvious (octave, perfect fifth, major third). Others probably took hundreds or thousands of years to be discovered.
But keep in mind that different cultures used different scales. It wasn't always the same. Some had minor seventh, others major seventh. Mesopotamians used a sharp fourth. This is still visible in different cultures today (blues/rock uses a lot of flat seventh!).
As for "why seven?": Since "mixing" major/minor sevenths/thirds/etc is very dissonant and weird, people ended up having seven notes regardless.
I would say that "note choice" was more cultural, but the options were obviously influenced by the harmonic series. After a while people started seeing patterns and those historical scales converged into the major scale we know today.
Later in the 1500s some geniuses found a way to transpose the scales but still maintain the ratio between notes, but without having to retune the instrument. The trick was to divide the octave in 12 notes but only use seven at a time [3]. It wasn't "perfect" like just intonation [4], but it was in the ballpark. That became the new normal. Equal temperament is not perfectly in tune with the harmonic series, but people got used to it (to the point that just intonation sounds "off" to a lot of musicians).
The harmonic series (of which the first three intervals are octaves, fifths, fourths in pythagorean tuning) is not arbitrary.
How you construct a more useful set of notes from the harmonic series is arbitrary. The 1.5^12 ~= 2^7 coincidence I note above allows you to construct a scale using octaves and fifths. You can just as easily do it with other similar coincidences, use fifths and fourths instead, etc etc.
Check the smoothness/roughness functions in my blog link - the reason you think e.g. a 3rd sounds as 'right' as a 7th is likely purely cultural. Other cultures have other scales. One of the Indian ragas has over 100 notes.
(I'm aware OP claims it's ALL learned, including the octave, but even if true I don't think that means it's all cultural).
An octave is the interval between one musical pitch and another with double its frequency, which isn’t arbitrary; however the number of notes within an octave (and, I think, their exact tuning) is arbitrary.
If you play two sine wave tones of different frequencies at the same time you get beats[0] caused by the alternating constructive and destructive interference.
Helmholtz hypothesized[1] that the dissonance of a pair of sine wave tones was related to these beats. Slow beats sound like a pleasant vibrato effect. Extremely fast beats are not perceived as beats at all, with only two separate tones heard. Only moderately fast beats sound dissonant.
This was confirmed experimentally[2] by Plomp and Levelt.
Sethares generalized this relationship to arbitrary sounds[3], finding that an amplitude-weighted sum of the consonance of all pairs of partials ("partial" meaning one of the sine waves that forms part of the waveform, as can be found by Fourier transform) well approximated perceived consonance.
Most Western musicals instruments are harmonic or approximately harmonic[4]. They produce a waveform with partials of frequencies that are an integer multiple of the lowest frequency partial (called the "fundamental").
Increasing pitch by an octave doubles the frequency of all partials. An integer multiplied by two is still an integer, so if you play harmonic notes separated by octaves the partials will overlap. All pairs of partials will be either identical or far apart, so none form dissonant beats. This maximizes consonance.
But music with only octave intervals would be very boring, so the octave in standard Western music theory is divided into 12 equal parts. This is an excellent choice for harmonic instruments, because it closely approximates several small-integer ratios. The interval of a "fifth" (actually seven steps away in the octave, but music theory uses strange numbering to simplify playing the most common musical styles) is a frequency ratio of 3:2. This results in half the partials overlapping, and the other half still being positioned so they avoid dissonant beats, so the fifth is also highly consonant.
Wikipedia has a graph comparing equal divisions of the octave with small-integer ratios:
You can see that 12 divisions has many useful approximations. It represents a good balance between complexity and musical utility, so I don't think it's surprising that it became the standard.
But note that small-integer ratios are only consonant with harmonic timbres! If the partials are not integer multiples of the fundamental, as is often the case in tuned percussion, you need a different tuning system. Indonesian classical music[5], which makes heavy use of tuned percussion, is famous for this. You can use Sethares' model to generate tuning systems suitable for arbitrary timbres, e.g. https://sethares.engr.wisc.edu/mp3s/morphine_crystal.html
I'm a musician but I was completely tone-deaf before I started studying, and took me a while to recognise even octaves. In my head it's still the "SomeWHERE Over the Rainbow" interval ¯\_(ツ)_/¯.
Said that, when playing two octaves together vs other intervals it's easy for a layman to notice how in tune and how consonant they are, because of beating.
> Said that, when playing two octaves together vs other intervals it's easy for a layman to notice how in tune and how consonant they are, because of beating.
I think the point was that this beating depends on the timbre of the instruments (e.g., the fact that overtones are integer multiples of the fundamental). For many percussion instruments this is not the case. You can actually shape a collection of bells so that octaves sound very dissonant (lots of beating) while a slightly different interval sounds consonant.
> The acoustic structure of octaves is always the same: The frequency of a note in one octave is half the frequency of the same note in the octave above
This is not a great definition; our understanding of frequencies came well after we were using the concept. The idea of a "string of half the length with the same tension" is much more natural, and is the easiest mechanical analogy. The discovery of this is very natural because you get sympathetic vibrations -- you can feel the other string moving when you get consonance across octaves. It shows up (along with fifths) in most other instruments -- blowing twice as hard in a flute or halving the tube size, or compressing the embouchure in brass instruments.
While I don't think it's intrinsic in the human brain, it is intrinsic in most musical instruments, and it's kind of intrinsic in the way sound works, so it would be very surprising if it didn't feature in the music made by a culture, and thus work its way into the appreciation of music in general. The tests described here are interesting but might just point to the fact that there is less instrumental music (produced or listened to) in this culture than in others.
I do need to add a section on cultural consonance/dissonance but..
Octaves are not cultural. Resonance is not just an aspect of music or even animals, but literally everything in nature that makes sound. Dropping a rock in water will create sound, and frequencies that resonate with each-other will stick around longer than those that fight, teaching us that these sounds belong together.
An octave is the simplest possible relationship between two frequencies, and it can be heard in almost every natural sound imaginable. We learn they belong together from almost every sound we hear.
As an aside, western tonality uses the twelfth root of 2 because if you just start hammering randomly on those tones, they'll get along extremely well regardless of whatever the biggest bassist vibration is. The big bassy frequency being the one that every other frequency needs to get along with (harder than it sounds). It's sort of a fudgy form of relativity that allows things to vibe together mostly, but not perfectly.
I feel looking for musical scores in peoples brains is a great place to start decoding. We clearly load data into a conscious signal somehow like CAN-bus but it must also be stored maybe on actin filaments in the dendritic spines as loops we load in? Also mentally we can change the music and add any number of instruments over the top change the tune and deviate the course of the melody, add vocal harmonies from crowds of people and violins etc. so where does that part happen?, in the signal?. can it be 'heard'? . and is the music 'composer' happening in the same part of the brain as the language generating part or is it just similar capabilities in a different part of the brain?. Decoding must be possible. I feel if they had like millions of brains all wired up to computers that were using AI to find match patterns . i.e. audio patterns from songs in brains data. they could brute force it? Is getting people to sing stuff and making assumptions weak science?
The concept of Octaves (as a lot of concepts in music) comes from physics. Because one of the most basic instruments you can have is a string under tension.
Pluck the string: note
Pluck the same string pinched in the middle: same note one octave higher
Not to mention Note + Note(octave higher) will have the harmonics overlapping, so it will sound better since there won't be any beating.
Your mind wants it to be simple. I heard of some country guitarists that de-tune their strings on purpose to get more chord richness. "Better" is in the ear of the auditor.
> so it will sound better since there won't be any beating
Unless presence or absence of beating is itself either a thing we learn to detect, or a thing we learn to associate with better-ness or worse-ness, rather than both being innate?
> Because one of the most basic instruments you can have is a string under tension.
I guess percussion instruments are even more basic. Just a piece of wood or of metal with a random shape. For percussion instruments (unless they are laboriously tuned to be harmonic), the octave is nothing special.
It reminds me of how tuning can depend on the era. Sometimes even on the composer. When you're used to hearing something in a particular way, the slightest change can be jarring.
Musical octaves are wired into physics. If a note is at frequency X Hz, then the note one octave above it is 2X Hz. Of course due to the harmonics of the string, a string vibrating at X is also vibrating at 2X, so when you play a note at X Hz, then you are also hearing the note one octave above at the same time.
So... it doesn't need to be learned... When you hear a middle A, you're also hearing the A above for sure. And when you hear the A above by itself, you're hearing one part of the A below.
Of course, understanding that what you're hearing is a called an octave is almost certainly learned. But in my experience, any one who is not tone deaf will naturally lower or raise their voice by an octave if you ask them to sing a song 'lower'.
I never understood octaves. Two sounds of different pitch that are supposed to sound "the same" somehow? Obviously if you shift a melody by multiplying every note by 2 will keep it sounding the same melodically, even though it's immediately obvious it's higher pitched. Beyond that, I have never been able to perceive the "sameness" between equivalent notes on different octaves. When I was younger and played adventure games, sometimes there would be a puzzle involving listening to a short melody and playing it back on some instrument. Literally the only way I could solve those was by analysing the original melody with a spectrometer, then analysing the instrument, and figuring what sequence of keys to press that way.
I read this yesterday and it really stayed with me, enough to find this comment again. Do two notes an octave apart not sound harmonious in a particular identifiable way to you?
From another view, it may help to think about it like this: Tones have a certain relationship with each other, and we split that into octaves in Western music to keep track of them. Just like the number 9 precedes 10, and 19 precedes 20, 10 and 20 are "the same number" in base 10 since that's where the pattern repeates. We could count in another number system, but the underlying physical truth of the numbers would be the same.
mmmm I'm skeptical of a singular study in which a certain tribe of people are asked to modulate a sound to an octave within their singing range disproving octave equivalence being hardwired.
Wanna know why? Our aural circuits have frequency binning, much like an FFT, and they "ring" at certain frequencies, and an octave is either 1/2 or 2x a given frequency/"note".
Closer examination of this system under stimulus is what I would consider a gold standard in this field of research.
The form of research undertaken in this study suggests many things, and the researchers have chosen a particular conclusion and it's not necessarily indicated by even their own data.
Given the current propensity for thinking that frequency detection is pre-cortex (lower level than speech decoding) and available rather early on in the evolutionary chain, I'm going to be interested in physical studies, not social science.
Perhaps it depends on the environmental noise you are exposed. In nature, most noise is percussion-like, non-rational harmonics, so octaves don't make much sens. OTOH listening to electric motors and combustion engines since the day you are born, makes you 'like' octaves, fifths and thirds, since these rotational machines generate these intervals while doing noise.
Notes separated by octaves are like higher and lower versions of the same note.
If a deep-voiced person and a high-voiced person sing the same tune together (without either straining), then they are singing one (or more) octaves apart.
Assuming they are singing in tune with each other, otherwise it will sound more like Chick Corea demonstrates here:
https://youtu.be/yfoxdFHG7Cw?t=371
> Notes separated by octaves are like higher and lower versions of the same note.
I don't know if this is an in-born ability or a cultural one, but I can say that I (for one) don't have any perceptual sense of "sameness" in an octave. To me, two keys with the same letter-name on a piano make two different notes, just like any other interval (though I'm not saying that the different intervals don't sound different). This may be a form of amusia, but I enjoy music, sing to myself, recognize melodies, and find the dissonant Happy Birthday in your link unpleasant (all of which contrasts strongly with the forms of amusia I've read about); on the other hand a lot of the things that people say about music don't mean much to me, except that in some cases I have a mathematical understanding which I can't really connect to a perception.
It was a surprise to me to learn that octaves have a physical. mathematical basis. I discovered this as a teen reading about how to make music on a computer, where the relationship of notes to frequency was explained. To my ear moving up an octave had always sounded like an arbitrary increase in pitch.
Did they try to sing along to the original notes? Perfect pitch needs to be practiced, but it should be immediately apparent when you sing out of tune if you're even slightly musically inclined.
I'm a little frustrated with the lack of discussion of frequencies in this article, which seems like a key bit of background to understanding this work. An "octave" is two sounds where one of them has exactly double the frequency of the other. I think it makes sense that octaves aren't hardwired, however it is fundamentally a mathematical property of frequencies.
I'm not sure how I feel about the thesis; can children tell if something is twice the size of another or do you need the concept of "two?" I think like other people I initially found this a little unlikely because octaves are innate physical and mathematical properties, but there are tons of physical/mathematical properties that require a lot of study to understand intuitively.
Seems like most of the disagreement in the comments here are due to people having different notions of the word octave. This should be no surprise as there are numerous interpretations of the word according to Merriam-Webster:
1: an 8-day period of observances beginning with a festival day
2a: a stanza of eight lines : OTTAVA RIMA
b: the first eight lines of an Italian sonnet
3a: a musical interval embracing eight diatonic degrees
b: a tone or note at this interval
c: the harmonic combination of two tones an octave apart
d: the whole series of notes, tones, or digitals comprised within this interval and forming the unit of the modern scale
e: an organ stop giving tones an octave above those corresponding to the keys
4: the interval between two frequencies (as in an electromagnetic spectrum) having a ratio of 2 to 1
The Tsimane man could probably identify the distinction between octaves of bird calls in his habitat. The experimenter would be in the opposite chair hearing only a handful of notes.
As a music teacher and music therapist, I have observed this behavior for three decades. It's especially true with people that have zero musical background.
I'm not too surprised by this. I've had awful relative pitch all my life, and just recently started ear training. I've also for fun read a bit about how absolute pitch develops in young children.[1]
Bit by bit, I'm realising how much of what's taken for granted by practitioners of Western music is in fact learned.
As a concrete example: ever since I was a child, I've heard people speak of major and minor triads as sounding happy or sad. That seems like something that's just obvious to a lot of people. It's never been obvious to me, and it lead me to think I was (at least partially) tone deaf; if I can't even hear something that is (evidently) that obvious to so many people, how can I ever get better at truly listening to music?
Now that I've started ear training, I'm beginning to mentally associate particular sounds with these chord qualities[2] to the point of actually being able to identify them better than a flipped coin would. (Which is a huge milestone for me!)
But it's taken a lot of practise. And I still don't hear them as "happy" and "sad". If I try to listen for "happy" and "sad" chords, I go back to mischaracterising these chord qualities just as much as before. I've simply never learned that association, for whatever reason.
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It goes on and on: I played a lot of piano (from sheet music) when I was young, and practised the C major scale up and down and up and down, resulting in the interval from the tonic to the major second being etched into my brain very strongly. And I suspect this is what has made it surprisingly hard for me to internally hear a half-step from the tonic. To my brain (before I started ear training), the minimal possible step from the tonic was to the major second. There just weren't any sounds my brain could produce in between.
Learned. With plenty of ear training, I'm now in addition learning to be able to reproduce a minor second too.
Even more speculatively, I've noticed there are some specific half-step intervals I hear as whole steps. These are from E to F, and from B to C. These are half steps, but they are outliers in how often I attempt to classify them as whole steps. I suspect this is because of said C major scale practise: somewhere in the back of my mind, I might associate these tones (in the absolute pitch sense) with steps of the C major scale and therefore think of them as whole steps. I don't have absolute pitch in any useful sense, but it wouldn't surprise me if my brain, somewhere in a back compartment somewhere, has retained the sound of these tones and associated them with adjacent steps in the scale I practised so much.
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Circling back to octaves: Yes, there are physical bases for considering octaves as a special case of some sort, namely that if you strip out the base resonance frequency of one tone on a string or wind instrument, what you're left with are its overtones, which also makes up its octave's frequencies.
But the fact that this relationship is something that makes these tones sound "the same" might very well be learned. The fact that this level of consonance is even desirable might very well be learned. I can picture cultures in which that level of consonance is considered uninteresting, hollow, and weak. Why would one deliberately care to seek something like that?
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[1]: By listening a lot to music that extensively and unpredictably uses the 12 tones of Western music in A440 equal temperament tuning, children can learn to recognise these specific 12 tones in that tuning – but it won't help with any other frequencies!
[2]: I now hear major triads as a combination of hollow and triumphant, and minor triads as fuller and more epic. Some parts of it make sense, others do not.
> As a concrete example: ever since I was a child, I've heard people speak of major and minor triads as sounding happy or sad. That seems like something that's just obvious to a lot of people. It's never been obvious to me
I completely agree. It's not very difficult to string together minor chords in a way that sounds happy, and descending major chords can easily sound sad. If you just play a major or minor chord, I don't associate it with much of anything happy or sad.
Well, there's a lot more to those performances than just switching the scale, so I'm not sure it's a fair comparison. But no, without being told they were switched from major to minor, I'm not sure I would have picked that out.
I think of a major third as a 'brighter' kind of third than a minor third.
Similarly, a major second is a brighter second than a minor second, etc.
This idea generalises to the idea of modal brightness: modes (scales) with more major / augmented notes are 'brighter' than modes with fewer.
For example, Lydian (with its augmented fourth) is brighter than Ionian (major). Or Dorian (with its major sixth) is brighter than Aeolian.
The brightest (conventional) mode is the sixth mode of harmonic major, which goes: root, augmented 2nd, major 3rd, augmented 4th, augmented 5th, major 6th, major 7th.
The least bright (convential) mode is the seventh mode of harmonic minor, which goes: root, minor 2nd, minor 3rd, diminished 4th, diminished 5th, minor 6th, diminished 7th.
The paradigm I would have liked to see is where two tones are held constant and the participant moves a third tone to the location that sounds the best or the worst.
It is remarkable work. But I don't believe it. Octaves are just too basic a phenomena to be viewed as a cultural invention.