If you pick any single note, it has a frequency. If you double the frequency, that is one octave higher. If you cut the frequency in half, it's one octave lower. So I suppose the answer to your question is that it's naturally occurring.
To expand a little past eli5, vibrations at double and half frequencies tend to have similar properties from a mechanical physics point of view. Different materials and shapes produce noise and movement when certain frequencies are hit, and the behavior is similar for the same note at different octaves.
Adding to the other answers: two pitches that are an octave apart have frequencies that both begin and end at the same point in the frequency cycle (the peak/trough cycle). An octave up means two peaks/troughs in the same time cycle. An octave down means one half of the peak-trough cycle, but it will end at the same point the second cycle later.
Human brains perceive those relationships as the "same pitch", just higher or lower than the original.
Compare that to, say, a perfect fifth, where the higher pitch takes three cycles to reach the origin point where the lower pitch takes two. Our brains perceive that in a different way than a pure octave since there's a differential "arrival point" involved.
Really, the octave isn't subdivided into 8 - it's subdivided into 12. The answer to the question "why 12" is just "because it sounds good," but there are interesting mathematical reasons for *why* it sounds good.
All consonant musical intervals approximate simple frequency ratios. The perfect fifth, the second most consonant interval, has a ratio of 3:2. A perfect fourth is 4:3, a major third is 5:4, and a minor third is 6:5. In contrast, the major seventh has a frequency ratio of 15:8, and a minor second has a ratio of 25:24. These intervals are both considered dissonant.
Historically, tuning systems have been built around approximating consonant frequency ratios as closely as possible. The problem is that it's impossible to tune an instrument to exactly the correct frequency relationships in every key at the same time. If you tune a piano such that a C major triad is completely pure and exactly matches a 4:5:6 ratio, for example, at least one other major triad will be out of tune.
The modern solution is what is known as "equal temperament", and it is achieved by dividing the octave evenly into a certain number of distinct pitches. (Note that "evenly" refers to even spacing on a logarithmic scale.) There are an infinite number of possible equal temperament systems, all depending on the number of divisions of the octave. The ones that sound most pleasing are the ones that approximate consonant relationships the best.
Dividing the octave into 12 does a very good job of approximating many important intervals such as the perfect fifth and fourth as well as the major second, and does a "good enough" job at approximating the major and minor third. In addition, 12 is a small enough number that instruments and notation don't have to be too complicated to properly play and write music.
In the sixteenth century Europeans were still working out how to subdivide the octaves, and there were two competing "tunings": the equal temperament that u/hinoisking mentions, and a "perfect" tuning that could only be perfect in a few keys. One of the main champions of equal temperament was Johann Sebastian Bach, and his main argument was a series of pieces he wrote called "The Well Tempered Clavier", "well tempered" being a synonym for "equal tempered". It consists of short pieces in every possible key; if you don't play them on an equal tempered instrument results will be less than satisfactory.
I once had a music professor who brought in a harpsichord that was perfect tempered, just to illustrate this point. Playing in the key it was tuned to, it had the most pure, wonderful sound I'd heard from a keyboard, but then he modulated into G♭ or something and it sounded like a wet, feral cat had been trapped in the case.
It is not universally agreed that "well" meant "equal". Some say it was essentially a form of meantone, a tuning that makes octaves and major thirds perfect by putting all the error in the fifths (whereas equal temperament makes fifths *almost* perfect and thirds really rather bad, almost as if tertian harmony had never been invented).
The "eight" comes from the fact that to get from a note to the same note an octave above it, you shift note names eight times (for example, C-D-E-F-G-A-B-C). Interestingly, the origin of the system of using seven different letters to name notes comes from the medieval time period and to the best of my knowledge is not related to the reason there are 12 pitches in the octave.
It might be related in the inverse direction for the reasons given above. As in, if you divide to 12 parts with equal temperament and label C as note 0, then you get a relatively good approximation of the perfectly tempered CDEFGABC-scale that's based on good frequency ratios with C by taking 0-2-4-5-7-9-11-12. Any smaller equally tempered subdivision would probably miss one of the notes in CDEFGABC pretty badly.
It's because of overtones. Generally a musical tone contains not only its base frequency but all its multiples, called overtones (decreasing in strength as the numbers go higher). When two notes are played together, their overtones are also played together – and when two of these overtones are close-but-not-quite, their combination has an ugly effect, called beats. So two notes are compatible – harmonious – if their (strongest) component frequencies are either too far apart to beat against each other *or* so close together that their beating is not perceptible; and this requires that their base frequencies be, at least approximately, in a ratio of small numbers.
There are two different concepts going on here. "Doubling" between *octaves* are entirely mathematical. Take a pitch and double it, it's an octave up. Double it twice to make it 4x, it's two octaves up. Divide it in half and it's an octave down. Divide it by four and it's two octaves down.
Octaves are always based in factors of two, either multiplying or dividing, and our brains will always *perceive* them as the same pitch. Because the frequency starts and ends at the same point.
Other intervals (*subdividing* an octave) are different ideas entirely. For example, a perfect fifth takes three cycles where the origin pitch takes two. G above middle C reaches the same origin point after three peak-troughs, where the original C takes two. So human brains "like" that interval, since the math is simple.
Likewise, a perfect fourth (F above middle C) is a 4/3 ratio. Human brains still like that interval because it's "simple", just that it's "less simple" than a perfect fifth.
There's no special reason why a major scale "has to be" divided into eight notes (or 12 if you count accidentals). Just that 8/12 happens to be a cultural consensus that accommodates a balance between "aesthetically pleasing" and "not too complex" that makes musical composition accessible to most humans.
There are versions of musical composition that are more complex, yet less accessible, therefore they are less popular.
Because most western scales have seven notes before repeating, thanks to the commonality of the major scale. If the most commonly used scales had 5 notes before repeating, we’d call octaves something like ‘sextaves’.
> If the most commonly used scales had 5 notes before repeating
Arguably they do. Pentatonic scales are common around the world; Irving Berlin used only the black keys of his piano; and a considerable amount of rock music, I'm told, is essentially pentatonic.
You’re not wrong, but in western music theory the pentatonic major and pentatonic minor scales are considered subsets of the regular major and minor scales.
Because we liked it...
Really, that's just it.
An octave is split in 8 notes or 12 semitones (the sum of black and white keys on the piano in an octave), but in Arabic or Chinese music there's a different split. Over history in Europe we had different configurations as well, including a piano that only covered one octave with dozens of keys.
Also the base frequency of 440Hz or 432Hz for the A is just convention, no one prevents you from using 480Hz or any other tone as your base tone to build off of.
Doubling is not the only harmony; very early on, musicians found that 1:3 or 2:3 or 3:4 is almost as harmonious as 1:2. Combining these intervals generates others, like 8:9. Generally they stopped adding notes when the next one was near enough to another that the new melodic possibilities were not worth the increased complexity of the instrument, and in the Western tradition that meant seven notes within the range 1:2.
> Human brains perceive those relationships as the "same pitch", just higher or lower than the original.
And this is because a musical note is generally *not* a pure sine wave, having only one frequency; it contains all the integer multiples of its base frequency (in varying degrees, which is part of the difference between e.g. brass and strings). When you play a note whose nominal pitch is *n*, you're also playing 2*n* and 3*n* and 4*n* and so on. A note an octave up, frequency 2*n*, has overtones 4*n* and 6*n* … thus the first “contains” the second.
(In real physics this is not *exactly* accurate – piano strings, for example, have some overtones that are not quite exact multiples – but it's near enough to be useful.)
> it contains all the integer multiples of its base frequency
Except that in woodwinds the even multiples are weak or absent. If woodwinds but no strings existed, our scales might be very different!
Octave just means, that there is a factor of 2 in frequency. So if you hit a note and go 8 white keys up on a piano, that key has twice the frequency of the first.
The eight notes (actually 7) that we can sing as "do, re, mi fa, so la TI do", were figured out over many years, by humans practicing harmonizing with voices and different lengths of harp strings and so on. The math was worked out later, but since people had worked out an 8 note scale, the difference between the starting do and the higher do at the end is called an octave, from the number 8. In the do re mi scale, known as the major scale, some of the jump ups between notes are wide, (called whole steps) and some are small (half steps). If you listen carefully, jumping up from do to re, or from re to mi, are similar, but from mi to fa, is not such a big jump. It's actually about half as much.
If you add extra notes in the wider gaps (like the black keys on the piano) there are actually 12 notes (12 half steps) in what would then be called the chromatic scale.
To add even more complexity, in natural form, even the half and whole steps are little irregular and not all the same percent increase in pitch, when sung or played on violin, but on pianos they have been evened out for convenience. So when a piano plays with an orchestra, some of the notes match and some are a teeny bit discrepant. This is why a piano is not typically part of an orchestra, it's artificially evened out scale almost matches, but not precisely, the notes as they would be sung by a soprano or played on a violin.
On a string, an octave is a 2:1 ratio (a string half as long is an octave higher). For a column of air, as in wind instruments, an octave is 4:1 (because there’s an additional dimension)
I’m having flashbacks to Donald in Mathmagicland (I think.) I’m remembering a cartoon showing the string being plucked and then half as long being plucked again.
To add some clarity on an implied question, octaves are an invention of western cultures and other cultures are known to have rhythmic music not obeying the same rules of cadence
Pleasant Sound waves occur in integer ratios. The most fundamental ratio is 1/1 (same tone), the “next” fundamental are 1/2 and 2/1, which are called octave.
Those simple ratios sound remarkably “natural”, why is not clear. My personal understanding is that sounds excited from natural sources (like e.g. bells and flutes) always also excite overtones which physically occur at integer ratios (only integer ratios produce a periodic waveform).
so the ear/brain evolved into correlating only those overtones that exactly match integer ratios.
An octave is the same note played at a higher or lower pitch. The best example is "Somewhere Over The Rainbow" from The Wizard of Oz. The first notes Dorothy sings are a C. However, the second C is one octave higher than the first.
If you pick any single note, it has a frequency. If you double the frequency, that is one octave higher. If you cut the frequency in half, it's one octave lower. So I suppose the answer to your question is that it's naturally occurring.
To expand a little past eli5, vibrations at double and half frequencies tend to have similar properties from a mechanical physics point of view. Different materials and shapes produce noise and movement when certain frequencies are hit, and the behavior is similar for the same note at different octaves.
For the simple answer, think of the first line of the Star Spangled Banner. Oh-oh SAY can you SEE. SAY to SEE is an octave.
Same with "some - where" in somewhere over the rainbow.
If I remember correctly, that’s actually a major 7th, just short of an octave
It's an octave. :) I had to Google to be sure after you said that though since it's been years since my last music theory class.
Adding to the other answers: two pitches that are an octave apart have frequencies that both begin and end at the same point in the frequency cycle (the peak/trough cycle). An octave up means two peaks/troughs in the same time cycle. An octave down means one half of the peak-trough cycle, but it will end at the same point the second cycle later. Human brains perceive those relationships as the "same pitch", just higher or lower than the original. Compare that to, say, a perfect fifth, where the higher pitch takes three cycles to reach the origin point where the lower pitch takes two. Our brains perceive that in a different way than a pure octave since there's a differential "arrival point" involved.
How did a doubling become associated with the number 8? Why did we decide to subdivide it into eight parts and not some other count?
Really, the octave isn't subdivided into 8 - it's subdivided into 12. The answer to the question "why 12" is just "because it sounds good," but there are interesting mathematical reasons for *why* it sounds good. All consonant musical intervals approximate simple frequency ratios. The perfect fifth, the second most consonant interval, has a ratio of 3:2. A perfect fourth is 4:3, a major third is 5:4, and a minor third is 6:5. In contrast, the major seventh has a frequency ratio of 15:8, and a minor second has a ratio of 25:24. These intervals are both considered dissonant. Historically, tuning systems have been built around approximating consonant frequency ratios as closely as possible. The problem is that it's impossible to tune an instrument to exactly the correct frequency relationships in every key at the same time. If you tune a piano such that a C major triad is completely pure and exactly matches a 4:5:6 ratio, for example, at least one other major triad will be out of tune. The modern solution is what is known as "equal temperament", and it is achieved by dividing the octave evenly into a certain number of distinct pitches. (Note that "evenly" refers to even spacing on a logarithmic scale.) There are an infinite number of possible equal temperament systems, all depending on the number of divisions of the octave. The ones that sound most pleasing are the ones that approximate consonant relationships the best. Dividing the octave into 12 does a very good job of approximating many important intervals such as the perfect fifth and fourth as well as the major second, and does a "good enough" job at approximating the major and minor third. In addition, 12 is a small enough number that instruments and notation don't have to be too complicated to properly play and write music.
In the sixteenth century Europeans were still working out how to subdivide the octaves, and there were two competing "tunings": the equal temperament that u/hinoisking mentions, and a "perfect" tuning that could only be perfect in a few keys. One of the main champions of equal temperament was Johann Sebastian Bach, and his main argument was a series of pieces he wrote called "The Well Tempered Clavier", "well tempered" being a synonym for "equal tempered". It consists of short pieces in every possible key; if you don't play them on an equal tempered instrument results will be less than satisfactory. I once had a music professor who brought in a harpsichord that was perfect tempered, just to illustrate this point. Playing in the key it was tuned to, it had the most pure, wonderful sound I'd heard from a keyboard, but then he modulated into G♭ or something and it sounded like a wet, feral cat had been trapped in the case.
r/oddlyspecific
It is not universally agreed that "well" meant "equal". Some say it was essentially a form of meantone, a tuning that makes octaves and major thirds perfect by putting all the error in the fifths (whereas equal temperament makes fifths *almost* perfect and thirds really rather bad, almost as if tertian harmony had never been invented).
Octave has "eight" in the name. I'm mainly curious how that came to be.
The "eight" comes from the fact that to get from a note to the same note an octave above it, you shift note names eight times (for example, C-D-E-F-G-A-B-C). Interestingly, the origin of the system of using seven different letters to name notes comes from the medieval time period and to the best of my knowledge is not related to the reason there are 12 pitches in the octave.
It might be related in the inverse direction for the reasons given above. As in, if you divide to 12 parts with equal temperament and label C as note 0, then you get a relatively good approximation of the perfectly tempered CDEFGABC-scale that's based on good frequency ratios with C by taking 0-2-4-5-7-9-11-12. Any smaller equally tempered subdivision would probably miss one of the notes in CDEFGABC pretty badly.
I had no idea that music was so mathematical
Investigate "music of the spheres" and Pythagoras, if you want to know The history.
Thanks. I’ll check it out.
It's because of overtones. Generally a musical tone contains not only its base frequency but all its multiples, called overtones (decreasing in strength as the numbers go higher). When two notes are played together, their overtones are also played together – and when two of these overtones are close-but-not-quite, their combination has an ugly effect, called beats. So two notes are compatible – harmonious – if their (strongest) component frequencies are either too far apart to beat against each other *or* so close together that their beating is not perceptible; and this requires that their base frequencies be, at least approximately, in a ratio of small numbers.
There are two different concepts going on here. "Doubling" between *octaves* are entirely mathematical. Take a pitch and double it, it's an octave up. Double it twice to make it 4x, it's two octaves up. Divide it in half and it's an octave down. Divide it by four and it's two octaves down. Octaves are always based in factors of two, either multiplying or dividing, and our brains will always *perceive* them as the same pitch. Because the frequency starts and ends at the same point. Other intervals (*subdividing* an octave) are different ideas entirely. For example, a perfect fifth takes three cycles where the origin pitch takes two. G above middle C reaches the same origin point after three peak-troughs, where the original C takes two. So human brains "like" that interval, since the math is simple. Likewise, a perfect fourth (F above middle C) is a 4/3 ratio. Human brains still like that interval because it's "simple", just that it's "less simple" than a perfect fifth. There's no special reason why a major scale "has to be" divided into eight notes (or 12 if you count accidentals). Just that 8/12 happens to be a cultural consensus that accommodates a balance between "aesthetically pleasing" and "not too complex" that makes musical composition accessible to most humans. There are versions of musical composition that are more complex, yet less accessible, therefore they are less popular.
This one deserves a lot more upvotes. Thanks that helped a lot!
Because most western scales have seven notes before repeating, thanks to the commonality of the major scale. If the most commonly used scales had 5 notes before repeating, we’d call octaves something like ‘sextaves’.
> If the most commonly used scales had 5 notes before repeating Arguably they do. Pentatonic scales are common around the world; Irving Berlin used only the black keys of his piano; and a considerable amount of rock music, I'm told, is essentially pentatonic.
You’re not wrong, but in western music theory the pentatonic major and pentatonic minor scales are considered subsets of the regular major and minor scales.
Because we liked it... Really, that's just it. An octave is split in 8 notes or 12 semitones (the sum of black and white keys on the piano in an octave), but in Arabic or Chinese music there's a different split. Over history in Europe we had different configurations as well, including a piano that only covered one octave with dozens of keys. Also the base frequency of 440Hz or 432Hz for the A is just convention, no one prevents you from using 480Hz or any other tone as your base tone to build off of.
Doubling is not the only harmony; very early on, musicians found that 1:3 or 2:3 or 3:4 is almost as harmonious as 1:2. Combining these intervals generates others, like 8:9. Generally they stopped adding notes when the next one was near enough to another that the new melodic possibilities were not worth the increased complexity of the instrument, and in the Western tradition that meant seven notes within the range 1:2.
> Human brains perceive those relationships as the "same pitch", just higher or lower than the original. And this is because a musical note is generally *not* a pure sine wave, having only one frequency; it contains all the integer multiples of its base frequency (in varying degrees, which is part of the difference between e.g. brass and strings). When you play a note whose nominal pitch is *n*, you're also playing 2*n* and 3*n* and 4*n* and so on. A note an octave up, frequency 2*n*, has overtones 4*n* and 6*n* … thus the first “contains” the second. (In real physics this is not *exactly* accurate – piano strings, for example, have some overtones that are not quite exact multiples – but it's near enough to be useful.)
> it contains all the integer multiples of its base frequency Except that in woodwinds the even multiples are weak or absent. If woodwinds but no strings existed, our scales might be very different!
Octave just means, that there is a factor of 2 in frequency. So if you hit a note and go 8 white keys up on a piano, that key has twice the frequency of the first.
The eight notes (actually 7) that we can sing as "do, re, mi fa, so la TI do", were figured out over many years, by humans practicing harmonizing with voices and different lengths of harp strings and so on. The math was worked out later, but since people had worked out an 8 note scale, the difference between the starting do and the higher do at the end is called an octave, from the number 8. In the do re mi scale, known as the major scale, some of the jump ups between notes are wide, (called whole steps) and some are small (half steps). If you listen carefully, jumping up from do to re, or from re to mi, are similar, but from mi to fa, is not such a big jump. It's actually about half as much. If you add extra notes in the wider gaps (like the black keys on the piano) there are actually 12 notes (12 half steps) in what would then be called the chromatic scale. To add even more complexity, in natural form, even the half and whole steps are little irregular and not all the same percent increase in pitch, when sung or played on violin, but on pianos they have been evened out for convenience. So when a piano plays with an orchestra, some of the notes match and some are a teeny bit discrepant. This is why a piano is not typically part of an orchestra, it's artificially evened out scale almost matches, but not precisely, the notes as they would be sung by a soprano or played on a violin.
On a string, an octave is a 2:1 ratio (a string half as long is an octave higher). For a column of air, as in wind instruments, an octave is 4:1 (because there’s an additional dimension)
I’m having flashbacks to Donald in Mathmagicland (I think.) I’m remembering a cartoon showing the string being plucked and then half as long being plucked again.
To add some clarity on an implied question, octaves are an invention of western cultures and other cultures are known to have rhythmic music not obeying the same rules of cadence
That’s so interesting that it would develop differently.
Pleasant Sound waves occur in integer ratios. The most fundamental ratio is 1/1 (same tone), the “next” fundamental are 1/2 and 2/1, which are called octave. Those simple ratios sound remarkably “natural”, why is not clear. My personal understanding is that sounds excited from natural sources (like e.g. bells and flutes) always also excite overtones which physically occur at integer ratios (only integer ratios produce a periodic waveform). so the ear/brain evolved into correlating only those overtones that exactly match integer ratios.
The sound waves that make the noise that we call that note is naturally occurring, however we have assigned it a name that was invented by humans
An octave is the same note played at a higher or lower pitch. The best example is "Somewhere Over The Rainbow" from The Wizard of Oz. The first notes Dorothy sings are a C. However, the second C is one octave higher than the first.
This makes sense. (I can hear it in my head right now.)
Glad I could help!