10 ms·

Here's my layman explanation: Pick a frequency. Let's say you picked 400 hertz. If you multiply this frequency by a simple fraction (1/2, 1/3, 2/5, etc) you ge

by pubby 7y ago

Here's my layman explanation: Pick a frequency. Let's say you picked 400 hertz. If you multiply this frequency by a simple fraction (1/2, 1/3, 2/5, etc) you get a new frequency that harmonizes with your original frequency. That means they sound nice when played together. The simpler the fraction, the better the two frequencies will sound (1/2 sounds nicer than 7/13, for example). If you pick several of these fractions between 1 and 2 (such as 3/2, 4/3, 5/4, 2/1, etc) you create what's called a scale. All the frequencies in the scale harmonize with the starting frequency, but they don't necessarily harmonize with each other. Musicians don't always want to play with the same starting frequency so they invented "equal temperament". The idea behind equal temperament is to create a scale using logarithms/exponents instead of fractions. Because it's logarithmic, any frequency in the scale can be used as a starting point and you'll get the same result.

jacquesm 7y ago
I think you meant 'each other' rather than 'themselves' at the end of paragraph 2.
- pubby 7y ago
  Fixed, thanks.
codeulike 7y ago
That means they sound nice when played together And why's that? The little vibrating hair cells in your cochlea, and overtones from the lower note matching up with the higher note.
- codeulike 7y ago
  ... and why did we in the west end up with a 12-note scale? 12 equal divisions of the octave lets you approximate a lot of the interesting ratios 2/3, 3/4 etc (only approximate, see 'even tempered') without the notes being too close together. I think I read the next optimum division would be to divide the octave into 43, but then you're getting far too muddy. ... and if its 12 equal semitones, why have some of them got proper names (CDE) and some treated as variants (sharp/flat). Or in other words, why the white keys and black keys on the piano? Well lets take the 'foundation' note of your piece of music - this would, more or less, be the note that gets involved the most, although its a bit more complicated than that. Building from that note, and lets say we've chosen C to make things simpler, if you wanted to choose other semitones from the 12 available that would let you build lots of nice ratios starting from C and the note thats 3/2 above it which would be G, then you more or less end up with the white keys on the piano, and you leave the black ones out (mostly). Thats called the C-major scale. There's also a minor version where you choose slightly different semitones to get a 'sadder' sound, and thats C-minor. The names of the notes (CDEFGAB) are the white keys on a piano are based on the C-major scale, with the sharps and flats defined in relation to them. Some sort of legacy naming convention that is now baked into musical notation at the lowest level and makes everything much less clear than it should be. Because it you want to transpose up or down and use a different foundation note (as happens literally all the time) then you have to use a mix of notes with proper names CDE and the sharps and flats (a mix of white and black piano keys) and thats when it gets confusing. Well, unless you know what you're doing I suppose, and then its not confusing. In summary: the major and minor scales are kindof optimum selections of notes from the 12 available to make your song sound fairly good. Like a kindof best practice. But the names of the notes are ridiculous. Its as if instead of the digits 0123456789 we had some weird number system where 3 didnt exist and we called it '4 flat' and 6 was replaced by '5 major' for no good reason.
cousin_it 7y ago
Ready to get your mind blown? Pure sine waves don't harmonize. If you play a sine wave at 400Hz, and continuously vary the frequency of another sine wave from say 700 to 900Hz, you won't hear any special consonance at 800. It will sound just as ugly as the neighbors. What really harmonizes is the overtone series. The human voice, and instruments imitating it, have overtones at integer multiples of the main frequency. For example, if I sing a note at 400Hz, it will consist of a sum of sine waves at 400, 800, 1200 etc. When two such notes are sounding at the same time, and their overtone series partially match up - that's when you hear harmony. It's easy to see that it happens at small integer ratios. The guy who came up with this idea (Sethares) also came up with an easy way to test it. He synthesized bell-like sounds whose overtone series aren't exactly integers. And sure enough, melodies with integer ratios of pitches sound horrible when played on that instrument, but melodies with tweaked ratios sound perfectly fine. EDIT: Thank you HN! I believed this for years, but after writing this comment and getting some replies I went and checked, and it's not completely true. Matching overtones play a role, but simple frequency ratios sometimes work even without overtones, and there are proposed explanations for that. https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2607353/#idm139704480569216title https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2607353/#idm139...
- onnodigcomplex 7y ago
  If I open 2 tabs of https://www.szynalski.com/tone-generator/ https://www.szynalski.com/tone-generator/ and then listen to 440+880, and then change 880 to 850 it is a world of difference. I would definitely describe that difference as dissonance and consonance. Now the overtone series IS important and is not always 'simple ratios', a good example in a real instrument is the strong minor third overtone of a carillon, and as expected writing in major for that instrument is hard.
  - cousin_it 7y ago
    That sine wave sounds a bit "fuzzy" to me, maybe the generator adds a small amount of overtones or aliasing. I tried another generator (http://onlinetonegenerator.com/ http://onlinetonegenerator.com/) and the consonance feels weaker.
phlakaton 7y ago
To fill in the bit about temperament: The problem is, once you construct a diatonic scale using these "Pythagorean" ratios, trying to play a diatonic scale that starts and ends on a different pitch, but using the same set of pitches, will have intervals that aren't in tune and sound terrible. Musicians and theorists struggled with this for years, coming up with various compromises that sounded good in some "keys" and awful in others, until math provided the ultimate compromise: base your logarithmic scale on twelfth roots of two and every key will be exactly the same more-or-less-in-tune.
- lebuffon 7y ago
  In my musical training I was taught that musicians figured it out in the west by trial error. Bach then wrote the Well Tempered Klavier to demonstrate pieces in all 12 major and 12 minor keys, in 1722. There is a reference here however to a chinese mathematician who worked it out in 1584. I don't know if his work made it across the continent and influenced anyone. It is not likely that many musicians would have understood the math in the 16th century. https://en.wikipedia.org/wiki/Well_temperament#targetText=Origins,(Duffin%202007%2C%2037) https://en.wikipedia.org/wiki/Well_temperament#targetText=Or....
  - IggleSniggle 7y ago
    Notably, well-temperament is not equal-temperament. Also under-appreciated is that you can change the pitch of each strike of a key on a clavichord by striking harder/softer or pressing into the key. It uses a “hammer-on” like action, so if you press into the key after striking a note, you will increase the tension on the string and raise the pitch. For me, realizing this gave me a whole new appreciation for CPE Bach’s music. There’s probably a dissertation in that somewhere for somebody. A klavier is not necessarily a clavichord, but an organ has some interesting tuning tricks up its sleeves with its various stops.
    - jacquesm 7y ago
      Organs are not really tuned with the stops, the stops engage or disengage banks ('ranks' in organ terminology) of related pipes. The only way this would affect tuning is if you have a secondary rank with an alternate tuning, you'd have to disable one rank and enable another. Organs are tuned directly at the individual pipes (just like string instruments are tuned at the strings). Each pipe, depending on the kind of pipe used has a slide that goes in and out on one end to match the pipe length or, alternatively, a tiny tongue of metal that is rolled up or extended. Tuning (or 'voicing') an organ is super labor intensive and time consuming. For a large organ it can take weeks.
dahart 7y ago
It’s worth noting that equal temperament predates the concept of a logarithm function, so the methods they used were both clever and imprecise. Also worth nothing that mean temperament was an intermediate step between Pythagorean tuning and equal temperament, musicians didn’t jump directly from simple fraction into equal temperament.
TheOtherHobbes 7y ago
Your lay explanation doesn't account for some important fundamentals - like the subtle distinctions between the different historic tunings, the fact that pure equal temperament is only really a thing on electronic instruments, the need for stretch tuning on some acoustic instruments but not others, the origins of the melodic minor, the harmonic minor, and the modes, the use of the dissonant major second in many folk musics around the world... and that's not even getting into the complications of composing in different styles. It's not technically wrong to say that equal temperament is based on the 12th root of 2. But it's also not a complete description of real world tunings, especially in acoustic performance. Generally, math turns out to be a bad way to understand music. There are elements in music that look like math, but the similarities turn out to be superficial and massively oversimplified. If you take them too literally you run into serious conceptual problems almost immediately. In fact the defining feature of music is that it always slips through any simple bounded conceptual model. Music simply isn't simple. That's what makes it so interesting.
marius_k 7y ago
I really like vihart's visual explanation [0]. I don't even know how to summarize it in a self contained way:). [0] https://www.youtube.com/watch?v=i_0DXxNeaQ0 https://www.youtube.com/watch?v=i_0DXxNeaQ0
analog31 7y ago
Temperament is irrelevant to most music, even if you're talking about western music using the 12 tone scale. The exceptions are keyboards and electronic music. The simple reason is that most instruments can't even be played consistently in tune, to the point where you could figure out if they're playing in any temperament at all. String instruments are tuned by pure intervals across the open strings, then the player has to figure out how the other notes should sound. They will push notes up and down to make them sound more "right" in the immediate context. Wind instruments are a bag of compromises. A large percentage of musician jokes are about intonation. Early keyboard instruments were tuned in simple temperaments that a musician could learn how to do quickly. A harpsichord had to be tuned before every performance. Equal temperament required an instrument that stayed in tune long enough to make it worth hiring an expert to tune it.
- kmill 7y ago
  > String instruments are tuned by pure intervals across the open strings Depending on what you mean by a 'pure interval,' it might come as a surprise that string instrumentalists tend to tune their fifths narrower than 3/2 (and apparently sometimes narrower than a 12-EDO perfect fifth!). This is so the perfect fifth above the highest string is tuned correctly as the major third (and some octaves) above the lowest string. Otherwise, the interval would be a Pythagorean major third (81/64) which is somewhat dissonant. (In "How Equal Temperament Ruined Harmony," Duffin recounts how in the late 1800s, even the best piano tuners in Britain were unable to get exact equal temperament, being off by about 1 cent per note in such a way that favored common keys. And, even so, equal temperament for pianos was not popular until the 1910s -- non-equal temperaments were favored due to their sound rather than just their practicality.)
  - analog31 7y ago
    I'm not surprised at all. I was taught to use perfect intervals myself, and it's how I've always seen it taught. But tuning is a very interesting topic. These days I play double bass in a jazz band, so of course every single instrument has its own tuning quirks. An amusing anecdote: I played in a band, and the drummer complimented my intonation. I asked him how a drummer knows anything about intonation. He said: "My college major was trombone."
sopooneo 7y ago
> If you pick several of these fractions between 1 and 2 (such as 3/2, 4/3, 5/4, 2/1, etc A point of strong debate I’ve had with mathematical, but non musical friends: is a scale an ordered or unordered set of such fractions?
kmill 7y ago
This doesn't really explain any of the music theory that's described on the linked page, unless you're meaning you're explaining a theoretical basis for harmony itself? The kind of music theory on the page is about triadic harmony, scales, the ways in which one triad moves to the next, non-chord tones, melody, and rhythm. I do like thinking about tuning systems, however. - If you have purely harmonic instruments, like bowed strings or the human voice (due to mode locking), there's some justification to try messing around with extended just intonation systems (ones where the ratios use prime factors that go beyond 2, 3, and 5). Ben Johnston has a workable notation for this. But, even after having played around with this for a while, I still struggle to hear things involving 7 or 11 as being in tune! - For slightly inharmonic instruments, like pianos or plucked strings, just intonation seems to make a bit less sense... though La Monte Young's "The Well-Tuned Piano" with it's 2,3,7-based tuning does work pretty well. (And as someone else pointed out, piano tuners compensate for inharmonicity even for 12-EDO tuning by making all the intervals just a bit wider. So twelfth roots of two don't completely explain things.) - I'd be interested in seeing how highly inharmonic instruments, like bells, might be tuned to take advantage of their own unique harmonies. It might be that a "5/4" ratio means "take the fifth harmonic of this bell, then find a bell for which that harmonic is the 4th harmonic." Equal temperament, by the way, apparently wasn't popular until around 1900-ish, and so-called equal temperaments before that tended to be various kinds of unequal but "circular" temperaments that worked well enough for every key, yet still preferred common keys. See [1]. 12-tones is an approximation of tuning systems that had been long used by singers and string players. A C# is slightly lower in pitch than a Db (in cents, this is roughly 87 cents vs 109 cents; musicians should be able to perceive deviations of 5 cents, for comparison). In fact, there were early experiments in having split black keys to be able to have both pitches on a keyboard. In [1], the author argues that 1/6-comma meantone is a good approximation for this system, which can be explained as 55-EDO. This makes sweeter major thirds (closer to 5/4 than 12-EDO's 81/64) at the expense of more dissonant perfect fifths, which tends to work out OK for post-medieval western harmonic practice. [1] Ross Duffin, "How Equal Temperament Ruined Harmony"
tripzilch 7y ago
But the really weird thing is that, any ratio that is close to a simple fraction like (say) 3:2, but not quite, is in fact a much less simple fraction, maybe 67:23 or something. Yet we don't need the simple ratios be entirely exact, they can be off a little and you get used to it and it's fine. That's why equal temperament even works. So our brains like simple ratios but also like ratios that are almost kinda like simple ratios but not quite. How does that work? Isn't any ratio close to a "simple ratio" when you leave some wiggle room like this?

by pubby 7y ago

jacquesm 7y ago

I think you meant 'each other' rather than 'themselves' at the end of paragraph 2.

pubby 7y ago
Fixed, thanks.

codeulike 7y ago

That means they sound nice when played together And why's that? The little vibrating hair cells in your cochlea, and overtones from the lower note matching up with the higher note.

codeulike 7y ago
... and why did we in the west end up with a 12-note scale? 12 equal divisions of the octave lets you approximate a lot of the interesting ratios 2/3, 3/4 etc (only approximate, see 'even tempered') without the notes being too close together. I think I read the next optimum division would be to divide the octave into 43, but then you're getting far too muddy. ... and if its 12 equal semitones, why have some of them got proper names (CDE) and some treated as variants (sharp/flat). Or in other words, why the white keys and black keys on the piano? Well lets take the 'foundation' note of your piece of music - this would, more or less, be the note that gets involved the most, although its a bit more complicated than that. Building from that note, and lets say we've chosen C to make things simpler, if you wanted to choose other semitones from the 12 available that would let you build lots of nice ratios starting from C and the note thats 3/2 above it which would be G, then you more or less end up with the white keys on the piano, and you leave the black ones out (mostly). Thats called the C-major scale. There's also a minor version where you choose slightly different semitones to get a 'sadder' sound, and thats C-minor. The names of the notes (CDEFGAB) are the white keys on a piano are based on the C-major scale, with the sharps and flats defined in relation to them. Some sort of legacy naming convention that is now baked into musical notation at the lowest level and makes everything much less clear than it should be. Because it you want to transpose up or down and use a different foundation note (as happens literally all the time) then you have to use a mix of notes with proper names CDE and the sharps and flats (a mix of white and black piano keys) and thats when it gets confusing. Well, unless you know what you're doing I suppose, and then its not confusing. In summary: the major and minor scales are kindof optimum selections of notes from the 12 available to make your song sound fairly good. Like a kindof best practice. But the names of the notes are ridiculous. Its as if instead of the digits 0123456789 we had some weird number system where 3 didnt exist and we called it '4 flat' and 6 was replaced by '5 major' for no good reason.

cousin_it 7y ago

Ready to get your mind blown? Pure sine waves don't harmonize. If you play a sine wave at 400Hz, and continuously vary the frequency of another sine wave from say 700 to 900Hz, you won't hear any special consonance at 800. It will sound just as ugly as the neighbors. What really harmonizes is the overtone series. The human voice, and instruments imitating it, have overtones at integer multiples of the main frequency. For example, if I sing a note at 400Hz, it will consist of a sum of sine waves at 400, 800, 1200 etc. When two such notes are sounding at the same time, and their overtone series partially match up - that's when you hear harmony. It's easy to see that it happens at small integer ratios. The guy who came up with this idea (Sethares) also came up with an easy way to test it. He synthesized bell-like sounds whose overtone series aren't exactly integers. And sure enough, melodies with integer ratios of pitches sound horrible when played on that instrument, but melodies with tweaked ratios sound perfectly fine. EDIT: Thank you HN! I believed this for years, but after writing this comment and getting some replies I went and checked, and it's not completely true. Matching overtones play a role, but simple frequency ratios sometimes work even without overtones, and there are proposed explanations for that. https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2607353/#idm139704480569216title https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2607353/#idm139...

onnodigcomplex 7y ago
If I open 2 tabs of https://www.szynalski.com/tone-generator/ https://www.szynalski.com/tone-generator/ and then listen to 440+880, and then change 880 to 850 it is a world of difference. I would definitely describe that difference as dissonance and consonance. Now the overtone series IS important and is not always 'simple ratios', a good example in a real instrument is the strong minor third overtone of a carillon, and as expected writing in major for that instrument is hard.
- cousin_it 7y ago
  That sine wave sounds a bit "fuzzy" to me, maybe the generator adds a small amount of overtones or aliasing. I tried another generator (http://onlinetonegenerator.com/ http://onlinetonegenerator.com/) and the consonance feels weaker.

phlakaton 7y ago

To fill in the bit about temperament: The problem is, once you construct a diatonic scale using these "Pythagorean" ratios, trying to play a diatonic scale that starts and ends on a different pitch, but using the same set of pitches, will have intervals that aren't in tune and sound terrible. Musicians and theorists struggled with this for years, coming up with various compromises that sounded good in some "keys" and awful in others, until math provided the ultimate compromise: base your logarithmic scale on twelfth roots of two and every key will be exactly the same more-or-less-in-tune.

lebuffon 7y ago
In my musical training I was taught that musicians figured it out in the west by trial error. Bach then wrote the Well Tempered Klavier to demonstrate pieces in all 12 major and 12 minor keys, in 1722. There is a reference here however to a chinese mathematician who worked it out in 1584. I don't know if his work made it across the continent and influenced anyone. It is not likely that many musicians would have understood the math in the 16th century. https://en.wikipedia.org/wiki/Well_temperament#targetText=Origins,(Duffin%202007%2C%2037) https://en.wikipedia.org/wiki/Well_temperament#targetText=Or....
- IggleSniggle 7y ago
  Notably, well-temperament is not equal-temperament. Also under-appreciated is that you can change the pitch of each strike of a key on a clavichord by striking harder/softer or pressing into the key. It uses a “hammer-on” like action, so if you press into the key after striking a note, you will increase the tension on the string and raise the pitch. For me, realizing this gave me a whole new appreciation for CPE Bach’s music. There’s probably a dissertation in that somewhere for somebody. A klavier is not necessarily a clavichord, but an organ has some interesting tuning tricks up its sleeves with its various stops.
  - jacquesm 7y ago
    Organs are not really tuned with the stops, the stops engage or disengage banks ('ranks' in organ terminology) of related pipes. The only way this would affect tuning is if you have a secondary rank with an alternate tuning, you'd have to disable one rank and enable another. Organs are tuned directly at the individual pipes (just like string instruments are tuned at the strings). Each pipe, depending on the kind of pipe used has a slide that goes in and out on one end to match the pipe length or, alternatively, a tiny tongue of metal that is rolled up or extended. Tuning (or 'voicing') an organ is super labor intensive and time consuming. For a large organ it can take weeks.

dahart 7y ago

It’s worth noting that equal temperament predates the concept of a logarithm function, so the methods they used were both clever and imprecise. Also worth nothing that mean temperament was an intermediate step between Pythagorean tuning and equal temperament, musicians didn’t jump directly from simple fraction into equal temperament.

TheOtherHobbes 7y ago

Your lay explanation doesn't account for some important fundamentals - like the subtle distinctions between the different historic tunings, the fact that pure equal temperament is only really a thing on electronic instruments, the need for stretch tuning on some acoustic instruments but not others, the origins of the melodic minor, the harmonic minor, and the modes, the use of the dissonant major second in many folk musics around the world... and that's not even getting into the complications of composing in different styles. It's not technically wrong to say that equal temperament is based on the 12th root of 2. But it's also not a complete description of real world tunings, especially in acoustic performance. Generally, math turns out to be a bad way to understand music. There are elements in music that look like math, but the similarities turn out to be superficial and massively oversimplified. If you take them too literally you run into serious conceptual problems almost immediately. In fact the defining feature of music is that it always slips through any simple bounded conceptual model. Music simply isn't simple. That's what makes it so interesting.

marius_k 7y ago

I really like vihart's visual explanation [0]. I don't even know how to summarize it in a self contained way:). [0] https://www.youtube.com/watch?v=i_0DXxNeaQ0 https://www.youtube.com/watch?v=i_0DXxNeaQ0

analog31 7y ago

Temperament is irrelevant to most music, even if you're talking about western music using the 12 tone scale. The exceptions are keyboards and electronic music. The simple reason is that most instruments can't even be played consistently in tune, to the point where you could figure out if they're playing in any temperament at all. String instruments are tuned by pure intervals across the open strings, then the player has to figure out how the other notes should sound. They will push notes up and down to make them sound more "right" in the immediate context. Wind instruments are a bag of compromises. A large percentage of musician jokes are about intonation. Early keyboard instruments were tuned in simple temperaments that a musician could learn how to do quickly. A harpsichord had to be tuned before every performance. Equal temperament required an instrument that stayed in tune long enough to make it worth hiring an expert to tune it.

kmill 7y ago
> String instruments are tuned by pure intervals across the open strings Depending on what you mean by a 'pure interval,' it might come as a surprise that string instrumentalists tend to tune their fifths narrower than 3/2 (and apparently sometimes narrower than a 12-EDO perfect fifth!). This is so the perfect fifth above the highest string is tuned correctly as the major third (and some octaves) above the lowest string. Otherwise, the interval would be a Pythagorean major third (81/64) which is somewhat dissonant. (In "How Equal Temperament Ruined Harmony," Duffin recounts how in the late 1800s, even the best piano tuners in Britain were unable to get exact equal temperament, being off by about 1 cent per note in such a way that favored common keys. And, even so, equal temperament for pianos was not popular until the 1910s -- non-equal temperaments were favored due to their sound rather than just their practicality.)
- analog31 7y ago
  I'm not surprised at all. I was taught to use perfect intervals myself, and it's how I've always seen it taught. But tuning is a very interesting topic. These days I play double bass in a jazz band, so of course every single instrument has its own tuning quirks. An amusing anecdote: I played in a band, and the drummer complimented my intonation. I asked him how a drummer knows anything about intonation. He said: "My college major was trombone."

sopooneo 7y ago

> If you pick several of these fractions between 1 and 2 (such as 3/2, 4/3, 5/4, 2/1, etc A point of strong debate I’ve had with mathematical, but non musical friends: is a scale an ordered or unordered set of such fractions?

kmill 7y ago

This doesn't really explain any of the music theory that's described on the linked page, unless you're meaning you're explaining a theoretical basis for harmony itself? The kind of music theory on the page is about triadic harmony, scales, the ways in which one triad moves to the next, non-chord tones, melody, and rhythm. I do like thinking about tuning systems, however. - If you have purely harmonic instruments, like bowed strings or the human voice (due to mode locking), there's some justification to try messing around with extended just intonation systems (ones where the ratios use prime factors that go beyond 2, 3, and 5). Ben Johnston has a workable notation for this. But, even after having played around with this for a while, I still struggle to hear things involving 7 or 11 as being in tune! - For slightly inharmonic instruments, like pianos or plucked strings, just intonation seems to make a bit less sense... though La Monte Young's "The Well-Tuned Piano" with it's 2,3,7-based tuning does work pretty well. (And as someone else pointed out, piano tuners compensate for inharmonicity even for 12-EDO tuning by making all the intervals just a bit wider. So twelfth roots of two don't completely explain things.) - I'd be interested in seeing how highly inharmonic instruments, like bells, might be tuned to take advantage of their own unique harmonies. It might be that a "5/4" ratio means "take the fifth harmonic of this bell, then find a bell for which that harmonic is the 4th harmonic." Equal temperament, by the way, apparently wasn't popular until around 1900-ish, and so-called equal temperaments before that tended to be various kinds of unequal but "circular" temperaments that worked well enough for every key, yet still preferred common keys. See [1]. 12-tones is an approximation of tuning systems that had been long used by singers and string players. A C# is slightly lower in pitch than a Db (in cents, this is roughly 87 cents vs 109 cents; musicians should be able to perceive deviations of 5 cents, for comparison). In fact, there were early experiments in having split black keys to be able to have both pitches on a keyboard. In [1], the author argues that 1/6-comma meantone is a good approximation for this system, which can be explained as 55-EDO. This makes sweeter major thirds (closer to 5/4 than 12-EDO's 81/64) at the expense of more dissonant perfect fifths, which tends to work out OK for post-medieval western harmonic practice. [1] Ross Duffin, "How Equal Temperament Ruined Harmony"

tripzilch 7y ago

But the really weird thing is that, any ratio that is close to a simple fraction like (say) 3:2, but not quite, is in fact a much less simple fraction, maybe 67:23 or something. Yet we don't need the simple ratios be entirely exact, they can be off a little and you get used to it and it's fine. That's why equal temperament even works. So our brains like simple ratios but also like ratios that are almost kinda like simple ratios but not quite. How does that work? Isn't any ratio close to a "simple ratio" when you leave some wiggle room like this?