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The paradox is that you can't create a theory of music whose notes are both (a) evenly spaced and (b) contain the integer ratios. You want (a) because it you g
by perihelions 4y ago
The paradox is that you can't create a theory of music whose notes are both (a) evenly spaced and (b) contain the integer ratios.
You want (a) because it you gives you nice algebraic properties (the music structure is invariant under frequency shifts). You want (b) because small-integer ratios are pleasant sounding -- partly culturally-acquired taste, partly because physics gives musical instruments acoustic spectra in integral multiples of a fundamental frequency: f, 2f, 3f ... nf. Small-integer ratios are naturally occurring and very recognizable.
Modern tuning (C-f "12-TET" in the article) almost, approximately satisfies (a) and (b) simultaneously. "12" means there's twelve tones between f and 2f; the ratio between adjacent tones is defined to be 2^{1/12}. This tuning can't contain both f and 3f (so it fails (b)), but it *can* contain f and 2^{19/12}f ~ 2.9966f, which is actually close enough to 3f to be indistinguishable. (Almost works!) But as you build ratios out of larger integers, it audibly falls apart. The closest you can get to (5/3)f is 2^{9/12}f = ~1.6818f, which is already 10% of the way to the next note. And it rapidly gets worse.
This is why two on-paper-identical notes can end up audibly different, depending on what key you're starting with (and hence how they are approached). There's tension internal to music theory itself.
- deleted 4y ago[deleted]
- noncovalence 4y agoIn the context of the difference between D# and Eb, 19-TET is very interesting to play around with. It adds an extra black key between every pair of white keys, and most songs intended for 12-TET still work fine, as long as you play sharps and flats as written. If you play a D# instead of an Eb, you suddenly get a very different sounding interval.
- bonzini 4y agoAlso as long as sharps and flats are written in a very pedantic manner. For example a diminished C chord only sounds "right" if it's notated as C-E♭-G♭-B♭♭ rather than C-E♭-G♭-A. On top of this, harmony may or may not work the same in 19-TET and 12-TET. With the same example of diminished chords, the diminished chord does not divide the octave in four equal parts in 19-TET. Adim and Cdim are enharmonic in 12-TET, but Adim in 19-TET is A-C-E♭-G♭♭; that is, only C and E♭ are the same.
- sobjornstad 4y agoWait, how do you play B♭♭ differently than A on the 19-TET keyboard? There isn't a separate key for that is there?
- bonzini 4y agoIn 19-TET, B♭♭ is an A#, which is two "steps" below B. BTW I was wrong in that Adim is A-C-E♭-G♭. Still not enharmonic to Cdim, and probably diminished chords sound weird in general because G♭-A is four 19-TET steps, which is a 19-TET step smaller than a minor third. It probably sounds halfway between a semidiminished chord and a regular diminished chord.
- bambax 4y agoExcellent explanation! It's not certain though that (a) is as desirable as we make it out to be. We accept that transposition is transparent but it could not be. Keys used to have a meaning attached to them and weren't interchangeable. The direction we have chosen made us lose that and it's a little bit sad IMHO.
- deleted 4y ago[deleted]
- mort96 4y agoOkay but in 12-TET, there are only 12 notes. D# and E♭ are the same note, because there is only one note between D and E. On paper and in practice, the note between D and E is the same whether you write it as D# or as E♭. A piano doesn't know how the note is written in the sheet music. EDIT:to be clear, I'm not disagreeing with most of what you're saying. 12-TET can't represent the desirable perfect fractions, and in a system which can (such as a just intonation system), the starting point does matter. And maybe a vocalist or a violinist would play D# and E♭ subtly differently, I don't know. My main point is just that in a whole lot of contexts, such as when playing a piano, there is no difference between the notes. Your comment made it look like there's always a difference between theory and practice which makes D# and E♭ different in practice, when that's often not the case. We do use 12-TET in practice.
- aikinai 4y agoA piano doesn’t know the difference and can’t differentiate them (on the fly), but a violinist certainly can and does. Most instruments have real time manual control over intonation and skilled musicians will bend pitch to best meet the current key and context.
- mort96 4y agoRight, I made an edit which accounts for that while you wrote your comment. It's an important detail.
- deleted 4y ago[deleted]
- xattt 4y agoWould there be a specific notation for the merry-middle in-between note (D# and E)?
- wizofaus 4y agoYou mean D "three quarter" sharp? The name is a bit illogical because it's really "a sharp and a half", or "sharpened three quarters of a tone". The usual representation looks like a sharp with three vertical bars, and there's a unicode symbol for it (tried to cut and paste but no luck). Microtonality is really annoying on a piano though. As it happens I've been trying to work out what exact intervals are used for the two-chord leitmotif heard in "The Sandman" series, I'm not sure if they're regular microtones or just some sort of eerie detuning (surprisingly I can't find any discussion of it online either).
- 323 4y agoIn Go, and Chess, there are a number of "rules": you should never do this (move the same chess piece twice in the opening), you should do that, ... And then AlphaGo appeared and dismissed all this and did just the right thing for the particular game being played. Know the rules, but if you are an expert you can break the rules. I wonder if AI will do the same thing in music, it will use the "perfect" tuning suitable for a particular piece of music and dismiss this idea of a universal tuning scale.
- nateburke 4y agoWhat would the AI's reward function be?
- 323 4y agoSong gets into Billboard Top 100? Song view count on YouTube? And for earlier stages you can have human raters or similarity with past successful songs.
- the_third_wave 4y agoFeedback from a neural link which indicates satisfaction in the listener?
- xyzzy4747 4y agoIt won’t get terminated on AWS.
- deleted 4y ago[deleted]
- magicalhippo 4y agoYour post reminded me of a video[1] Adam Neely made where he explored some (to me) weird tunings, starting with one where A = 432Hz. As someone who hasn't taken any musical theory or similar, it was quite interesting to hear. [1]: https://www.youtube.com/watch?v=ghUs-84NAAU https://www.youtube.com/watch?v=ghUs-84NAAU
- 4y ago
- BurningFrog 4y agoOn a physical piano you have to make these tuning compromises. A computer generated piano performance could pretty easily pick versions of each note that are in harmony with the other notes played at that moment. I wonder if that would be worth doing? Has it maybe already been done?
- mandmandam 4y agoThis might be the closest to what you're looking for; it was linked in tfa: https://oddsound.com/ https://oddsound.com/
- coldtea 4y agoThough, this is more of a "toolset" to do custom tunings and apply them at various times in a DAW, than something actually implementing what the parent asked for. In other words, it's something someone might use to implement what they asked for - but also lots of other things besides, and it's not meant specifically for that purpose.
- shadowofneptune 4y agoTuning an instrument to the rest of the instruments in the ensemble is pretty common outside of Western music. Could be done with a normal piano.
- BurningFrog 4y agoI'm talking about tuning the individual "keys" of the piano to harmonize with the other keys being played at every moment.
- nsv 4y agoWell, pianos are not as easy to tune as some other instruments. But you're right that it could be done.
- pclmulqdq 4y agoWestern music involves a lot of instruments with fixed tunings, like oboes and trumpets, which are made in equal temperament.
- emerged 4y ago
- evrydayhustling 4y agoNow that we can have electronic instruments that "tune" themselves, could we compute song-optimal tunings that preserve the intervals used most in that song? Does this have a name? As a guitarist we often swap guitars or retune to make certain songs easier to play, or to be able to get a certain tamber put of the note. But I never considered it as a way to address temperament. It's interesting to think how much of music theory emerges out of reconciliation with available instruments, as opposed to reconciliation with the ear.
- ajross 4y ago> Now that we can have electronic instruments that "tune" themselves, could we compute song-optimal tunings that preserve the intervals used most in that song? We've had self-tuning instruments for thousands of years. Vocal harmony has almost always been perfectly tuned for its key. Likewise orchestral strings are fretless and can produce perfect intervals. Equitemperment was an innovation in the 17th century because it approximated the perfect intervals very well ("sounded good") but also permitted the ability to simultaneously represent scales based on every note in the circle of fifths ("sounded interesting"). But the "real" chords were always (well, since the late middle ages) understood to be integer ratios.
- tengwar2 4y agoBrass instruments (not just the trombone) can have micro-adjustments in pitch through the embouchure (lip position/tension) of the musician.
- analog31 4y agoAlso, the tubes for the individual valves have their own tuning slides. A trumpet will typically have a little thumb-operated lever for one of those slides, to help with some of the notes. I saw a video of a tuba solo, and the tubist was working the tuning slides almost as much as the valves.
- tengwar2 4y ago
- wyager 4y ago> You want (b) because small-integer ratios are pleasant sounding -- partly culturally-acquired taste, partly because physics gives musical instruments acoustic spectra in integral multiples of a fundamental frequency I'd say it's more likely because intermodulation distortion between frequencies with low-complexity fractions tends to be low-frequency.
- MichaelDickens 4y ago> The paradox is that you can't create a theory of music whose notes are both (a) evenly spaced and (b) contain the integer ratios. I don't know much about this, but isn't (b) impossible even if you satisfy (a)? There is no sequence of numbers that contains any arbitrary integer ratio because there are infinitely many possible ratios but only finitely many ratios you can make out of a sequence of numbers. (Obviously some ratios like 2:1 and 3:1 are more important than, say, 52697:16427. 12-TET chooses to permit 2^n:1 at the cost of all other ratios, which seems like a good tradeoff to me.)
- perihelions 4y agoThat's true! (a) makes it much more restrictive though: you can't even have {f, 2f, 3f} simultaneously. (If 2f = a^m f and 3f = a^n f, then 2^n = a^{mn} = 3^m, which has no nonzero solutions. Equal temperament contains *no* integer ratios at all, other than whole-number multiples).