Show HN: Modern music is slightly off tune, here is an instrument that isn't
mhauru.org
mhauru.org
Also, while I agree that things like aesthetics of modulation are one reason why just intonation is such a marginal thing, another big reason is the difficulty of making and tuning instruments for just intonation, when there's in principle an infinity of tones within an interval, and the whole system changes when you change key. I thought this latter reason is an unfortunate one, and something we could try to overcome with digital technology, hence Jintone.
The link opens with the help overlay of the instrument. It reads like a blog text explaining tuning systems and just intonation. If you would rather poke at things than read, then click the X on top left to close the overlay and just play with the thing.
Source code is at github.com/mhauru/Jintone. Comments are very welcome.
or are you referring to the idea that people can be trained to hear these kinds of differences?
A string, pipe, or other resonating component of a musical instrument, if it resonates at some frequency X, will also resonate at frequencies 2X, 3X, 4X, etc. (Picture a taut string: you can get a standing wave where the whole thing vibrates back and forth, or you can get a standing wave where there's a node in the middle and it looks like the left and right sides are vibrating in opposite directions, or you can get a standing wave where there are two nodes and three components, or so forth.)
These frequencies are exact pitches in just intonation. Let's say that you play a C at octave 3. Its overtones are C at octave 4 (twice the pitch), G at octave 4 (3x the pitch, or 3/2 the pitch of C4), C at octave 5 (4x the pitch), E at octave 5 (5x the pitch, or 5/2 the pitch of C5).
When you play a note on any instrument except a pure sine generator, there are natural overtones because of the construction of the instrument. A guitar or a piano basically introduces random noise into the string when you pluck/hammer it, and the standing waves stay around and resonate, producing the "bright" sound of these instruments from their overtones.
If you play a chord on an instrument, the overtones of the various notes in the chord will line up - if you play a C3 major chord, both the C3 and the G3 have overtones at G4, both the C3 and the E3 have overtones at E5, etc. If you're playing an instrument in C-major just intonation, those notes will line up perfectly. (Moreover, for e.g. a piano with the sustain pedal down, the strings for those upper keys will resonate by receiving the vibrations in the air, even if you're not playing them!) If the instrument is in equal temperament, they won't exactly line up. That is perceptible, even if only slightly, to the non-musical listener.
Of course, the tradeoff is that if you play a C# major chord on an instrument tuned to C major just intonation, they'll line up even worse than they would in equal temperament. Which is why "modern" instruments - which is to say, from at least around Bach's time - have avoided tuning to a just intonation.
There are reasons grounded in the physics of soundwaves for why frequencies related by simple ratios sound "special" when played together. As geofft explained, they naturally arise in any physical instrument due to how things like strings (guitar, piano, etc.) or air in a tube (wind instruments, etc.) vibrate. Another way to look at this is to think of two waves with wavelenghts that are related in say a 3/2 ratio (wavelength of a wave is essentially just 1 over the frequency of the wave, so you can think in terms of either, but wavelengths are easier to visualise). If you combine two sine waves with such wavelenghts by summing them up, they form a very clear repeating structure, where the crests and valleys of the sum-wave have a regular pattern to them that repeats: https://www.wolframalpha.com/input/?i=plot+sin%28x%29+%2B+si... Whereas if the wavelenghts of the two waves being combined are not related to each other by a simple ratio, the crests and valleys of the sum-wave keep moving and changing instead of repeating: https://www.wolframalpha.com/input/?i=plot+sin%28x%29+%2B+si... When heard as sound, the fixed repeating structure of the combined wave when the two component waves "line up" with each other has a special quality to it, that we would call harmonious or consonant.
So in the above sense, and because of overtones as explained by geofft, just intonation really is special among various tuning systems. Of course there's still the separate question of if the difference between just intonation and 12-EDO (what most music uses) is big enough for the average listener to notice. This you can test for yourself by playing on Jintone e.g. a just intonation major chord, for instance 1/1, 5/4, and 3/2 together, and then the same chord in 12-EDO by clicking the piano keys directly below the tones you just played, so C, E, and G. The difference isn't like night and day, but the latter pretty clearly has an unstable wobbliness in it that's not there in the just intonation version, a difference that I think most people would detect. If you want to make the difference super obvious, click the gear icon on top right and choose the sawtooth waveform, and repeat the exercise. The timbre of the sawtooth waveform makes the difference more obvious for exactly the reason geofft was talking about: It's a waveform that has a lot of very strong overtones in it (kinda the opposite of a pure sine wave), and in the just intonation chord the overtones of the three tones in the chord match exactly.
of course people can hear and distinguish mathematically pure intervals especially if demonstrated in these ways.
you said,
> This brings some practical musical advantages, but even the average listener can hear the difference, the loss of purity of harmony
which I interpreted as meaning that average listeners are both hearing and actively aware of this difference in their day-to-day exposure to equal-tempered music.
I am very skeptical of this claim, but perhaps this is not what you intended to imply? I can understand why my question would lead to yours and geofft's answers.
I would hypothesise that even the average listener might enjoy some pieces of music somewhat more if they were played in just intonation (depends of course massively on the piece whether this is even realistic or a good idea, but given a suitable piece), but I doubt they would be concious of what's making the difference. Not saying I have any evidence of this, but that's my guesstimate of what the answer to the question "how much of a difference would just intonation vs 12-EDO make for the average listener" would look like.
I also wonder how much continuous pitch instruments do this, when playing pieces of music and with ensembles that would allow adjusting to just intonation. If I walk up to a good violinist and ask them to play a simple melody on a single string, do they gravitate towards just intonation, or does muscle memory make them place their fingers for 12-EDO? What about a superbly good violinist?
The guitar is aspirationally tuned in equal temperament — note that one can transpose a piece up by playing in a higher position and it ought to sound the same. In practice, though, I would find myself often bending notes to make them sound better (closer to just), especially major 3rds. Sometimes depending on the key of the piece I was playing, I might even tune a string slightly off so that the major 3rd would sound better. E.g. when playing a piece in D, I might tune the top E string so that the F# (the major 3rd) on the second fret sounded good.
The trumpet is definitely not tuned to equal temperament — but it's sort-of "continuous-pitch" because the player can bend ("lip") notes slightly flat or sharp, depending on the skill of the player and the properties of the instrument and mouthpiece. In practice, I'd find myself lipping to a greater or lesser extend depending on the key I was playing in, which has more to do with the trumpet's physical limitations (it's impossible to actually make it perfectly in tune in all keys) than the details of which tuning system was chosen. Once I know I have to lip notes into tune anyway, I choose the tuning based on what sounds good, rather than aiming for 2^(7/12) or whatever.
Finally I'll note that even the piano, the prototypical example of an equal-temperament instrument, is not tuned according to equal temperament, because the vibrating strings' stiffness means that the overtones are not harmonic. (They're slightly sharp relative to harmonic.) That means that for a piano to sound in tune with itself, pitches are slightly stretched (flat at the low end, sharp at the high end).
Am I missing something? Do I need to get a spectrum analyzer to see the difference?
To hear the difference, try playing a just intonation C major chord, by clicking the red dots labeled 1/1, 5/4, and 3/2 (hold Shift for sustain to have them ring at the same time), and then the same chord in 12-EDO by clicking the piano keys directly below the tones you just played, so C, E, and G. The difference isn't massive, but you can probably hear it. If you want to make it more obvious, click the gear icon on top right and choose the sawtooth waveform, and try the same thing again.
You can also try playing a 12-EDO E from the piano keys and a 5/4 from the just intonation dots. These are the same tone in the two different tuning systems, and you can hear the "beat" between them, the slow oscillation that comes from them not being quite at the same frequency.
https://www.youtube.com/watch?v=0i68ifRkitA
—and "modern" equal temperament:
https://www.youtube.com/watch?v=HEbi-7tPaqo
As much as people describe just intonation as pure, etc, at least in this example it just sounds wrong to me.
As far as I understand, just intonation requires instruments to be tuned differently for different keys and stops "working" if a piece strays from the key (which music from Bach's time will often do).
Equal temperament is our workaround for the physics which doesn't quite match the 12-tone system.
You can change the frequency of the root tone in the settings though (gear icon, top right). You have to specify it in Hz which is a bit inconvenient, sorry about that. To compute the frequency in Hz for any tone on the usual piano keys, compute 2^(n/12) * 440, where n is the number of semitones above (n>0) or below (n<0) the mid A that you want the root tone to be. If that's not clear, let me know and I can give you list of frequencies for various tones.
And changing keys mid-song is kind of passé, at least in popular music, isn't it? At least, the common practice of modulating up a key for a repeat chorus is a cliché.
Edit: Then again, just now I was listening to a song that, while it doesn't have the clichéd upward key change, does move between two major keys. I'm guessing this song, as written, couldn't be played in just intonation. https://music.youtube.com/watch?v=ElsdmXU8dUw And yes, I know I have questionable taste in music.
Not dissing this, it's a cool experiment and I like the visualization, but there's a reason strict just intonation hasn't been used for 500 years.
Of course, you can't do this for every progression you might enjoy playing, but as an example this one works out quite nicely, with no serious harmonic ambiguity or conflict.