[openrisk]

Besides geometry there is not a lot of music either, in the sense that even this simple symmetry is kinda fake, effectively a forced resolution of an essentially unsolvable "problem": hammering the intervals into place to inject some "logic" into the task of dividing the octave in heptatonic scales. Despite the unrepentant Pythogereans across all ages, our musical brain is not mathematical except in a very loose way.

Besides aesthetics, though, there might be educational value given that the equal temperament tuning is a cornerstone of western music education.

[PaulDavisThe1st]

> the task of dividing the octave in heptatonic scales.

that's not task. The octave has already been divided into twelve tones here.

The task is to pick sets of those twelve tones to serve some aesthetic purpose. The sets may be of various sizes, though 7 is common in a lot of european music.

If the task was to generate heptatonic scales from within the octave, there are a huge number of possibilities not described here, and most of them are rarely used (though many more than the ones based on a 12TET system are).

[openrisk]

> The task is to pick sets of those twelve tones to serve some aesthetic purpose

Aesthetic in the musical or visual sense? The visual aspect is based on the Z12 symmetry and it is pleasant - like all symmetries.

The question is what does the visual experience have to do with the music experience?

The first disconnect with the musical experience is that the 12TET itself is not what people would, e.g., choose to sing in [1].

The second disconnect is that the Greek modes of the major scale are not remotely covering all the scales people enjoy, even adopting a Eurocentric point [2].

[1] https://music.stackexchange.com/questions/41383/do-capable-h...

[2] https://en.wikipedia.org/wiki/Harmonic_minor_scale

[HPsquared]

On singers and violinists adjusting their harmony to just intonation by fine-tuning to zero beat frequency, I wonder if anyone has made a keyboard that can do that.

[PaulDavisThe1st]

If you mean a keyboard which includes a mechanism for causing strings to vibrate, you can tune ANY such keyboard to use just intonation.

What you cannot do is modulate between keys with a keyboard tuned to just intonation: it would have to be retuned for every key change. The scope of the mechanism that would be required to do this has not been implemented since the harpsichord was invented.

There are synthesizers that can be retuned in this way, because there is no physical mechanism to adjust. The results are ... odd. It is still challenging to play them because in addition to performing the notes, you need to signal the key change/retuning points.

Also, when singers and violinists do this, they are not "fine tuning to zero beat frequency". Either you sing in just intonation, in which case you cannot modulate between keys (because the Nth note of the scale has a different frequency depending on the root note), or you sing in some tempered scale (in which the frequencies of the notes have been adjusted to make modulation possible).

[black_knight]

Eivind Groven developed a mechanical piano for playing in just intonation: http://www.joranrudi.no/mediefiler/The%20Just%20Intonation%2...

[PaulDavisThe1st]

Ah yes, of course. 36T ... increasing the number of pitches per octave is a different approach to the problem, and works (at some cost to the performer :)

[bkazez]

This is not true.

Singers and violinists can and do adjust intonation so each chord sounds (justly) in tune. The exception is if they were trained with equal tempered instruments (which is common nowadays - see Duffin, “How Equal Temperament Ruined Harmony”) or if they are playing with pre-quantized (fretted/keyed) instruments, in which case they would match the existing temperaments.

So the linked article, while it shows some beautiful shapes linked to 12s, has nothing to do with actually (justly) in tune music.

Source: master’s degree in the topic; am a professional singers specializing in music written before equal temperament was invented

[Tor3]

I go to a lot of choir concerts. What I've found is that I much prefer (good) choirs singing without accompanying instruments, because when there are instruments involved the harmonies always fall into equal temperament. There is a quality when they sing a cappella which simply isn't there when they don't.

[tugu77]

For a professional musician you are oddly singer/violin focused. Any instrument which can physically detune while playing has their musicians do this. On wind instruments it's via the mouthpiece, any string instrument beyond violin has some flexibility etc. It's only the piano that doesn't, essentially everybody else does.

But in practise, for many music styles, it doesn't really matter. Music is so much more than whether some chord is pitch perfect in tune.

Source: Jazz musician on 6 instrument types part time professional for 25 years (other part is software engineer).

[PaulDavisThe1st]

I did speak a bit loosely and too simplistically. What I was trying to get to was the point you're making which is the same point the GP was making: when performers have the ability (because of the instrument they are using, including the voice), they will adjust to reduce beating.

My words on this were wrong and misleading.