[mhax]

That book looks really interesting - thanks for the recommendation.

What still twists my melon about harmony is the Pythagorean Comma.

Harmony equating to integer ratios seems so right... But the fact that a perfect 5th on a piano isn't really a perfect 5th troubles me at some existential level.

[luch]

The comma is not a big deal : it's a problem caused by tone equalizing (aka temperament). Think of instruments as ladders : they have all the same steps, but start at different heights. When combining several types of instruments, the resulting steps are not aligned.

Equal temperament solve that problem by normalizing the steps' absolute height. It's a trade-off between the instrument's perfect harmony, and the orchestra's overall one.

[haberman]

I don't mean to be rude, but I'm not sure you understand the comma. It has nothing to do with combining types of instruments. It's a problem that exists even with a single instrument.

The problem is that, even on a single instrument (say a piano), if you start on some note (say C0), going up seven octaves should land you on exactly the same note as going up twelve perfect fifths (C7). The problem is that the precise tuning of that C7 is different depending on if you followed the perfect octaves or perfect fifths.

If you followed perfect octaves, which have a ratio of 2/1 (or just 2), C7 should have a tuning of precisely (2^7)·C0 = 128·C0. If you followed perfect fifths, which have a ratio of 3/2, C7 should have a tuning of precisely ((3/2)^12)·C0 = ~129.74·C0. The difference between these two is the Pythagorean comma.

Equal temperament solves this problem by keeping the octaves perfect (2:1), but compromising on everything else. It defines a half-step as the 12th power of 2, so a fifth ends up being a little narrower than a perfect 3:2 perfect fifth.

[dcsommer]

I'd recommend "How Equal Temperament Ruined Harmony" if you are interested in the nuances and trade offs of the imperfections of tuning.

[BurningFrog]

I understand that pianos have to pick a frequency for each key.

I wonder if there is any effort in electronic instruments to produce "better" frequencies on the fly? I'll explain:

On a piano hitting C+G plays something close to a perfect fifth.

An electronic instrument could move the G slightly to produce a perfect fifth when that key combination is played.

[mhax]

It's not just a matter of 'better' frequencies. It's trivial to have a softsynth or something similar set up to use just intonation (and many modern composers have experimented with this), but that has practical problems of it's own. Problems which were the very reason that equal temperement took over in the first place. This page has some wonderful examples http://www.nesssoftware.com/home/asn/homepage/teaching/exp-l...

[jpitz]

You cant do that, without a large effort in music education or ensembles of all electronic instruments implementing this feature. Everyone else will be out of tune.

Edit: you could do it, but you'd better implement it on a feedback system that can re-tune in a hundred milliseconds or so.

[BurningFrog]

I'm just talking about an individual keyboard, though if all instruments are digital, which is pretty common these days, you could have a whole orchestra dynamically adjusting pitch to make "perfect" harmonies.

What I wonder is mostly if that would sound any better.

[graycat]

Nice. I'd go with the 2^7 and not the (3/2)^(12).

> It defines a half-step as the 12th power of 2

Uh, instead, "12th root of 2".

[foobarrio]

I've always seen equal temperament similarly to digitizing or any other form quantization. It's "close enough". Any type of dissonance heard in what are supposed to be perfect intervals is like grain in film.

[m-photonic]

The Pythagorean comma is nothing to worry about -- it's tiny and more or less negligible. It's the syntonic comma that's the killer.

The fifths in equal temperament are completely fine, but the thirds are really quite a ways off from the harmonic 5/4 major third. I would recommend experimenting with the traditional meantone tuning system if you're interested in this -- it offers almost as much flexibility as equal temperament (with 6 adjacent keys on the circle of fifths being free of wolf intervals), and it lets you play with "real thirds".

[im3w1l]

Just intonation is just a theory. In practice slightly detuned intervals sound good. Imo they can even sound better than when they are perfectly in tune.

Sometimes musicians deliberately use detuning to create a sound, for instance in the "chorus effect".

http://forum.audacityteam.org/viewtopic.php?f=42&t=68007 http://www.dancemusicproduction.com/forum/index.php?showtopi...

EDIT: Another interesting fact is that pianos aren't actually tuned to equal temperament. The intervals are tuned slightly bigger than it would dictate, to accommodate pianos' slightly inharmonic overtone series.

http://en.wikipedia.org/wiki/Piano_tuning#Stretch

[analog31]

In my view all it means is that an approximation of perfection is good enough. And the notion of temperament only really applies to keyboard instruments. Strings are tuned to perfect intervals, and even 'good' intonation of fingered notes is a matter of interpretation. Wind instruments are a hodgepodge of compromises.

[yellowapple]

> Wind instruments are a hodgepodge of compromises.

It depends on the wind instrument and the setting.

For woodwinds, yes, this is typically true; it's very difficult to adjust the tuning of a particular note on-the-fly.

For brass, however, this becomes less and less true; more sophisticated brass instruments will have easily-accessible "tuning slides" (in addition to the main one) meant to adjust the tuning of a particular note, making them well-suited for situations where absolute tuning is necessary (it's really useful for brass-only ensembles) and where temperament is dominant (like orchestral and symphonic settings).

This is taken to the extreme with the trombone, which is basically just a giant tuning slide with a mouthpiece and bell, allowing for effectively-unhindered tuning flexibility akin to that of a chamber instrument or a human voice.

[zwieback]

It was all too easy for me to adjust the tuning of any particular note on my oboe, unfortunately usually in the wrong direction. The violin players in front of me, who had much better pitch recognition than me, would shoot me dirty looks and I'd get my pencil out to mark up the sheet music with arrows pointing up or down.

[yellowapple]

Ah yes, forgot about oboe (and bassoon and English horn and double-reeds in general), which tend to be much more sensitive about embouchure when it comes to pitch if I remember right.

[analog31]

All good points. I played flute at one time, and there were notes that had to be shoved around a bit, unless hidden in a fast passage, such as C sharp. And I'm told that french horn players use the hand in the bell as a tuning slide.

[yellowapple]

> And I'm told that french horn players use the hand in the bell as a tuning slide.

You can do this, since how far your hand is in the bell can determine pitch up to even a whole semitone or more (I forgot which way, though, but I think it goes flat as you shove your hand in further; it's been a few years since I've played French horn on any semblance of a routine basis, and I don't have one on hand (those things are damn expensive...)). However, this will also affect tone; the further you shove your hand in, the more muffled it'll sound.

French horn players usually opt for double-horns (F/B-flat) in order to avoid needing to use their bell-hands for tuning; since both "sides" (the F side and the B-flat side) of each rotor can be tuned independently of one another, it means that a skilled French horn player who's learned how to transpose F-keyed notes to B-flat-keyed fingerings (which is actually pretty easy if you already know how to play a bass-clef'd valved brass instrument like a baritone or euphonium or tuba, since F-key and bass-clef C-key have the same note positions on the staff, so you just need to be aware of various differences in accidentals and you can transpose very easily) can pick either an F-side fingering or a B-flat-side fingering for a given note depending on which one is closer to the desired pitch.

[meggar]

I think there is a similar situation with guitar tuning where the G string can either be tuned to a 4th above the D string, or a Maj3 below B string, which are not exactly equal, so guitar chords are usually a tiny bit out of tune.

[neonscribe]

No, the Maj3 interval between the 2nd and 3rd strings on the guitar is unrelated to this issue. It's just the fret spacing on the guitar that causes it to have equal temperament and therefore makes chords slightly "out of tune".

[elihu]

Incidentally, a just (non-tempered) guitar fingerboard might look something like this:

http://jsnow.bootlegether.net/cbg/fingerboard_at_a.svg

I swapped fingerboards on an old Harmony Archtone with one based on the above design. Admittedly, 20 notes per octave may have been a bit much; it's kind of awkward to play. The next one I do will be simpler. Sounds nice, though.

[gnaritas]

Just remove the frets, problem solved.

[elihu]

Fretless is difficult to play accurately, especially for chords. Also, you get a louder sound from frets because your fingers aren't muting the strings quite so much.

[bsder]

22 frets can make some of the extended harmonies much, much closer to ideal.

https://www.youtube.com/watch?v=qHHv3mwJTlg