According to wikipedia, it's a neural phenomenon. It works VERY nicely when notes share common harmonics, but it isn't neccesary, it works without that.A Thing of Eternity wrote:Interesting, what I'm assuming causes those "phantom notes" is that A and B both have harmonics that line up, thus doubling and becoming louder? I was studying a way to construct a totally usable Just Intonation system for pianos, and just to play in the key of C minor, and to use all possible Major, Minor, Flat Seven, Sus 4 and Add 9 chord I had to use something like 19 keys per octave - because a perfect 5th from a certain note wouldn't line up exactly with what that note should have been tuned to in relationship to the root... I think I ended up with 4 different versions of F, but some notes needed no duplicates. I only did it on paper though, I'll maybe send you my math and results some time so you can tell me if I'm blowing hot air!Schu wrote:Ah, now just intonation is an area where I genuinely do know my shit
One of the most fun things you can do with just intonation (that you can also do with 12 tone equal temperament but it doesn't sound anywhere near as good) is build chords using addition and difference tones: when a sound of frequency a and another of frequency b are heard, other tones of frequencies a-b and a+b are heard, along with, to a lesser degree, all frequencies xa+yb so long as x and y are integers. This leads to my favourite method of chor construction: the Fibonacci sequence! Since everything in it is addition, it's naturally self-reinforcing with addition and subtraction tones. a good 1 1 2 3 5 8 sounds like an exceptionally pure major chord, and rather nice sounding fully diminished seventh chords, typically considered a device for modulation rather than a genuine chord, can be made with 7 5 12 17 or 3 7 10 17 etc. etc.
JI is something I've only been learning about for about a year, and just from that short speach I can tell that you know vastly more than I do on this subject.
I've actually tried for each mode to construct a useable JI system once (and for the general use of classical music, so different chords of course). I'll tell you how I went with just ionian (major), and maybe later for aeolian (minor):
c is defined as 1:1 (or 2:1 or 2^integer:1, you get the idea). From now on I'm going to be lazy and use fractions instead of ratios.
g is always 3/2, e is always 5/3, b is always 15/8, so that's the easy stuff out of the way.
a, f and d depend very much on context. f is almost always 4/3 except for when it's part of a dominant 7th chord, when it's 21/16. d is usually 9/8, but things get tricky when you get a d minor chord. I'm often tempted to consider it a septimal chord and call it 9/8:21/16:27/16 = 6:7:9 but the usual is to make it a regular minor chord of 10/9:4/3:5/3 = 1/6:1/5:1/4. So D can be 10/9 in some contexts. In one of my favourite little cadences, I need D to be 8/7 too. A is usually 5/3 except for a few chords where it's 27/16.
The accidentals - I'm just calling them all sharp, because # is less confusing than b. c# can be 135/128, 25/24 and 17/16. d# can be 75/64 or 6/5. f# can be 7/5, 25/16, sometimes 10/7 too, but it's usually 45/32. g# is either 25/16, 8/5, or if it's being used in a diminished chord, 51/32. a# probably has the most possibilities: 9/5, 16/8, 7/4, 225/128 and a couple of others. That's about 30 different notes, and that's being pretty conservative really, I think I forgot a few of the more obscure ones and I've not bothered with some of the possibilities of stationary modulation.
Yeah, I've been toying with JI for about 7 years