Music + Maths + physics


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Schu
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Music + Maths + physics

Post by Schu »

Just splitting from here: viewtopic.php?p=53001#p53001" onclick="window.open(this.href);return false;
A Thing of Eternity wrote:
Schu wrote:Ah, now just intonation is an area where I genuinely do know my shit :P

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.
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!

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.
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.

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 :P
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Re: Music + Maths + physics

Post by SadisticCynic »

:shock:

I'll just stick with hitting things... (drums I mean).
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Re: Music + Maths + physics

Post by Schu »

Drummers aren't real musicians ;)
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Re: Music + Maths + physics

Post by A Thing of Eternity »

Schu wrote:
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.
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 :P

Very cool, I'm assuming you chose those more extreme ratios by ear? Or was that accomplished through research into what other people have attempted. One of my problems is that I am very weak on my music theory, if I want to know the minor third of say, F, I have to literally write down the notes and count over to the result, or take a year doing so in my head.



What I tried to do was very different, and until I have a cheap piano or fully tunable keyboard it will remain un-tested. I'd love to hear your thoughts on this in theory though, it's just something I came up with one day when I was thinking about how fretless players are completely unhindered, they can play any pitch they want... so what if I had the ability to use any pitch necessary on a keyed instrument? For something that could switch keys this would require far too many notes per octave, but for just one scale in one key... might be possible. This systen would use just under 2 octaves on a piano to represent one octave, the leftover keys I'm sure could be put to work on quartertones or something.



What I did was I assembled the C minor scale using regular, widely accepted ratios for all the harmonies. Then what I did was I figured out what the actual frequency of C was, and I applied the ratios to that freqency, which told me what exact freqency all the harmonies were.



Then - to make perfectly in tune chords, I treated the root note of each chord as 1:1 and based all the other harmonies off that. In some cases a chord's notes needed zero adjustment (I don't have to use my ears, because I figured out all the exact freqeuncies, so it either matchs or it doesn't), but in some cases one, or two, notes would have to have a different frequency... now, what that new note's relationship back to C is - I have no clue. But, by figuring out all the notes for every single chord I planned on using based on only the notes in the chord, I think I might have created something usable. Some extra notes were just completely outside the scale, I think I added a B to use as a 5th. F had four completely different versions (some notes had 2, or 3 or just 1), and I came up with a way to name them:

F = that note in perfect relationship back to C (or as a root in it's own chord)

F5 = F used as a fifth in a chord

F3 = F used as a third

F2 (or maybe it was 4... can't remember) = obviously, used as a fourth or a ninth/second



I'll have to dig out my papers, it has all the exact frequencies for each note, and all the ratios are really simple and obvious, they just relate to the root of each chord rather than the root of the scale. When I post my note chart this may be a little clearer if it isn't now.
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Re: Music + Maths + physics

Post by Schu »

The more extreme-seeming ratios are actually not that extreme when you take into account a bit of theory - a lot of them are "artificial leading notes" like 225/128 - B is 15/8, and a#, as a leading note to b, is 15:16 of that, so 225/128. These kind of uses of harmony crop up all the time in classical music.

One of my pet ideas is to engineer a piano that, in a way, can be "stopped" like a fretless string instrument. My idea is that you have a a few templates so that each chord you could need is covered by at least one of the templates. Switching between these templates could be done by some special pedal.

These templates would stop the string at a certain point determined by the template, and when you wanted to change keys (maybe also done by some kind of pedal), the template(s) would shift over to the spot required, tuning in the new key.

This way, modulations would be done in equal temperament, but everything else could be done by just intonation.

Well, in a minor scale, f used as a 5th should be either 27:20 or 4:3 depending on how you do it, used as a regular note in relationship to C it should be 4:3, used as a 3rd it should be still 4:3 (or in some nasty progressions, possibly 27:20) and as a 9th/2nd it should be 27:20. Used as a 7th, it should be 21:16. I'd certainly be interested in seeing your maths, because what I've just written is obviously based on my experience, which is mainly with classical and/or modal music, so your calculations might be an interesting window into another implementation of harmony :D
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Re: Music + Maths + physics

Post by A Thing of Eternity »

Schu wrote:The more extreme-seeming ratios are actually not that extreme when you take into account a bit of theory - a lot of them are "artificial leading notes" like 225/128 - B is 15/8, and a#, as a leading note to b, is 15:16 of that, so 225/128. These kind of uses of harmony crop up all the time in classical music.

One of my pet ideas is to engineer a piano that, in a way, can be "stopped" like a fretless string instrument. My idea is that you have a a few templates so that each chord you could need is covered by at least one of the templates. Switching between these templates could be done by some special pedal.

These templates would stop the string at a certain point determined by the template, and when you wanted to change keys (maybe also done by some kind of pedal), the template(s) would shift over to the spot required, tuning in the new key.

This way, modulations would be done in equal temperament, but everything else could be done by just intonation.

Well, in a minor scale, f used as a 5th should be either 27:20 or 4:3 depending on how you do it, used as a regular note in relationship to C it should be 4:3, used as a 3rd it should be still 4:3 (or in some nasty progressions, possibly 27:20) and as a 9th/2nd it should be 27:20. Used as a 7th, it should be 21:16. I'd certainly be interested in seeing your maths, because what I've just written is obviously based on my experience, which is mainly with classical and/or modal music, so your calculations might be an interesting window into another implementation of harmony :D
I'll find it and type it up for you. Some of the harmonies I quoted are sure to have been BS, because like I said, I have a hard time remembering things like what chord in the key of C minor would have an F as a fifth in it. Hopefully everything will make more sense when I post the numbers!

I like your ideas by the way, and I've read of similar attempts with electronic keyboards that are intelligent, and swap between exact values for notes depending on what chord/scale it feels you playing in.
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Re: Music + Maths + physics

Post by Schu »

Yeah, I've read about those kinds of keyboards too, in fact I've also seen one in action. But since I have an acoustic instrument fetish, I'd want mine to be acoustic :P
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Re: Music + Maths + physics

Post by A Thing of Eternity »

Baraka Bryan wrote:
Schu wrote:Yeah, I've read about those kinds of keyboards too, in fact I've also seen one in action. But since I have an acoustic instrument fetish, I'd want mine to be acoustic :P
considering the tension levels of piano strings though, I can't see a 'stopping' system being possible. a digital system is probably the only way it could happen without re-tuning each time :P
Not if the stop came from both sides and pinched the string at a given point. Even a single stop on the top or bottom should do the trick if enough pressure is applied (it's position would have to be altered to take into account the stretching of the string).

This would be a pain in the ass to design and build, but anyone who's ever looked at the mechanics of a single piano hammer/key can testify that it's already the most complicated pain in the ass thing in the world!

I love accoustic instruments too (or electrified acoustics, such as good old electric guitars), I'm working on a design for something like a piano/harp, but instead of hammers hitting the strings a magnetic feild causes the strings to resonate. Even the prototype is going to be expensive, but it should sound pretty cool.

Have you ever checked out the Bazantar? http://www.youtube.com/watch?v=crSi9IxPfYA
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Re: Music + Maths + physics

Post by Schu »

Whoooa, that thing has a metric fuckton of sympathetic strings! And yeah, you're exactly right about the piano, it would be a pinching stop, not a violin-style stop. Your piano/harp sounds awesome, very sci fi :D
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Re: Music + Maths + physics

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(I was looking into that Chapman Stick thingie yesterday after watching the baliset clip on YouTube. Kewl. But REALLY expensive. :shock: )
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Re: Music + Maths + physics

Post by A Thing of Eternity »

I contacted the inventor about purchasing one of the modules so that I could mount it onto an upright bass, but he never got back to me. I did purchase his album off iTunes (couldn't find the actual CD for sale online), and it's really good, but a lot of it is just odd westernized sitar playing (only every second song or so is Bazantar).

Schu, speaking of classical (kinda) and J.I. - have you heard this fellow? Michael Harrison: http://www.youtube.com/watch?v=2ieHZ5qmJZI

He has a website that explains some of the bizzare tuning he uses, I love his work.

I'm still working on typing up my math so you can see what my system is, but it will be a bit hard to read, because only some (the root scale itself) of the deviations from 12TET are in cents, the extra notes are all just in hz.
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Re: Music + Maths + physics

Post by Schu »

I'm sure I'll be able to decypher it :) and yeah, I've heard some of harrison's stuff! My favourite JI composer is Terry Riley though.
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Re: Music + Maths + physics

Post by A Thing of Eternity »

Schu wrote:I'm sure I'll be able to decypher it :) and yeah, I've heard some of harrison's stuff! My favourite JI composer is Terry Riley though.
What style/instruments does Riley do? Living in vancouver with all the Indians is what got me interested in JI, their music theory is so much simpler and more complex than ours that I just couldn't stay away.

Also, do mind if I ask how old you are and what your musical background is? I'm always curious about people on the net that I can't see.
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Re: Music + Maths + physics

Post by Freakzilla »

http://www.pbs.org/wgbh/nova/musicminds/" onclick="window.open(this.href);return false;

I saw this guy on The Daily Show last night, looks like an interesting Nova.
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Re: Music + Maths + physics

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Most I've heard of Riley is chamber music (string quartets, piano work (ET and JI). He's worked a lot with the Kronos Quartet, who are good with non-traditional music.

Nice like, FreakZ :D
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Re: Music + Maths + physics

Post by A Thing of Eternity »

Schu wrote:Most I've heard of Riley is chamber music (string quartets, piano work (ET and JI). He's worked a lot with the Kronos Quartet, who are good with non-traditional music.

Nice like, FreakZ :D
Cool, I'll look those up.

I should have a full explanation of what I did to get my note system typed up this weekend, I would definitely apreciate any input you have, as I am planning on buying a tune-able keyboard soon. You clearly have vastly more experience both with JI in theory and in practice, whereas I'm just a heavy metal guitarist (who only learned the names of all the notes after 7 years of playing so that I could become a teacher. I'd just never needed them before that) who has developed a taste for exotic intervals.

I bought a fretless bass around Xmas and it has really helped to show me what Just intervals sound like, and it helps a lot visually too, for example I can actually see that I'm playing a major third a fair bit flat to be Just - but because I've so limited experience with JI I have problems with the harmonies that are naturally more dissonant, such as the minor third - I know it should be just sharp of 12TET, but it always beats no matter where I place it. I think what I might need to do is skip the piano lessons I was planning to take, and take cello lessons instead so that I have someone right in front of me helping teach me how the intervals should sound.
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Re: Music + Maths + physics

Post by A Thing of Eternity »

I FINALLY managed to get my intonation math into a format I can post. I hope this makes sense, I didn't proof read it!

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Re: Music + Maths + physics

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Pythagoras would praise you. :D

(I might when/if I understand it. ;) )
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Re: Music + Maths + physics

Post by A Thing of Eternity »

SandChigger wrote:Pythagoras would praise you. :D

(I might when/if I understand it. ;) )
Nah, he'd say that his system using a circle of Just fifth intervals is better than my system...

... of course his system fucks you up pretty good when you come to the last note and find it slightly higher in pitch than the octave of your root... that part didn't work out so well!

Both systems are Just intonation, he just bases each note off of a "circle" of 5th intevals, and I base each note off of the root note of the scale only, and then add a pile of secondary notes that are based not on the root of the scale, but the root of each chord.
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Re: Music + Maths + physics

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:shock:

"I sense the mirth is strong in this one." :lol:
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Re: Music + Maths + physics

Post by A Thing of Eternity »

:D Once you learn the basics of JI it's actually fairly simple to work with... on paper anyways.

When I actually pick up my fretless bass the only two intervals I'm able to improve on over our current note system (12 tone equal temperament, AKA 12TET) is a 5th (tiny improvement, 12TET has 5ths almost in tune, and of course, if I change the 5ths I change the 4ths, same thing) and major 3rds (gigantic improvement as 12TET has that one totally out of tune). I cannot for the life of me get minor thirds any better (worse in my hands actually), same with the other intervals that come up often. I can see on paper what I should do to the notes, but making it happen isn't working out so well. :P

Or am I just rambling gibberish? I don't know that much music theory, so I have a hard time communicating it sometimes!
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Re: Music + Maths + physics

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Well ... I don't know any, so let's hope Schubert pops back in again soon. :)

(Haven't checked, is he back playing over on Dumb Novels? :roll: Gotta get him banned from there, too, huh Freak? :P )
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Re: Music + Maths + physics

Post by Schu »

I still post here, only sporadically. And I always lurk a little. And still not banned from DN :P

First thing about this is that 6/5 from C is Eb, not D#, 6/5 is a minor 3rd, not an augmented 2nd. Same goes for Bb, not A# and Ab, not G#. Other than that it all makes sense, and is a pretty good way of doing this kinda thing. Interesting that all your chords were minor+2 chords. I guess one thing I can point out that's a little implausible is that you don't normally get any harmony with 7th harmonics unless you use, say, dominant 7th chords or diminished chords etc, so having an entire chord based on 7/4 out of the blue doesn't quite make sense, and doesn't fit well with anything else. More plausible, if you're working with Bb/A#, for it to be 9/5 or possibly 16/9.

Other than that, it looks pretty good :)
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Re: Music + Maths + physics

Post by SandChigger »

schubert rulz! :P

(No idea what either of you is on about except in a very general way, but it's impressive anyway. :) )
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Re: Music + Maths + physics

Post by A Thing of Eternity »

Schu wrote:I still post here, only sporadically. And I always lurk a little. And still not banned from DN :P

First thing about this is that 6/5 from C is Eb, not D#, 6/5 is a minor 3rd, not an augmented 2nd. Same goes for Bb, not A# and Ab, not G#. Other than that it all makes sense, and is a pretty good way of doing this kinda thing. Interesting that all your chords were minor+2 chords. I guess one thing I can point out that's a little implausible is that you don't normally get any harmony with 7th harmonics unless you use, say, dominant 7th chords or diminished chords etc, so having an entire chord based on 7/4 out of the blue doesn't quite make sense, and doesn't fit well with anything else. More plausible, if you're working with Bb/A#, for it to be 9/5 or possibly 16/9.

Other than that, it looks pretty good :)
Thanks for the feedback, I'll have to digest what you said because it doesn't come to me quickly (understanding the theory, I have to sit and write it out or think very slowly).

I do a LOT of +2 chords, I think of it more in terms of stacked 5ths., but I do occasionally use major +2 as well, not just minor.

Which chord did I base on 7/4 that doesn't make sense? I thought I just went up and down the scale using all the usual major and minor chords and just added the 2s for good measure? I've written pieces that use all these chords, but not necessarily with the 2s (those aren't meant to be in every chord, just optional if you occasionally want them).
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