Someone help my poor brain!

Paul,
I'm not quite sure what diagrams you mean. Can you post an image or a link?
Tim

Better buckle up, Paul... it's a bumpy ride into this territory.

A guy named Leonhard Euler, a mathematician, decided to look into the mathematical relationships between tones. He figured that since tones are communicated through vibrations, you could figure out the mathematical relationship between two tones as a ratio... for example, 2:3 (where the first pitch vibrates twice for every time the second pitch vibrates 3 times). So far so good, and not so very different from Pythagoras.

Then he set about to order sets of tones by level of complexity:

1st level = 1:1 ratio, or unison tones
2nd level = 1:2, or octaves
3rd level = 1:3... octave & a fifth
4th level = 1:2^x (raising a power of 2), additional octaves
5th level = 1:5, reasoning that this must be more complex than 1:8, which is 1:2^3

From the 5th level, he noted that any odd number relationship to the original tone will result in a new sound... 1:3 is not the same as 1:2; 1:4 sort of IS the same sound as 1:2; 1:5 gives us a new sound; 1:6 is an octave of 1:3, 1:7 is the next new sound, etc.

So far so good. Now we get into more elaborate sets of sound, where the second frequency is not a pure multiple of the first... call the first x, and the second y. If both x and y are prime numbers, you get a sound distinct from any other set of x/y numbers.

How to logically relate those numbers? Enter the lattice... we can create a vector based on the interaction of two tones, with the vector representing the view you're taking in finding unique tone combinations (either odd or prime). Lattices are drawn using vector math - I won't go into that here, but essentially if you go in one direction you'll find an interval like 3:2 (a perfect fifth), and if you go in the other direction along the same line you'll get the inversion of that interval - 4:3, or a perfect fourth.

Using these vectors, we can map out the possible combinations of tones within a set of limits. Some decent illustrations (fairly well annotated, although it's not always easy reading) can be found here:

http://sonic-arts.org/dict/lattice.htm

Using the simplest diagram, the 5-limit set, you can see that from the origin you move in any direction to find x:y, and in the opposite direction you'll move to x:2y (or 2x:y). 3:2 in one direction becomes 4:3 in the other... 5:4 becomes 8:5.

Intellectually, it's a great game, and it can be carried out to extremes. After looking at the diagrams, I'm sure you can envision those based on the combinations of sounds possible with seventh chords!

I've never tried to actually apply it to composition or improvisation, though... simply knowing the interval inversions seems to provide enough sonic material for me :)

Tom

*tries to stop his head spinning*

Better buckle up, Paul... it's a bumpy ride into this territory.

A guy named Leonhard Euler, a mathematician, decided to look into the mathematical relationships between tones. He figured that since tones are communicated through vibrations, you could figure out the mathematical relationship between two tones as a ratio... for example, 2:3 (where the first pitch vibrates twice for every time the second pitch vibrates 3 times). So far so good, and not so very different from Pythagoras.

Then he set about to order sets of tones by level of complexity:

1st level = 1:1 ratio, or unison tones
2nd level = 1:2, or octaves
3rd level = 1:3... octave & a fifth
4th level = 1:2^x (raising a power of 2), additional octaves
5th level = 1:5, reasoning that this must be more complex than 1:8, which is 1:2^3

From the 5th level, he noted that any odd number relationship to the original tone will result in a new sound... 1:3 is not the same as 1:2; 1:4 sort of IS the same sound as 1:2; 1:5 gives us a new sound; 1:6 is an octave of 1:3, 1:7 is the next new sound, etc.

So far so good. Now we get into more elaborate sets of sound, where the second frequency is not a pure multiple of the first... call the first x, and the second y. If both x and y are prime numbers, you get a sound distinct from any other set of x/y numbers.

How to logically relate those numbers? Enter the lattice... we can create a vector based on the interaction of two tones, with the vector representing the view you're taking in finding unique tone combinations (either odd or prime). Lattices are drawn using vector math - I won't go into that here, but essentially if you go in one direction you'll find an interval like 3:2 (a perfect fifth), and if you go in the other direction along the same line you'll get the inversion of that interval - 4:3, or a perfect fourth.

Using these vectors, we can map out the possible combinations of tones within a set of limits. Some decent illustrations (fairly well annotated, although it's not always easy reading) can be found here:

http://sonic-arts.org/dict/lattice.htm

Using the simplest diagram, the 5-limit set, you can see that from the origin you move in any direction to find x:y, and in the opposite direction you'll move to x:2y (or 2x:y). 3:2 in one direction becomes 4:3 in the other... 5:4 becomes 8:5.

Intellectually, it's a great game, and it can be carried out to extremes. After looking at the diagrams, I'm sure you can envision those based on the combinations of sounds possible with seventh chords!

I've never tried to actually apply it to composition or improvisation, though... simply knowing the interval inversions seems to provide enough sonic material for me :)

Tom
ARRRRGH! OMG! I read that and the one poor brain cell I had left imploded. MEDIC!!

Tom, I hereby promote you to the rank of moderator. I just looked at a few or your nice, detailed post and can't help but smile. Talk about deep on theory! I thought I was gettin' making a chart with each note's frequency across the fretboard...

Actually, I figured it out myself near the end of yesterday, but thanks anyway. I hadn't yet realized that going the opposite direction from the origin gave you an inversion of the interval, so you enlightened me a bit more.

someone help MY poor brain!!! ow , headache. :x

Noteboat, they cant make you moderator, you're overqualified.

Noteboat, they cant make you moderator, you're overqualified.
Yeah - that's what we said (j/k)

Mind-blowing stuff. Doesn't the 3:2 trace exactly back to Pythagoras in that reducing the length of your string by that ratio eventually brings you back to the same note seven octaves higher? And - is that why piano's have seven octaves on the keyboard or is it just coincidence?

A :-)

Yeah, it does trace back to Pythagoras... except for the comma (which I think we went into on a different thread).

The size of the piano keyboard is strictly coincidence, in relation to Pythagoras though... keyboards used to have fewer keys. I think it's got more to do with the fact that the entire range of a modern symphony orchestra can be replicated on a piano keyboard.

Plus it's about the largest size the average person could handle.

There are some bigger ones, though... Boesendorfer makes one with 97 keys. Problem is, nobody writes music that uses those extra notes!

That's less than an octave more than usual. I wonder how useful that could be.

I read somewhere that the extra notes in the lower register on the 97 key piano are there mainly because they give the piano more resonance and not so much for playing.

They might do that - depends on the dampers (how it's built, and how it's pedaled). It turns out there must be some music for it, though... from the Bosendorfer website, model 290:

It is the only concert grand in the world to have nine sub-bass notes, down to bottom C, giving it a full eight octave compass keyboard. These extra notes enable some compositions to be accurately performed, which were originally scored with lower notes, by composers such as Bartók, Debussy, Ravel and Busoni.

I'll be curious to find some of those pieces. I'll bet they use the lower note as the root of a left hand octave, and the upper note is used alone on most pianos.