Universal Property of Music Discovered

I'm kind of a peeping tom on the discussions here; some I find interesting and I keep coming back. I'll start my question with the article I found:

http://www.sciencedaily.com/releases/2011/03/110325102008.htm

and the beginning of the text:

"Researchers at the Institute for Logic, Language and Computation (ILLC) of the University of Amsterdam have discovered a universal property of musical scales. Until now it was assumed that the only thing scales throughout the world have in common is the octave.

The many hundreds of scales, however, seem to possess a deeper commonality: if their tones are compared in a two- or three-dimensional way by means of a coordinate system, they form convex or star-convex structures. Convex structures are patterns without indentations or holes, such as a circle, square or oval."

It goes on a bit, although it is pretty short and doesn't really describe how they mathematically describe this universal property except to say that they used an Euler Lattice, and that through that they found that most scales used in music the world over form star-convex structures.

I am decent with basic math, but this stuff is a bit outside my experience. Do any of you know what the researchers referenced in this article are talking about? And can you explain it?

I am curious about this, because I consider myself a wanna-be composer, and if I can find out what this concept actually implies musically, I may find it useful.

--- In tuning@yahoogroups.com, "bigAndrewM" <bigandrewm@...> wrote:

> I am decent with basic math, but this stuff is a bit outside my experience. Do any of you know what the researchers referenced in this article are talking about? And can you explain it?

They were talking about something that was pretty well known around these parts, actually. You can explore it yourself using Scala if the scale is 5-limit. In Scala, load a 5-limit scale from the Scala scale directory--for example, ptolemy.scl. Now go to the pull-down menu under Analyze and go to "Lattice and player" on the menu, and show the lattice. Note that you can draw a polygon around the lattice--this is called the convex hull--and that there are no new lattice points inside this polygon. This means the scale is convex.

Note also that it doesn't matter whether you choose the square or triangular arrangement of dots--convexity is independent of that sort of thing--it is an "affine" property.

> I am curious about this, because I consider myself a wanna-be composer, and if I can find out what this concept actually implies musically, I may find it useful.

It basically implies, musically, that the notes of the scale are linked together in a way which is efficient in terms of allowing for dyads and triads.

Hi :) Do you understand tonal lattices?

They took 998 Just Intonation scales from the Scala archive that are formed by combinations of the intervals 3/2, 5/4 and 7/4, and analysed the shapes of the scales as visualised by a tonal lattice. They discovered most were convex (blob shaped) and almost all were star convex (spiky like a star but not too spiky) with the tonic often being at the centre of the star. Then they show that these structures indicate consonance with the tonic, this being a 'universal property of music' :O like wow ... It's sort-of-kind-of obvious that JI scales are this shape since they are usually built outwards from the tonic to a particular harmonic distance.

MatC

I haven't studied Euler lattices. I have studied some just intonation and for me, simply describing the notes in the scale is sufficient for me to write music.

If this study is based on just intonation-based scales, and if this star-convex shape implies that all notes relate to the tonic, this is definitely no surprise and I don't see how the title "universal property of music discovered" can be legitimate, because this stuff is already known. I also can't read Dutch, so I can't really follow the website that is in the references. It may be the whoever wrote that little article exaggerated the importance of the Dutch study.

But I would also be highly surprised if all of the scales that they analyzed were in just intonation. It's true that many scales of world music can be analyzed that way, but not all. There's the obvious tempered scales of western music, there's gamelan music, and there's music based on purely melodic and not harmonic constructs like some Native American music which (while perhaps conforming to some general shape of pentatonic scales) are simply irrational and not meant to be analyzed in terms of harmonic implication.

Andrew

--- In tuning@yahoogroups.com, "ixlramp" <ixlramp@...> wrote:
>
> Hi :) Do you understand tonal lattices?
>
> They took 998 Just Intonation scales from the Scala archive that are formed by combinations of the intervals 3/2, 5/4 and 7/4, and analysed the shapes of the scales as visualised by a tonal lattice. They discovered most were convex (blob shaped) and almost all were star convex (spiky like a star but not too spiky) with the tonic often being at the centre of the star. Then they show that these structures indicate consonance with the tonic, this being a 'universal property of music' :O like wow ... It's sort-of-kind-of obvious that JI scales are this shape since they are usually built outwards from the tonic to a particular harmonic distance.
>
> MatC
>

Only if the scale is 5-limit? Odd. (pun intended)

I haven't really explored that functionality of Scala. Perhaps I should tinker with it.

Andrew

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
> They were talking about something that was pretty well known around these parts, actually. You can explore it yourself using Scala if the scale is 5-limit. In Scala, load a 5-limit scale from the Scala scale directory--for example, ptolemy.scl. Now go to the pull-down menu under Analyze and go to "Lattice and player" on the menu, and show the lattice. Note that you can draw a polygon around the lattice--this is called the convex hull--and that there are no new lattice points inside this polygon. This means the scale is convex.
>
> Note also that it doesn't matter whether you choose the square or triangular arrangement of dots--convexity is independent of that sort of thing--it is an "affine" property.
>
> > I am curious about this, because I consider myself a wanna-be composer, and if I can find out what this concept actually implies musically, I may find it useful.
>
> It basically implies, musically, that the notes of the scale are linked together in a way which is efficient in terms of allowing for dyads and triads.
>

--- In tuning@yahoogroups.com, "bigAndrewM" <bigandrewm@...> wrote:
>
> Only if the scale is 5-limit? Odd. (pun intended)

No, 2 with any two odd primes will work the same.

On 30 March 2011 11:33, bigAndrewM <bigandrewm@...> wrote:

> If this study is based on just intonation-based scales, and if this star-convex shape implies that all notes relate to the tonic, this is definitely no surprise and I don't see how the title "universal property of music discovered" can be legitimate, because this stuff is already known. I also can't read Dutch, so I can't really follow the website that is in the references. It may be the whoever wrote that little article exaggerated the importance of the Dutch study.

The Science Daily article isn't very good, and certainly exaggerates
the importance. The original paper, in English, is here:

http://staff.science.uva.nl/~rens/convex_scales.pdf

> But I would also be highly surprised if all of the scales that they analyzed were in just intonation. It's true that many scales of world music can be analyzed that way, but not all. There's the obvious tempered scales of western music, there's gamelan music, and there's music based on purely melodic and not harmonic constructs like some Native American music which (while perhaps conforming to some general shape of pentatonic scales) are simply irrational and not meant to be analyzed in terms of harmonic implication.

They only analyzed just intonation scales because they only defined
the property for just intonation.

Graham

Aha!

Thanks for the reference.

And thanks for the input, everyone!

Andrew