Music Theory

[Borodog]

Note: I am not a music theorist nor even a particularly good musician (I play drums). Hopefully this post, which will consist mainly of my musings about scales, will incite someone who knows what the hell they're talking to step up and comment, because I think it's a fascinating subject.

A few years back I finally broke down and bought a guitar, as it was always something I'd wanted to play. I noodled around with my sister's guitar when I was a kid, and learned to pick out a few riffs, chords, etc., but didn't stick with it. So I finally bought a guitar and started taking some lessons, learning tablature, etc. I immediately became fascinated with the physics of the thing and figured out how the 12 note scale works.

A tone that is an octave above another tone that is the same note has exactly twice the frequency of the first. I am not sure why the ear hears these as the "same" note, though. It does not seem obvious to me why this should be true.

In any event, an octave, the frequency range from say f to 2f in western music is broken up into 12 intervals. These tones are not evenly spaced in frequency, but rather (at least on the guitar and other fretted instruments) each is a fixed fraction higher in frequency than the previous note; specifically, the twelfth root of 2, or slightly less than 6%.

It occured to me that (obviously) there was no particular reason to have 12 tones instead of 8, 10, 13 or any other number. This immediately made me wonder why we have twelve. Much hunch was that there have been (and almost certainly are) many differetn scales that use a different number of intervals, and that the 12 tone scale defeated them in the market of musical competition, much like gold defeated many other contenders to become the world's money by the 19th century. This occured for a simple reason, that being that gold makes the best possible money (at least commodity money; if you want to argue how great fiat currency is, take it to politics) of all the contenders (it is in limited supply, the supply does not change by much over short periods of time, it is maleable, easily divisible and recombinable, when you divide and combine it it is still gold, it is virtually indestructable, visually distinctive and impossible to counterfeit, and has a high value to weight ratio. What would be the properties of the 12 tone scale that would make it defeat all comers?

My instinct is that it must somehow come down to the fact that twelve has the largest number of integer divisors:
# # of int. divisors
6 4
7 2
8 4
9 3
10 4
11 2
12 6
13 2
14 4
15 4
16 5
17 2
18 6

But I have no idea why this would actually make the 12 tone scale sound "better" than the other options.

My strong hunch is that 24 tone music (8 integer divisors) would sound extremely beautiful. My guess why we don't actually use 24 tone music (or 18 tone, which according to my theory would work as well as 12 tone) is that the improvement in the quality of music is not warranted by the added complexity, or possibly that the trade-off in limited range (try to imagine a piano with twice as many keys, or a guitar with twice as many frets) is not worth it. I have since found out that 24 tone music is very popular in muslim music, and I would very much like to hear some of it.

So what prompted this thread is that the other day I was thinking about these things and decided to read up on music theory and the mathematics of music on wikipedia, and came across what I have determined to be the most incomprehensible wiki article I've ever read:

http://en.wikipedia.org/wiki/Mathema...musical_scales

It turns out that the 12 tone scale that I had figured out (with frequencies increasing by a factor of the twelfth root of 2 from note to note) is not the only 12 tone scale. It's what is called an "equal temperment" scale. There are many, many, many different 12 tone scales. There's "Pythagorean tuning", "just intonation", there are "well tempered" scales, "regular temperment" scales, the aforementioned "equal temperment" scales, "meantone temperment" scales, and God knows what else.

There's no real moral to this story, other than that I find the subject fascinating, and the wiki articles have not really helped much. I'm hoping that someone will come along and drop some knowledge, and that others will find it interesting as well.

[turnipmonster]

if you are interested in this stuff, I highly recommend "The Harmonic Experience" by W.A. Mathieu, in which he attempts to explain a lot of fundamental issues behind music as well as tuning (just intonation vs. equal temperament).

I really cannot recommend this book highly enough. I have been a working musician for most of my life and this book changed the way I listen to, and play, music.

[turnipmonster]

couple thoughts and some brief explanations I hope will help.

as you mention, most western music uses a 12 tone system with seven note scales. there are certainly exceptions to this, especially in jazz and blues. there are also lots of microtonalists (musicians who play and compose using more than 12 tones) in western music, they play special instruments and have festivals and concerts and internet forums etc.

there is lots of music in other parts of the world that uses other pitch collections and/or scales, indian classical music probably being the most well known example. indian classical singing is extremely nuanced and pitch sensitive.

in western music, the pitch collection we use is derived from higher partials in the harmonic series. in the book I recommended, there are many exercises for learning to hear and sing these for yourself. if you tuned a piano based on the higher partials of the harmonic series, you would have yourself piano tuned in what we call "just intonation".

your music would sound great, as long as you stayed in the same key you tuned the piano in. if you tried to modulate, say up a major 3rd, your piano wouldn't sound so good anymore.

as western music developed and got more complex, composers wanted to write elaborate modulations. as we found out earlier, on an instrument like the piano these modulations won't sound so good. therefore, musicians and piano tuners got together and figured out a compromise, namely a tuning system where none of the notes are exactly in tune with one another, but they are all damn close. the advantage to this system (called "equal temperament") is the notes' tuning is perfectly symmetrical around the octave, and that means we can modulate from key to key and our piano will sound great (well, slightly less than great, but our ear will accept the comparatively smaller degree of out-of-tuneness).

hope this helps

[Borodog]

It does. Please feel free to elaborate more.

[NickMPK]

There are sort of two different issues involved in why we have a 12-tone scale. One involves the natural way that harmonics are generates, and the other involves man's imperfect attempts to harness these natural harmonics.

The different piano tunings you were reading about are basically an evolving compromise between getting perfect natural harmonics and building an instrument that is practical to play. As you will learn if you study natural harmonics, the note we call "G" should not exactly be the same frequency when you are playing the key of "C" versus when you are playing in the key of "D" or "G". Most early instruments were designed to play in a single key, and thus didn't have tuning controversies. But pianos were built to accomodate a wider range of repertoire, including songs that modulated keys within the song, so they had to be able to play the same note in many different keys without retuning all the time. The different tuning systems are basically debates about whether "G" should be closer to the "G" in the key of "C", or close to "G" in the key of "D", etc. Some tuning systems sound equally dissonant no matter what you are playing, while others sound less dissonant in some keys than others.

I think the first issue, natural harmonics, is one of great mysteries of human life. I haven't heard any convincing explanation for why we derive pleasure from experiencing the geometric division of sound waves. Unlike other things that humans enjoy (food, sex, sleep, love, security, competition, etc.), I don't see any evolutionary advantage to our enjoyment of music.

In any case, as you mentioned, an octave is the division of a sound wave in half, and for some reason, it sounds "the same" in some sense as the original sound wave. This "sameness" of the octave appears in the music of all cultures, AFAIK.

Other harmonies are derived from other geometric divisions. For the most part, the smaller the division, the more "dissonant" the note sounds. If you divide the note in thirds, you get an "open fifth", basically G in the key of C. This is the most fundamental harmony, and was more or less the only harmony used in medieval music.

Divide a note in fifths, and you get a "major third" (E in the key of C), which is the other note in a "major chord", which for some reason we identify as sounding "happy". The use of thirds is basically the extent of harmony you get in Renaissance music.

Classical era music moved beyond this into more dissonant sounding harmonies, such as the seventh, the ninth, the eleventh, etc. (only odd numbers generate new harmonies, even ones just generate existing harmonies at a higher octave). They use all the notes in "scale" of a given key, but not all the notes available on piano. For example, you will rarely hear C# played during a song in the key of C, unless the song has modulated into a key in which C# is part of the scale.

It took until the 20th century for composers like Schoenberg to start fully using a 12-tone scale, and this music still sounds very dissonant to people who are not heavily acclimated to it. (Just as classical music would probably have sounded dissonant to medieval people.)

As the divisions get smaller, the notes sound more dissonant because they come from overtones (sounds waves which are naturally being divided when the base note is played) which are harder to hear in nature. Later 20th century composers like Harvey Parch have used 13-tone scales, for which they had use computers or synthsizers, or manufacture their own instruments.

In response your 24-tone scale idea, the music would sound very dissonant because it is such a radical departure from what we are used to. Most people can't hear the consonance in 12-tone music. Anything more than incremental additions to our harmonic vocabulary would probably sound so alien as to be indecipharable.

[burningyen]

[ QUOTE ]
Harry Partch

[/ QUOTE ]

FYP. I always found his stuff really interesting, but never quite worked up the motivation to hunt down recordings. Looks like it'll be my next YouTube hunt.

As some of my guitarist acquaintances say, intonation is for pussies.

[NickMPK]

[ QUOTE ]
[ QUOTE ]
Harry Partch

[/ QUOTE ]

FYP. I always found his stuff really interesting, but never quite worked up the motivation to hunt down recordings. Looks like it'll be my next YouTube hunt.

As some of my guitarist acquaintances say, intonation is for pussies.

[/ QUOTE ]

Good catch...I've actually never listed to him, just read about him.

[burningyen]

In my college electronic music class I saw some old film clips of some of his work being performed. It's a bizarre mix of primitive yet alien sounds.

[secret.asian.man]

FWIW i was a serious classical musician for many years, and I find very little appeal in 12 tone music. I have never ever ever met a non musician who heard 12 tone music and said "hey that sounds good!"

(btw I didn't read your post thoroughly so i apologize if I am completely missing your point, just thought I'd chime in )

[Redd]

[ QUOTE ]
I think the first issue, natural harmonics, is one of great mysteries of human life.

[/ QUOTE ]

I hope this isn't derailing the thread, but the reason that 'harmonics' occur is actually more a function of math than biology. The frequency domain is a pretty abstract concept that takes a while to wrap your head around; I'll do my best to summarize why harmonics sound consonant.

If we were to generate a single, time-invariant 'note' (ie, a truly periodic, nondecaying sinusoid that had no beginning or end), we would create only a single piece of frequency content on a graph (aka our 'fundamental frequency'). This is the example I found on a website for a 200Hz sine wave 'note':


The mathematical rule is that an infinitely long signal in time creates a finite frequency spectrum, and a finite signal in time creates an infinitely long frequency spectrum.

Since our music would be pretty terrible if it was a single note lasting constantly from the start until the end of time (and ofc physically impossible), all of our frequency content will extend from the 'fundamental' frequency for an infinite length of time at a regular interval. There's a long, calculusy proof to prove this mathematically. The end result is that any finite signal we generate will repeat itself like this:


In sound waves, each of those peaks would correspond to the same note different octaves. Increasing octaves, in fact. And each of those peaks is generated when you play a single note. For instance, playing a low E also creates a high E, and a high high E, on for infinity diminishing octaves of E. So when you play a high E on top of the low E, the high E is already present in the sound. Our ears are better at math than we are, because they can effortlessly realize that the two identical notes are present, and this is heard as harmony.