Klarinet Archive - Posting 000241.txt from 2004/07
From: Umar Goldeli <umar@-----.com>
Subj: Re: [kl] Re: Music Theory
Date: Tue, 6 Jul 2004 19:18:02 -0400
Thank you all for your input - greatly appreciated - after reading your
responses and a little bit more reading - it "clicked" (I think!)
While I know this is all not going to change and I should just get used to
it - it makes me feel a lot better knowing that there's a reason behind it
all.
Summary:
* The human ear does not care about absolute frequencies - it cares about
relative frequencies and their relationships to each other. As such -
there are a great number of different tuning methodologies and scales
including some weird ones with 43 divisions etc... you name it.
* The question of why and what types of frequency relationships appeal to
the human ear is a HUGE topic (psychoacoustics) - but there are a great
many theories including the incompatibility of certain waves/frequencies
travelling together down certain nerve paths - dissonance etc. I'll take
this one as a given - but it looks like a very interesting field of study.
;)
* Western music is based on the 12 divisions. Historically "found" by ear
and due to traditions etc etc.
To bring it into focus a little better and derive it:
* Define steps between the 12 divisions as semitones, and a series of 2,
2, 1, 2, 2, 2, 1 will give you a major scale. Why? Psychoacoustics again -
the human ear likes it that way.
* Beginning with this idea/goal, to mathematically derive the major scale,
define f as the root frequency, and 2f as the same note in the next
register/octave - etc etc - (full derivation at
http://home.austin.rr.com/jmjensen/musicTheory.html) - eventually to
arrive at the Just scale ratios:
(C-D-E-F-G-A-B) = [ 1 9/8 5/4 4/3 3/2 5/3 15/8 ]
However, this doesn't lend itself to changing keys etc - so we go for 12
Tone Equal Temperment and approximate to the exacts with a geometric
progression (which works because the human ear generally can't perceive
pitch differences less than 5 cents).
(Amusing to note that the largest errors are at E and A)
* Now here's the good bit about why A-G etc.. To derive the remainder of
the major scales, start with C major, and begin each subsequent scale on
the "5th" of each - we get:
C maj - 0 sharps
G maj - 1 sharp - F# (meaning F-sharp)
D maj - 2 sharps - F#, C#
A maj - 3 sharps - F#, C#, G#
E maj - 4 sharps - F#, C#, G#, D#
B maj - 5 sharps - F#, C#, G#, D#, A#
F# maj - 6 sharps - F#, C#, G#, D#, A#, E#
C# maj - 7 sharps - F#, C#, G#, D#, A#, E#, B#
(Similarly with the flats - but going to a 5th *below*)
Now if we had the "Umar Notation" from A to F with all sharps in between -
this would make it very painful and we wouldn't get pretty patterns and it
just wouldn't work.
So from: {A,A#,B,B#,C,C#,D,D#,E,E#,F,F#}, using the
tone,tone,semitone,tone,tone,tone,semitone stepping to be kind to
psychoacoustical theory:
A maj - 4 sharps: A,B,C,C#,D#,E#,F#
A# maj - 3 sharps: A#,B#,C#,D,E,F,A
B maj - 4 sharps: B,C,D,D#,E#,F#,A#
B# maj - 3 sharps: B#,C#,D#,E,F,A,B
C maj - 4 sharps: C,D,D#,F#,A#,B#
C# maj - 3 sharps: C#,D#,E#,F,A,B,C
D maj - 4 sharps: D,E,F,F#,A#,B#,C#
D# maj - 3 sharps: D#,E#,F#,A,B,C,D
etc etc...
While I haven't done the rest - nor worked out complications etc - clearly
it is not a winner! ;)
So in fact, the C major scale has it's roots in psychoacoustics, and can
be derived mathematically. As such, taking into acount the derivation of
the other major scales etc as well - it certainly doesn't look like a
coincidence - the white keys on a piano are thus the "naturals" - the
beginning of deriving the remainder of the scales - and someone obviously
has thought about this a great deal to work it out and the semitone
between b/c and e/f is the least painful way to make it all work.
So I'm glad the answer isn't 100% historical and in fact was the result of
a lot of logic. (Yes, that was never in doubt - but at least now *I*
understand :)
Of course, the above maybe complete rubbish and I may have gone about it
completely incorrectly - but it at least satisfes my curiosity at this
point and it's not something "just to shut up and learn it because that's
how it is". :)
Thanks again..
Cheers,
Umar.
> On 6 Jul, Elgenubi@-----.com wrote:
>
> > The following thoughts are not well researched, I admit, but maybe they
> > will suggest something about why a major scale is "in some sense, natural"
> > as Tony worded it.
>
> [snip]
>
> > If we consider a major scale as 8 notes in two groups with one step between
> > them, then it is step, step, half-step, and .... step, step, half-step.
> >
> > Or in another way: C, D, E, F, and...... G, A, B, C.
> >
> > Does this repeated pattern have anything to do with the 'natural' sound??
>
> Well, good try, but I don't think that's the reason.
>
> By 'in some sense', I meant that if you look the matter up in a music theory
> book, you find that there's a long and complicated discussion about why some
> intervals are consonant to us. (It turns out roughly to be because the notes
> of a consonant interval have harmonics in common.) Then, from the idea of
> consonance, we build up a set of notes that are more or less consonant, and
> they form a scale, which is therefore 'in that sense', natural.
>
> The thing is though, it's impossible to deal with such a complicated subject
> in a few lines of email, even if I were qualified to do it. You have to read
> about it, and study it.
>
> What small amount I know I've enjoyed finding out about. I recommend the
> process to you:-)
>
> > My second thought: What about other cultures? If one listed all the
> > 'natural' sounding scales used in different cultures, what are the
> > patterns, if any, that show up a lot?
>
> It's another big subject.
>
> Mike's paper, which I gave a URL for, has a lot of references.
>
> Tony
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