The Bis Key Chronicles

At work on the company musicians alias a fellow named Wolf asks:

Why is there no B# or E#? Or to put it another way; Why are there 7 notes A-G?

There’s 12 halftones in the western octave. I get that. It’s probably something biblical, based on 12 disciples, like the 12 hour clock and 12inch foot, and the 12 houses of Astrology (sic). But what I’m struggling with is that to my mathematical mind the 12half-tones would most neatly result in 6 full tones. A-F with of course a real honest to goodness B# and E#.

Not looking for the ‘what’ which is so excruciatingly outlined in all dissertations on musical theory.

Luv these kind of questions and the musician’s alias at work provides a lot of fodder for thought. Further came home to roost as I was handed the soprano sax solo to Holst’s “Venus” to sight read. Yikes!

Hal sez: “Because what sounds melodic to the human ear in terms of a standard C major scale is designed so that the notes intoned are all whole letters – CDEFGABC (As Vincent called out. Of course the number of scale degrees obviously matters here, too). Since the intervals between the notes of the diatonic are not all uniform in equal temperament—specifically, E-F and B-C are half steps—you end up with a half-step up being just the next letter in the scale, rather than “E#” or “B#”

However, I do confess confusion about why the major scale is standardized to “C” instead of “A” –wouldn’t it make a lot more sense to call your starting note “A”?

Chad adds: “My understanding from my music theory courses was that there are mathematical relationships between the frequencies of the pitches in our standard scale.   There is a definite relationship between a 1 and a 5.  Then the 5 has a relationship with the note that is a fifth above it.  I seem to remember that we get the standard scale using mathematical relationships until we get to semi-tones (some of which are used in Indian classical music).  Not sure why they stopped at half tones but it must have sounded good to someone at some point.

Copied the following from http://www.songtrellis.com/concepts/tone: “Once musical sounds were graphed the investigators noticed that the graphs that were produced were simpler than those for other sounds. Musical sounds had wave graphs where the distance between individual wave peaks was regular. They called this distance between peaks the wavelength of the sound. By experimenting further, they discovered that two musical sounds sounded best together, when their waveforms were exactly the right length so that their wave cycles would line up and begin at exactly the same moment every few cycles for each wave. Sounds that sound unmusical together almost never fall to zero at the same time and fall to zero at the same moment at irregular intervals, if they ever do.“

Dan replies, “I think that’s a piece of the best explanation I know of for why there are 12 halftones. If you look at the values of 2^(1/12), 2^(2/12), 2^(3/12), etc., you’ll see one that is approximately 4/3 (which also happens to be the major third) and one that is approximately 3/2 (which also happens to be the fifth): probably not a coincidence that these, along with the root, make up the major chord.

As I recall, if you look at evenly-spaced fractional powers of 2, twelfths happen to yield the two values which are closest to those two fractions, until you go up to a much higher denominator (like, somewhere around 50).

But, I don’t think that helps explain why there are 7 tones in the major scale; it’s really just the third and the fifth that are very close to simple fractions, and the rest of the notes don’t really line up with anything. So, I don’t actually know of a good hypothesis to answer the original question, except effectively “because people like the sound of the major scale”.

Wolf sez: “So what I’ve learnt is that I was really asking two questions;

1. Why are the standard western scales diatonic? B-C and  E-F being tonics, two being ‘dia’. The answer appears to be ‘coz it just sounds the best’

2. Why is the standard music notation heptatonic scale based and not normalized to either the Chromatic or the whole tone scale  ?

For example on the diatonic notation the scale of e is e f# g# a b c# d# [e]

Whereas if that was written on a whole tone notation that would be E F A A# B# C# D# [E]. However A + A# doesn’t fit even using flats E F A Bƅ Cƅ Dƅ Eƅ [E] as that results in a double E.

So I guess the answer to the second part is that the diatonic scale simply can’t be translated to a whole-tone notation without an ambiguous note.

The other option would be to use a chromatic notation with notes labeled A-L. Of course any western scale can be played on the 12 half-tones regardless of how they’re labeled.

The wave-theory/harmonics is an interesting avenue to follow. I’ll also look into that more. Especially in chord formation and keys.

Read more at: http://en.wikipedia.org/wiki/Wolf_fifth where you will find great charts like this one (below).

Are we having fun yet?  :O)