Natural half step

[Fusion]

Does anyone know why there is a half step between B-C and E-F? I think it has something to do with the mathematics of the equal tempered scale but can't remember.

Fusion

[This message has been edited by Fusion (edited May 25, 2000).]

[Fusion]

I think I found the answer; here it is for anyone who likes mathematics:

Pythagorus discovered that when you divide a string in two, it sounds an octave higher. Fret it one third up the neck (2/3rds of its original length) and it will sound the perfect fifth (from DO to SOL). Since frequency and wavelength are inversely proportional, the frequency of the fifth is 3/2 that of the fundamental (and the octave is twice).

From these two relationships it's possible to construct fractional relationships for all of the notes of the diatonic scale (DO RE MI...DO). For example, a perfect fourth (FA) is the same as an octave minus a fifth, so 2 / 1.5 = 4/3. So the wavelength of the fourth is 3/4 of the fundamental; it's fretted 1/4 up the neck.

Well, all of this is well and good until you change keys and things start sounding dissonant. Why? Because frequency is a logarithmic function, not a linear one. Just look at a guitar neck and see how the frets come closer together as you head up the neck (increase frequency).

If you start at C and go through the circle of fifths you should end up on C seven octaves higher. But it doesn't take a math genius to realize that there is no way that 3/2 taken to the 12th power will equal 2 to the 7th. Three to any (integer) power is going to be an odd number.

The pythagorean comma is the ratio of this inequality.

Because of this mismatch, pianos are tuned 'tempered' chromatically, with each half-step (e.g. F to F#) tuned the 12th root of 2 apart in frequency. This way, there is a logarithmic rise that doubles the frequency over the course of 12 chromatic semi-tones.

They don't miss by much: The 12th root of 2 taken to the 7th power is 1.4983 (SOL is 7 chromatic semi-tones, or half-steps, above DO), whereas the ratio of the perfect 5th is 1.5. But they miss by enough that if you tuned a piano with just temperament in 'A' it would sound off in 'Eb'.

What then happens to our perfect 4th and 5th? They aren't so perfect anymore. The 5th of A440 ideally should E660, but instead we have to multiply 440 by the 12th root of 2 seven times (there are 7 half-steps from A to E) yielding 659.26. It misses less than 1 cycle per second, but you can hear the miss if you listen closely, as a beat frequency.

Similarly, The 4th of A440 should be D586.67, but 5 half-steps later we land on 587.33. So the 5th will sound a smidge flat, and the 4th a tad sharp. Unless you're really paying attention, though, you probably won't even notice.

Imagine a string stretching from point 0 to point 1:

that when plucked will sound 'DO', a fundamental root.

Now let's divide the string into a half, a third, etc., and put numbers along the string inverse to these divisions (e.g. the length from 3 to 1 is 1/3 the length from 0 to 1; the string will vibrate 3 times as fast):


If 1:0 plays 'C', here are what some other intervals will sound:

To the right:

1:8 C, DO - 3rd octave
L = 1/8 of 1:0, f = 8

1:6 G, SOL - 2 octaves and a fifth
L = 1/6 of 1:0, f = 6

1:5 E, MI - 2 octaves and a major third
L = 1/5 of 1:0, f = 5

1:4 C, DO - 2nd octave
L = 1/4 of 1:0, f = 4

1:3 G, SOL - 1 octave and a fifth
L = 1/3 of 1:0, f =3

1:2 C, DO - 1st octave
L = 1/2 of 1:0, f = 2


To the left:
0:2 C, DO - 1st octave
L = 1/2 of 1:0, f = 2

0:3 G, SOL - the perfect fifth
L = 2/3 of 1:0, f = 1.5

0:4 F, FA - the perfect fourth
L = 3/4 of 1:0, f = 1.333

0:5 E, MI - the major third
L = 4/5 of 1:0, f = 1.25

0:6 Eb, MI(b) - the minor third
L = 5/6 of 1:0, f = 1.2


Notice how the notes taken to the right make up a very pleasing C major chord.

The Harmonic Mean

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Taking the arithmetic proportion of two notes will yield a third note called the harmonic mean. The arithmetic proportion is found by dividing the sum of two terms by 2. If C=1 (DO) and C an octave higher=2, the mean equals 3/2, a frequency that will sound 'G', the fifth (SOL).

Now we have three notes C,G, and C whose intervals are 1:3/2:2, which is equivalent (multiplying by 2) to 2:3:4 (See the numbered string).

Similarly, we can take the root and fifth, 2 and 3, and get the mean of 2 1/2 for intervals of (X2 again) 4:5:6. This is a major chord (DO-MI-SOL).

Continuing with the math, the whole diatonic scale was developed.

Cool, huh?

Fusion

[ghodaddyyo]

I think I have a headache and now I'm going a bit cross eyed. I'll never learn theory.http://www.zentao.com/ubb/smilies/frown.gif

[Fusion]

This isn't really theory but rather the mathematics of the musical scale. None of this is necessary to understand music theory but it is interesting for us math geeks.
I studied electrical engineering in college so I love this stuff ... I assume I'm not the only one http://www.zentao.com/ubb/smilies/confused.gif
Maybe StoneDragon needs to find me geek icon http://www.zentao.com/ubb/smilies/wavey.gif
Peace

[FUNKGRAVY]

DAMN and I thought Nuclear Medicine engineering was mentally demanding!http://www.zentao.com/ubb/smilies/eek.gif

[DocRoop]

especially being field service engineering

[Strat-n-Paul]

Well by day, I'm an accountant; Hey Fusion, if you put dollar signs in front of all that maybe I'll understand it!


http://www.zentao.com/ubb/smilies/explode.gif http://www.zentao.com/ubb/smilies/lol.gif

[Strat-n-Paul]

It would be interesting to see if Buzz Fietens tempered system helps solve any of the inherent tuning issues with guitars, or has anyone heard of that tempered nut called an earnavana nut? Anyone tried either of these?

[StoneDragon]

Unless you are going to play solo guitar and stay in one or two closely related keys at most, then there is no point in trying to re-temper anything. The equal temperment that we use was designed on purpose... it's the only way to even out the tuning discrepancies all the way across the board. Any other system leaves the dreaded "wolf" tone lurking somewhere.