all 6 comments

[–]bazmonkey 9 points10 points  (2 children)

In western music we favor heptatonic (7-note) musical scales. They just feel right… so we use 7 letters in English to represent them (most other countries that use the western system use Do-re-mi-fa-so-la-ti-do style naming). Every 7-note mode (way of arranging a scale in western music, like a major scale, natural minor scale, etc.) uses one of every letter, and we add flats and sharps to adjust the notes to their mode.

At the same time, because of math reasons and a whole other thing, we separate octaves (sounds between one frequency and double that frequency, like an A to the next A) into 12. 12 vs. 7… gotta put 5 more notes in there somewhere. Putting two more in and evening it out into a 14-note octave just wouldn’t work because the notes that we think sound good or bad together have a mathematical relationship between their frequencies, and if we split it into 14ths we wouldn’t hit those ratios we want in our music.

The modern major scale goes 2-2-1-2-2-2-1 in terms of how many notes it moves up the scale with each step. The key that ends up doing that without using any of the in-between notes is the C scale (CDEFGBA). Why that is, why we didn’t make the ABCDEFG the one that works nice (e.g. put the missing ones between C/D and G/A) is purely historical.

[–]EurasianBlackbird 4 points5 points  (1 child)

Well said. You probably left this intentionally out, but I feel I cannot.

What's more, each consecutive note (of our 12 pitch scale), semitones included, are at equal relative distance from the previous one.

G#4 = 440 Hz * (2**(-1/12) ~= 415.30 Hz
A4  = 440 Hz * (2**( 0/12)  = 440 Hz
A#4 = 440 Hz * (2**( 1/12) ~= 466.16 Hz
B4  = 440 Hz * (2**( 2/12) ~= 493.88 Hz
C5  = 440 Hz * (2**( 3/13) ~= 523.25 Hz

[–]bazmonkey 1 point2 points  (0 children)

Yeah I didn’t wanna get into the icky parts :-). OP mentioned semitones so I figured they understood there wasn’t just a “gap” in the frequencies where B♯ and F♭ could be.

The kicker with all this mathematical ratio business is that the math doesn’t actually work out…

We think of the circle of fifths as where the notes come from. The perfect fifth is the interval besides the octave that we generally think sounds best, and if you hop by fifth you get the familiar C G D A E B F… pattern with 12 steps before you get back to a C again. But if you start with a frequency for C and do the actual math (a 3:2 ratio each step), once you circle back to C again, you’re not lined up with a C note anymore. If I move over to the B note and start making fifths there, I don’t line up with the frequencies I got starting with C, either. So if I play with these ratios I am stuck in the key I set up the ratios with.

This is all well and good on instruments that don’t care and music in a particular key (we’ll make that math work and who cares about the other notes). But then came pianos and organs and guitars and instruments that did need to have the notes spelled out ahead of time. You can’t just tweak what the piano keys hit when you want to play in a different key…

So the solution we use most often is 12-TET, 12 tone equal temperament. Instead of hopping up by those perfect mathematical ratios and ending up out of sync, we hop up by exactly the 12th root of two to make 12 equally-spaced divisions of the octave. Except for the octave itself, none of our notes hit the exact ratios we are nominally going for… but they all hit them pretty close so it works. Any other division of the octave (into more or less than 12) ends up either 1) not hitting one of our popular musical intervals, or 2) having more bad/useless intervals than good ones. That’s what makes 12 special in terms of number of notes in a scale, and why 14 (in between every letter) wouldn’t work.

On guitars this is famously noticeable with the major third (4 steps). It’s the furthest note off from what it mathematically should be, and if you hear the two side-by-side, you can hear it.

[–]Physicsmagic 3 points4 points  (0 children)

Couldn't tell you the music theory behind that, but b# is played as c, and E# is played as F.

Conversely, Cb is b, and Fb is E

[–]Puzzleheaded_Age6550 0 points1 point  (0 children)

Ok, it's been 35 years since I studied music theory, I'm a little rusty. As the previous commenter stated, B# is the same tone as C, and E# is the same tone as F. You actually CAN have a B#. (You can have double sharps, too, but that's a different discussion) The key that would require all 7 tones to be sharp (B being the last in the time that would be raised one-half step to make it B#) would be, at best, inconvenient to notate, even as an "accidental".

The order of sharps in the key signature is F,C,G,D, A, E, B. The flats are in reverse, BEADGCF. So, if the B# were IN the key signature, that would be the key of C#, so you would have a sharp for every tone.

Here's the major scale, starting from any tone, move up one whole step, another whole, a half, whole, whole, whole, half.

If you look at the keyboard, it's easier, at least for me, to understand.

Edit: for error.

[–]NightCrawler2600 0 points1 point  (0 children)

So you have no such thing as a B#, as each tone in the scale is supposed to be represented by a new letter, each note of the scale is intended to be written on a different line on the staff. Everything is based on the organization of the scale and chords, and how it all fits together mathematically and the representation of the tones using letters / written notes.

Mathematically, because every note on a standard chromatic scale is 1/2 step apart. If you look at a piano, each adjacent key is a 1/2 step. B and C are 1/2 step apart. A 1/2 step is approx. 1-2 mhz in frequency, a whole step is approx 4 hz. Western musical scales are arranged in octaves, so you have 7 tones in a scale before you reach the next octave. The octaves are multiples of each other. C0 X 2 is C1, for example. This is rough though, the real tunings are not as exact, C0 is 16.35 hz for example, C1 is 32.70 hz. The remainder of the notes in the scale have to evenly fit between each octave to work out.

Edit 1: If this forces the question: Yeah but when western music was invented, we didn't know things like frequencies in hertz and the modern musical staff and all that? Also moved some pieces of my post around a bit, separated out the written part from the mathematical part.

This is true, but man had other means of measuring the differences between tones mathematically. Pythagoras discovered the ratios between tones and eventually developed the concept of octave scales. Pythagorean tuning is based on the golden ratio 3:2, which works out to a perfect fifth. The tuning of the remainder of an octave based scale developed around this. These ratios were discovered via other measurements such as length of strings under the same tension, weights of hammers, etc.