Where did these small number ratios "come from"? People have tried to come up with small and nice numbers for frequency ratios, and that was the smallest and nicest they could get. Wikipedia tells the history https://en.wikipedia.org/wiki/Just_intonation
A:B:C is a condensed way of listing the relationships of the frequencies of a three-note chord. All these mean the same ratio:
- 4 : 5 : 6
- 4Hz : 5Hz : 6Hz
- 400Hz : 500Hz : 600Hz
- 440Hz : 550Hz : 660Hz
- 444Hz : 555Hz : 666Hz
- 500Hz : 625Hz : 750Hz
- 600Hz : 750Hz : 900Hz
- 800Hz : 1000Hz : 1200Hz
- 1000Hz : 1250Hz : 1500Hz
For equal temperament a major chord would be expressed as:
Pretty nice and clean, huh? You just type in the semitone intervals and don't have to figure out any magical 4:5:6 things.
As approximate decimal numbers this is:
- 1.000000000 : 1.25992105 : 1.498307077
- 1000.00000Hz : 1259.92105Hz : 1498.307077Hz
The other ratios are similar. "10:12:15" looks and sounds cleaner than its equal-temperament counterpart
- 1 : 2^(3/12) : 2^(7/12)
- 1.000000000 : 1.189207115 : 1.498307077
- 1000.000000Hz : 1189.207115Hz : 1498.307077Hz
Diminished chord, "20:24:29" for very just intonation, or for equal temperament:
- 1 : 2^(3/12) : 2^(6/12)
- 1.000000000 : 1.189207115 : 1.414213562
- 440.000000Hz 523.251131Hz : 622.253967Hz
- 1000.000000Hz : 1189.207115Hz : 1414.213562Hz
What is this 2^(1/12) thing? It's two raised to the power of one twelfth, which is another way of saying twelfth root of two. Maybe you remember powers and roots from school maths? If not, power means multiplying something by itself several times. Lacking proper math typesetting facilities, it is sometimes written with the ^ sign, for example 4^2 = 4 * 4 = 16. In other words, that's the "square" of four, or four squared. Maybe you remember four times four equals sixteen?
Root is the other way around. Square root of 4 means, "what number raised to the power of two is 4". And that's two. Two times two is four.
Another way of writing square root of four is as a power, where you raise the number four to a power of one half.
When talking about music, the octave means a frequency ratio of two. If you raise a pitch an octave higher, its frequency doubles, i.e. it is multiplied by 2.
If you raise it two octaves, the frequency is multiplied by 2 twice. Three octaves, multiply by 2 three times, etc. How many octaves, so many times "* 2".
Ok. What about the twelfth root? The twelfth root of two is the ratio of a semitone interval in equal temperament. If you multiply a frequency by that number twelve times, you raise the pitch by 12 semitones, which is one octave.
As a decimal number the twelfth root of two is approximately 1.059463094359295. Try it: take a calculator or a spreadsheet and multiply a number by that twelve times.
In equal temperament, we can get the frequency ratio of any interval in terms of twelfth roots. And multiplications by twelfth roots can be combined in the same fractional number exponent, for example, 2^(3/12) is three semitones:
A diminished seventh chord splits the octave to four equal sized jumps:
But in just intonation, it ain't necessarily so - the intervals in a diminished chord are not the same size. 20:24:29 or 160:192:231 or whatever it is, it means that intervals won't be entirely symmetric across the octave. But equal temperament delivers! :)
This has implications for the whole "building beautiful chords from beautiful intervals" idea. If you plan to just play a single chord in a single key and not make drastic harmonic movements, having a key-specific tuning might be ok. But if you intend to have ambiguous harmony progressions with symmetric intervals and lots of jumps between keys, use equal temperament. YMMV, but for me, music gets boring if it stays basically in one chord or mode all the time. (Well ok, depending on the instrument you can fine-tune pitches as you go, and skilled instrumentalists and singers make such adjustments all the time anyway)