How are chord ratios developed exactly?

Question

• Major Chord 4:5:6
• Minor Chord 10:12:15
• Diminished Chord 20:24:29

Are 3 examples of triad chord ratios. My question is how are these ratios developed?

If we look at a Major chord in the key of C, it contains the intervals and ratios..

• Unison (C) 1:1
• Major Third (E) 5:4
• Perfect Fifth (G) 3:2

How do we go from 1:1, 5:4 and 3:2 to the three intervals combined in a chord becoming 4:5:6?

If you could also articulate the process for the other two mentioned chords it would gratefully appreciated.

Thank you.

Answer 1

If you multiply both sides of the ratio by the same factor, the ratio doesn't change, so 2:3 is the same as 4:6 (just takes twice as long). So, in the time it takes the root to oscillate 4 times, the Major third oscillates 5 times and the perfect fifth oscillates 6 times, giving us a combined ratio of 4:5:6.

More generally, we just need to put it in the form A1:B, A2:C. If we can get both the As to be the same number we can merge it into the form A:B:C. The easiest way to do this is to multiply both sides of A1:B by a factor of A2 and both sides of A2:C by a factor of A1. With a minor third and a perfect fifth we get the ratios 5:6 and 2:3. (5:6)×2=10:12; (2:3)×5=10:15; merge them together and we get the 10:12:15 ratio of a minor triad.

As for the diminished triad, the most accurate ratio I can find for a dim 5th is 64:77. Using the process above on 5:6 and 64:77 we get 320:384:462, but since all these are divisible by 2 it is equivalent to 160:192:231 (in the same way that 4:6 is equivalent to 2:3). As it turns out, 160:192:231 is actually the exact harmonic ratio of a diminished triad. Since we are already messing things up with equal temperament we approximate this by changing it to 160:192:232 (which is pretty negligible) and reducing by a factor of 8 to get 20:24:29.

Answer 2

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:

• 1 : 2^(4/12) : 2^(7/12)

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

Math lesson

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)

Answer 3

Historically the ratios that define intervals come from the natural harmonics of a linear vibrating system. The harmonic frequencies are related to the fundamental tone by the relation

fn = n*f1 (f1 = the fundamental frequency)

We get the following sequence

f1

2*f1 = octave

3*f1 = Defines the 5th, actually this is an octave and a 5th above f1, you can always divide by 2 any number of times to bring the tone into the octave for reference. 3*f1 / 2 = (3/2)*f1, hence the ratio 3/2.

5*f1 = defines the maj 3rd, 5*f1 / 2 / 2 = (5/4)*f1 this is 2 octaves and a third above the fundamental.

This accounts for the 1, 3, and 5. As for others, any pair of notes related by a ratio that has a power of 2 in the denominator come from this sequence. The perfect 4th does NOT. It can be seen in terms of its relation to the 1 as a 5th down. That is the one must be a ratio of 3/2 to the 4th below it. Bumping that up an octave and inverting gives 4/3.

Just scales are based on these ratios. the equal tempered tuning uses the 12th root of 2 as the ratio of a half step. This is an irrational number that cannot be exactly expressed. 12TET intervals will not be in perfect harmony like Just intervals. In fact for many acoustic instruments one wants to excite the open string sympathetic resonances for better tone and volume. Damping helps in that it creates a broad response for the instrument and a TET interval will likely excite these resonances in the same way as a Just interval.

The minor third, which you asked about, is not part of this sequence. One must simply lower it by an amount equal to a half step. The interval between the Maj 3rd and the P5 is in fact a min3. So one could start with a ratio of 5/4:3/2 which reduces to 6/5. This is not part of the harmonic series but is inferred by the 3rd and the 5th.

As for the chords, a previous answer describes the pattern x:y:z by multiplying through by the denominators to get a set of integers. I interpret your question to be "where does the ratios come from". If I'm wrong I hope my explanation at least helps a little. In my comment I mention why worry about these ratios? If one want 12TET tuning then the -3rd is simply 3 half tones or 2 raised to the 3/12 = 1/4 power. This cannot be expressed as a ratio in the first place.

Answer 4

This answer covers some of the ground.

Note that each of the frequency ratios is a ratio of integers which are not multiples of any prime above 5. So no ratio involving 29 is relevant, so 20-24-29 is not relevant as a tuning of a diminished triad.

How are frequency ratios developed? The guiding principle is that intervals between pitches sound in tune if the ratio of the pitches' frequencies is the ratio between small integers. This answer covers some of the theory here.

In Pythagoras's system, the octave is tuned to 1:2, the perfect fifth to 2:3, and every other interval is constructed by adding or subtracting intervals. Thus the major ninth is two perfect fifths, and thus 4:9; the major second or tone is a major ninth minus an octave, and thus 8:9, and a ditone is two tones or 64:81. Trouble is, by the Italian Renaissance, composers were using the ditone as an interval (i.e. making singers or players simultaneously sound two pitches which are that interval apart), and it sounded rough. Theorists such as Bartolomé Ramos de Pareja and Gioseffo Zarlino favoured instead the interval of frequency ratio 4:5. This is the just major third. It is smaller than a Pythagorean ditone by the interval 80:81, which is called the syntonic comma. (Thanks to user Richard for his answer which put me on to the theorist Ramos.)

Now let's return to the diminished triad mentioned in the OP. Say this triad is vii in a major key. C major, for example. Say we make C-G, F-c and G-d just perfect fifths 2:3, and C-E, F-A and G-B just major thirds 4:5. This entails the following tuning:

C D E F G A B c d e f 24 27 30 32 36 40 45 48 54 60 64

and our diminished triad B-d-f has frequency ratio 45:54:64. The B-d third is a just minor third, with frequency ratio 5:6, but the d-f third is a Pythagorean minor third, with frequency ratio 27:32.

On the other hand, say the diminished triad in question is ii in a minor key. A minor, for example. Say we make A-E and C-G just perfect fifths 2:3, A-D and B-E just perfect fourths 3:4, and A-C and D-F just minor thirds 5:6. This entails the following tuning:

A B C D E F G a 120 135 144 160 180 192 216 240

Now the B-D third is a Pythagorean minor third, and the D-F is a just minor third because I made it so. So our diminished triad B-d-f has frequency ratio 135:160:192.

Why do the two diminished triads have different tunings? Indeed, why do two minor thirds in the same scale have different sizes? It's because of the combo of factors:

• Just intonation tunes perfect fifths to the frequency ratio 2:3
• Just intonation tunes major thirds to the frequency ratio 4:5
• The scale to be tuned has 7 diatonic pitches per octave
• Among the intervals to be tuned are perfect fifths on 6 of these pitches, and major thirds on 3 of them

It turns out that there are so many fifths and thirds to tune, that tuning them all imposes too many constraints on too few variables, and something's got to give. Hence the aforementioned syntonic comma.