8
  • 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.

  • 1
    I am a bit confused by your question, and can only guess it's related to your last one. The ratios you are listing seem to be related to the natural harmonics of a linear system. This carries over to Just tuning, but NOT to equal tempered tuning. So why use them? – ggcg Sep 11 at 1:58
  • 2
    The only perfectly in tune interval in equal temperament is the octave. All others differ slightly from the perfect ratio from the harmonic series. They are VERY close approximations though and we often use their perfect intervals for analytical purposes. – WillRoss1 Sep 11 at 2:21
  • 2
    Some can hear the difference. But I guess my question was motivated by your chord construction question too. Does the ration really matter in terms of describing chord construction? – ggcg Sep 11 at 2:44
  • 1
    You are missing the minor third interval from the major chord. The interval between E and G is a minor third. – piiperi Sep 11 at 5:32
  • 2
    finally the question is: are the harmonic relations of music natural (physically) given or they a product of cultural invention and development. I tend to the 2nd interpretation. The physical and mathematical explanation are just the scientific reason for something like the chemical processes in our body for something that we call LOVE. But I like all the given answers. – Albrecht Hügli Sep 11 at 9:27
12

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.

7

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?

four to power of two

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.

square root of 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.

square root as fractional power

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.

to raise an octave, multiply the frequency 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".

raise 5 octaves, multiply by 2^5

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.

twelfth root of two

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.

twelfth root of two approximately

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:

three semitones interval frequency ratio

A diminished seventh chord splits the octave to four equal sized jumps:

four jumps to octave

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)

  • 3
    Gotta love how the just intonation diminished chord note ratios give two different ratios for the minor third: 5:6 and 24:29. There's probably a ghost of a wolf interval somewhere in there, isn't there? – Dekkadeci Sep 11 at 11:27
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    @Dekkadeci Ghosts and wolves go together with this stuff. Mystic things, numbers and ratios. This page gives 160:192:231 as a more accurate ratio pages.mtu.edu/~suits/chords.html but that's inconsistent about the minor third as well. :/ How are you supposed to make jazz without symmetric intervals. I'm wondering, is this numerology thing compatible with good music at all. ;) – piiperi Sep 11 at 14:09
  • Thank you for your answer piiperi. A difficulty i have in understand some answers including yours, is my lack of math skills so when i hear the "power of 2" or "1 : 2^(4/12) : 2^(7/12)" i have absolutely no idea what that means and dont expect you to teach me but im very grateful for your contribution and im sure others will benefit from it. Thank you – Seery Sep 12 at 19:42
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    @Seery it's not rocket science really. You can intuitively understand powers and roots with simple numbers, and if you toy around with a calculator for a while, you can understand the concepts for fractional or decimal numbers as well. If you intend to continue exploring intervals, understanding how the numbers work is kind of a must-have. – piiperi Sep 12 at 22:56
  • You're a gent for providing the math explanation thank you very much!! – Seery Sep 12 at 23:01
4

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.

  • Sure, you can express equal temperament ratios: 1 : 2^(4/12) : 2^(7/12). It's actually a lot nicer because you can just type in the semitone intervals! ;) – piiperi Sep 11 at 6:01
  • Perhaps I misspoke when I said it's not a ratio. What I meant was that the ratio of frequencies is not rational. I will fix. But there nothing "clean" about it. 12TET is, by its very nature dissonant and many people can hear it. – ggcg Sep 11 at 10:26
  • "many people can hear it", but these "many people" are still a minority. My bet is that 99% of the population don't know the difference between the 12TET and the Pythagorean scale. – mkorman Sep 11 at 16:44
  • I disagree. Perhaps it is rare to hear a single note out by a few cents but intervals in 12TET will have a noticeable dissonance relative to just intervals. You don't need to have perfect pitch to hear it. If you are ever trained on a fretless instrument like violin, cello, or classical bass you will not be able to un-hear it. – ggcg Sep 11 at 21:01
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    @ggcg thank you for your elaborate answer, very informative. My goal is to figure out the process of utilizing individual intervals to build a chord through taking the ratio of an interval from the interval ranking, adding an additional interval or more where we can create our own chords with as much consonance or dissonance as we desire, mapping out the spectrum of possible chords from most consonant to most dissonant with the aid of interval ranking with complete control. I want to build chords interval by interval using the interval ranking to determine the level of consonance/dissonance. – Seery Sep 12 at 19:38
3

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.

  • Very great answer and historic information. Very much appreciated Rosie. If i come with any further questions regarding your answer down the line of my research i will return to your post. Thank you for your time and effort! – Seery Sep 12 at 19:47
  • 1
    @Seery You're welcome. To some extent I was standing on the shoulders of other Music.SE contributors, and we're all standing on the shoulders of giants such as Pythagoras and those Renaissance theorists. – Rosie F Sep 13 at 6:25

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