# I have three questions on the process of multiplying interval ratios for chords?

Good day music lovers!

After the stupendous answers i received recently on this site, i have established the following..

To find the ratio of a chord

• You take two or more intervals
• Multiply each intervals ratios
• Combine the two or more resulting ratios which will supply you the chord ratio you have produced from the intervals selected.

My questions are as follows,

1. Firstly, are my calculations on the triads mentioned below correct, or have i made errors and if so which?

2. When you have two intervals or more that you are going to combine into a chord, how do you know how many times to multiply each interval ratio to give you the correct ratio of the chord you are building? e.g. in the major chord, 4:5 x ? - 2:3 x ? = 4:5:6?

3. When you take the two interval ratios of an augmented chord for example which is a major 3rd (ratio 4:5) and a minor 6th (ratio 5:8), the end result of multiplying those two ratios ends with 16:20:25:40 although the actual chord ratio is 16:20:25.. How do you know which number to omit from the ratio which in the augmented chord is the 40?

Here below are the triad chord ratio formulas in the key of C..

Major Chord (ratio 4:5:6)

Root (C) - Major 3rd (E) ratio 4:5 - Perfect 5th (G) ratio 2:3

Multiply ratio 4:5 x 1 = 4:5 - Multiply ratio 2:3 x 2 = 4:6

As the two intervals merge to produce the major chord so do the ratios, giving us a major chord ratio of 4:5:6

Minor Chord (ratio 10:12:15)

Root (C) - Minor 3rd (D#) ratio 5:6 - Perfect 5th (G) ratio 2:3

Multiply ratio 5:6 x 2 = 10:12 - Multiply ratio 2:3 x 5 = 10:15

As the two intervals merge to produce the minor chord so do the ratios, giving us a minor chord ratio of 10:12:15.

Diminished Chord (ratio 20:24:29)

Root (C) - Minor 3rd (D#) ratio 5:6 - Tritone (F#) ratio 64:77

Multiply ratio 5:6 x 64 = 320:384 - Mutliply ratio 64:77 x 6 = 384:462

As the two intervals merge to produce the diminished chord so do the ratios, giving us a diminished chord ratio of 320:384:462. Given the magnitude of this figure we can shrink it to its smallest figure by dividing it down x16, giving us the diminished chord ratio of 20:24:29.

Augmented Chord (ratio 16:20:25)

Root (C) - Major 3rd (E) ratio 4:5 - Minor 6th (G#) ratio 5:8

Mutiply ratio 4:5 x 4 = 16:20 - Multiply ratio 5:8 x 5 = 25:40

As the two intervals merge to produce the augmented chord so do the ratios, giving us an augmented chord ratio of 16.20:25.

Suspended 2 Chord (ratio 8:9:12)

Root (C) - Major 2nd (D) ratio 8:9 - Perfect 5th (G) ratio 2:3

Mutiply ratio 8:9 x 1 = 8:9 - Mutiply ratio 2:3 x 4 = 8:12

As the two intervals merge to produce the suspended 2nd chord so do the ratios, giving us a suspended 2nd chord ratio of 8.9:12.

Suspended 4 Chord (ratio 6:8:9)

Root (C) - Perfect 4th (F) ratio 3:4 - Perfect 5th (G) ratio 2:3

Mutiply ratio 3:4 x 2 = 6:8 - Mutiply ratio 2:3 x 3 = 6:9

As the two intervals merge to produce the suspended 4th chord so do the ratios, giving us a suspended 4th chord ratio of 6.8:9.

Many many thanks guys!

Seery

• I answered this in a comment under your last question, but it seems to have been deleted. You compare fractions by expressing them with the same denominator: 1/1 : 5/4 : 3/2 = 4/4 : 5/4 : 6/4 = 4:5:6. You multiply the numerator and denominator of 1/1 by 4 to get 4/4 and 3/2 by 2 to get 6/4 because you want to end up with the same denominator: 4. Sep 13, 2019 at 1:22
• Excuse my simpleness, what are you referring to exactly when you state you compare fractions? Sep 13, 2019 at 1:25
• Sorry, I should have mentioned that this answers question 2. Sep 13, 2019 at 1:26
• I'm very sorry Bob, my math education is deplorable and i can't understand what you're saying. Could you in layman terms explain, how do you find the number that you multiply each interval ratio? Lack of math skills is kicking my butt Sep 13, 2019 at 1:35

I can see why there might be some confusion from my answer to your previous question, as I took a few shortcuts. I'll try to put everything in terms of the generalized formula I mentioned.

We give each number a name in the format: A1:B, A2:C

So,
A1 = 4
B = 5
A2 = 2
C = 3

Now, we need A1 and A2 to be the same number, so we multiply A1 by A2 and multiply A2 by A1. But in order to keep the same ratios we also need to multiply B and C by the same amount as their respective A. So, all together, A1 and B are multiplied by A2 and A2 and C are multiplied by A1.

(A1 x A2): 4 x 2 = 8
(B x A2): 5 x 2 = 10
(A2 x A1): 2 x 4 = 8
(C x A1): 3 x 4 = 12

Now that A1 and A2 are the same (8), we can simplify by calling them both A

(A:B, A:C) 8:10, 8:12

Since A is the same number we can now merge them together

(A:B:C) 8:10:12

Now we find a common factor (a number that all of them can be divided evenly by), if possible, and reduce

8/2 = 4
10/2 = 5
12/2 = 6

And now we have out final answer: 4:5:6

Ratios: 5:6, 2:3

A1 = 5
B = 6
A2 = 2
C = 3

(A1 x A2): 5 x 2 = 10
(B x A2): 6 x 2 = 12
(A2 x A1): 2 x 5 = 10
(C x A1): 3 x 5 = 15

(A:B, A:C) 10:12, 10:15

(A:B:C) 10:12:15

Common factor: none

Ratios: 5:6, 64:92.4 (I was off the last time)

A1 = 5
B = 6
A2 = 64
C = 92.4

(A1 x A2): 5 x 64 = 320
(B x A2): 6 x 64 = 384
(A2 x A1): 64 x 5 = 320
(C x A1): 92.4 x 5 = 462

(A:B, A:C) 320:384, 320:462

(A:B:C) 320:384:462

Common factor: 16 (not quite, but we'll make it work)

320/16 = 20
384/16 = 24
462/16 = 28.875 (rounded to 29)

Ratios: 4:5, 16:25

A1 = 4
B = 5
A2 = 16
C = 25

(A1 x A2): 4 x 16 = 64
(B x A2): 5 x 16 = 80
(A2 x A1): 16 x 4 = 64
(C x A1): 25 x 4 = 100

(A:B, A:C) 64:80, 64:100

(A:B:C) 64:80:100

Common factor: 4

64/4 = 16
80/4 = 20
100/4 = 25

Ratios: 8:9, 2:3

A1 = 8
B = 9
A2 = 2
C = 3

(A1 x A2): 8 x 2 = 16
(B x A2): 9 x 2 = 18
(A2 x A1): 2 x 8 = 16
(C x A1): 3 x 8 = 24

(A:B, A:C) 16:18, 16:24

(A:B:C) 16:18:24

Common factor: 2

16/2 = 8
18/2 = 9
24/2 = 12

Ratios: 3:4, 2:3

A1 = 3
B = 4
A2 = 2
C = 3

(A1 x A2): 3 x 2 = 6
(B x A2): 4 x 2 = 8
(A2 x A1): 2 x 3 = 6
(C x A1): 3 x 3 = 9

(A:B, A:C) 6:8, 6:9

(A:B:C) 6:8:9

Common factor: none

• You caught my attention from the beginning Will and not many others have. This answer is absolutely beautiful and you must recognize that you are an outstanding teacher my friend. Thank you so so so much for this, it is explained incredibly well! I have a proposition and zero pressure behind it, would you be willing to aid me in my other questions in a more instant messaging form such as skype or any other messaging app? I messaged you in the chat but i don't know if you got it and if you would rather that, thats fine. Sep 14, 2019 at 1:10
• Thank you @Seery , that just made my day! I would be happy to help in any way I can (I really enjoy explaining things, if you couldn't tell 😉). I just saw your chat message and that seems like a great place to continue the discussion. Sep 14, 2019 at 3:49

Your calculation on the augmented chord shows the need for some type of temperament. There is no consistent choice of intervals that gives "just" intonation to all intervals in a scale. For example, the just third is defined by the ratio 5/4 and the just major second by 9/8. However combining two major seconds gives 81/64.

Some compromises have to be made. Check the mean-tone tunings or the Wendy Carlos tuning.

Equal temperament is often used so that all keys are easily reachable.

There is a practical example. Ordinary guitar tuning is E-A-D-G-B-E covering two octaves. The two E strings should have a ration of 4/1, however, should each interval be tuned exactly, one would get 4/3*4/3*4/3*5/4*4/4 which equals 320/81. I have known guitarists who tried tuning each interval by ear (one can hear the beats), but the final result (E) is out of tune.

There are some web sides worth search on the matter.

• Interesting. Am i fooling myself utilizing the interval ranking for direct reference in what intervals to use to produce desired consonance/dissonance within a chord? All i want is to know the levels of consonance/dissonance of intervals so i can measure the consonance/dissonance of a chord and adjust it to my preference. Can Just Intonation ratios not be utilized for that purpose even though im composing in 12TET? Sep 13, 2019 at 1:09
• Ludmila Ulehla has a book "Contemporary Harmony" which discusses the (approximate) dissonance composers assigned to teach chord. There's also a nice outline thereelscore.com/PortfolioStuff/PDFFiles/… here where dissonances are ranked according to the period (common practice, late romantic, post romantic, impressionistic....)
– ttw
Sep 13, 2019 at 1:50
• Thanks for the link ttw, i found the section you are referring to but it does not answer the question "can just intonation interval ranking be directly applied to 12tet intervals consonance and dissonance ranking". Thank you regardless. I just asked a question addressing this question specifically if you're interested music.stackexchange.com/questions/89641/… Sep 13, 2019 at 2:01