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Good day guys!

Thanks to the gentleman @willross1 and others contributions, i have now discovered the formula behind turning interval ratios into chord ratios, which allows me to determine the level of consonance/dissonance of a chord based off my choice of intervals. I will demonstrate this formula below for context and reference.

"Formula for forming a chord ratio through interval ratios"

i. Select two or more intervals relative to your level of desired consonance/dissonance. In this example for a Major Chord, we will utilize the intervals Major 3rd (4:5) and Perfect 5th (2:3).

ii. The first ratio (4:5) is multiplied by the second ratios (2:3) first number which is 2.

⦁ 4 x 2 = 8

⦁ 5 x 2 = 10

Then

i. The second ratio (2:3) is multiplied by the first ratios (4:5) first number which is 4.

⦁ 2 x 4 = 8

⦁ 3 x 4 = 12

Our ratios for both intervals after being mutliplied between each other, now go from 4:5 and 2:3 to being 8:10 and 8:12 and since both ratios hold the same first number, we can summarize the combined ratios to 8:10:12 which now becomes our Major Chord ratio.

We can further reduce this chord ratio if possible, by finding a common factor (a single number that can produce each of the three numbers) between the three numbers in our chord ratio (8:10:12) which would be "2" and divide each of the three numbers by it.

⦁ 8 ÷ 2 = 4

⦁ 10 ÷ 2 = 5

⦁ 12 ÷ 2 = 6

Our final chord ratio (Major Chord) has now become 4:5:6.

Here are the following questions..

Question 1 - We know the process for finding the chord ratio of two intervals which equals a triad. How do we approach the formula to deal with more than two intervals (e.g. 3 intervals) when we want to find the chord ratio for example of a Major 6th Chord which contains the intervals Major 3rd (4:5) - Perfect 5th (2:3) and Major 6th (3:5)?

This question revolves around finding the formula of a chord ratio when i utilise not only 2 or 3 intervals but 4,5,6,etc. Also, if you could demonstrate the formula in a similar manner to the one demonstrated in this post, it would be most practical.

Question 2 - If we agree that the consonance of an interval stems from the idea of "the lower the ratio, the more consonant the interval", how does this concept apply to situations where for example a minor chord with the ratio of 10:12:15 would be more consonant than a sus 2 chord with the chord ratio of 16:18:24 based off the minor chord having a smaller ratio, but once you find the common factor of a sus 2 chord ratio, the sus 2 chord ratio now becomes 8:9:12 which is now a smaller ratio than the minor chord?

With these scenarios how can one determine which chord is more consonant than the other based off the chord ratios?

Question 3 - If intervals are composed from root to note relationships such as a root and major 3rd producing a Major 3rd interval, once you add a second interval in the process of building a chord, how do the interval relationships begin to act within that chord.

Allow me to elaborate,

We know when building a Major Triad that the first interval (Major 3rd) level of consonance/dissonance is relative to the root. Now, as shown in the image below, if we now add a perfect 5th on top of the major 3rd, is the perfect 5ths level of consonance/dissonance relative to the root also or the major 3rd below it?

If you would care to elaborate on this concept beyond my initial query, I'm sure it would be of significant use. This information will prove useful once i start stacking intervals and need to understand their consonance/dissonance relationships.

Question 3 Chord Visual

Question 4 - What was the logic or process behind the interval ratios that were selected to build major,minor,diminished,augmented,sus2 and sus4 triads? Why were those specific intervals utilized to build each specific triad?

These are the only question i believe i have on this specific topic but should any others of importance arise, i will include them in the post edit. These questions are of immense value to me so thank you very much for your contribution guys.

  • 2
    Don't understand a lot of what you're doing, but bear in mind that often, voicings do not simply go upwards. For example, C9 often isn't played sequentially as C E G Bb D, so intervals may be different from your original blueprint. Hope this helps in some way. – Tim Sep 19 at 11:42
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    "the lower the ratio, the more consonant the interval" - do you have a source for that. I'm surprised to see it expressed that way. Also, what is a "lower" ratio? A ratio expressed in small numbers, or a ratio where the proportion is smaler? You could sat that a 10:11 ratio is smaller than 2:3 because the notes are closer to each other in the former. – mkorman Sep 19 at 15:42
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    On a different note, I'm surprised that you want to sudy consonance/dissonance from the frequency point of view. Most people study it from the interval point of view. After all, there are only 10 different intervals and then the octave. – mkorman Sep 19 at 15:43
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    @mkorman By "lower" he doesn't mean smaller, he means simpler – WillRoss1 Sep 19 at 16:59
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    @Seery The more frequently the waveforms line up, the simpler the ratio. – WillRoss1 Sep 20 at 16:08
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Question 1

This question revolves around finding the formula of a chord ratio when i utilise not only 2 or 3 intervals but 4,5,6,etc.

Answer 1:

The more intervals you introduce the more difficult this becomes. The process, however, is exactly the same, just with more numbers.

This is the general formula: A1:B, A2:C, A3:D

Now we need A1, A2 and A3 to be the same, so we multiply A1 by (A2 x A3), multiply A2 by (A1 x A3) and multiply A3 by (A1 x A2). Just as with 2 intervals we also need to multiply B, C and D by the same number as their respective A.

A1 x (A2 x A3)
B x (A2 x A3)
A2 x (A1 x A3)
C x (A1 x A3)
A3 x (A1 x A2)
D x (A1 x A2)

From there we merge and reduce, just as before.

Let's use this to figure out a Maj7:

Ratios: 4:5, 2:3, 8:15

A1 = 4
B = 5
A2 = 2
C = 3
A3 = 8
D = 15

(A1 x (A2 x A3)): 4 x (16) = 64
(B x (A2 x A3)): 5 x (16) = 80
(A2 x (A1 x A3)): 2 x (32) = 64
(C x (A1 x A3)): 3 x (32) = 96
(A3 x (A1 x A2)): 8 x (8) = 64
(D x (A1 x A2)): 15 x (8) = 120

64:80, 64:96, 64:120

Merge: 64:80:96:120

Common factor: 8

64/8 = 8
80/8 = 10
96/8 = 12
120/8 = 15

Combined Ratio: 8:10:12:15

For 4 intervals:

A1:B, A2:C, A3:D, A4:E

A1 x (A2 x A3 x A4)
B x (A2 x A3 x A4)
A2 x (A1 x A3 x A4)
C x (A1 x A3 x A4)
A3 x (A1 x A2 x A4)
D x (A1 x A2 x A4)
A4 x (A1 x A2 x A3)
E x (A1 x A2 x A3)

A Major 9th:

First, we need to figure out the ratio of a 9th, since it doesn't seem to be explicitly listed anywhere online. Fortunately, its pretty easy: a 9th is 1 octave up from a 2nd, a 2nd has a ratio of 8:9, adding an octave to any pitch doubles it's frequency, the root doesn't change so it remains an 8, the 2nd moves up an octave, doubles and becomes an 18 meaning the interval ratio of a 9th is 8:18. Having a common factor of 2 we can then reduce it to 4:9)

Ratios: 4:5, 2:3, 8:9, 4:9

A1 = 4
B = 5
A2 = 2
C = 3
A3 = 8
D = 15
A4 = 4
E = 9

(A1 x (A2 x A3 x A4)): 4 x (2 x 8 x 4) = 256
(B x (A2 x A3 x A4)): 5 x (2 x 8 x 4) = 320
(A2 x (A1 x A3 x A4)): 2 x (4 x 8 x 4) = 256
(C x (A1 x A3 x A4)): 3 x (4 x 8 x 4) = 384
(A3 x (A1 x A2 x A4)): 8 x (4 x 2 x 4) = 256
(D x (A1 x A2 x A4)): 15 x (4 x 2 x 4) = 480
(A4 x (A1 x A2 x A3)): 4 x (4 x 2 x 8) = 256
(E x (A1 x A2 x A3)): 9 x (4 x 2 x 8) = 576

256:320, 256:384, 256:480, 256:576

Merge: 256:320:384:480:576

Common factor: 32

256/32 = 8
320/32 = 10
384/32 = 12
480/32 = 15
576/32 = 18

Combined Ratio: 8:10:12:15:18

Now, just for fun, a Dominant 7 #9 11 b13:

#9 has a ratio of 5:12 (aug 2nd (min 3rd), 5:6, plus one octave)

11 has a ratio of 3:8 (perfect 4th, 3:4, plus one octave)

b13 has a ratio of 5:16 (minor 6th, 5:8, plus one octave)

Ratios: 4:5, 2:3, 5:9, 5:12, 3:8, 5:16

A1 = 4
B = 5
A2 = 2
C = 3
A3 = 5
D = 9
A4 = 5
E = 12
A5 = 3
F = 8
A6 = 5
G = 16

(A1 x (A2 x A3 x A4 x A5 x A6)): 4 x (2 x 5 x 5 x 3 x 5) = 3000
(B x (A2 x A3 x A4 x A5 x A6)): 5 x (2 x 5 x 5 x 3 x 5) = 3750
(A2 x (A1 x A3 x A4 x A5 x A6)): 2 x (4 x 5 x 5 x 3 x 5) = 3000
(C x (A1 x A3 x A4 x A5 x A6)): 3 x (4 x 5 x 5 x 3 x 5) = 4500
(A3 x (A1 x A2 x A4 x A5 x A6)): 5 x (4 x 2 x 5 x 3 x 5) = 3000
(D x (A1 x A2 x A4 x A5 x A6)): 9 x (4 x 2 x 5 x 3 x 5) = 5400
(A4 x (A1 x A2 x A3 x A5 x A6)): 5 x (4 x 2 x 5 x 3 x 5) = 3000
(E x (A1 x A2 x A3 x A5 x A6)): 12 x (4 x 2 x 5 x 3 x 5) = 7200
(A5 x (A1 x A2 x A3 x A4 x A6)): 3 x (4 x 2 x 5 x 5 x 5) = 3000
(F x (A1 x A2 x A3 x A4 x A6)): 8 x (4 x 2 x 5 x 5 x 5) = 8000
(A6 x (A1 x A2 x A3 x A4 x A5)): 5 x (4 x 2 x 5 x 5 x 3) = 3000
(G x (A1 x A2 x A3 x A4 x A5)): 16 x (4 x 2 x 5 x 5 x 3) = 9600

3000:3750, 3000:4500, 3000:5400, 3000:7200, 3000:8000, 3000:9600

Merge: 3000:3750:4500:5400:7200:8000:9600

Common Factor: 50

3000/50 = 60
3750/50 = 75
4500/50 = 90
5400/50 = 108
7200/50 = 144
8000/50 = 160
9600/50 = 192

Combined Ratio: 60:75:90:108:144:160:192


Question 2

...how does this concept apply to situations where for example a minor chord with the ratio of 10:12:15 would be more consonant than a sus 2 chord with the chord ratio of 16:18:24 based off the minor chord having a smaller ratio, but once you find the common factor of a sus 2 chord ratio, the sus 2 chord ratio now becomes 8:9:12 which is now a smaller ratio than the minor chord?

Answer 2:

Consonance and dissonance can be subjective given the context, so it can be difficult to directly compare things. That said, a Sus2 chord may contain one interval that is more dissonant (major 2nd) than a major or minor 3rd, but it's perfect 4th is more consonant so it balances out. Personally, I would consider a Sus2 more consonant than a minor. To illustrate this, play a Sus2 resolving to a minor (CSus2 -> Cm) and you will notice it getting darker, but resolve it to a major and it gets brighter.


Question 3

If intervals are composed from root to note relationships such as a root and major 3rd producing a Major 3rd interval, once you add a second interval in the process of building a chord, how do the interval relationships begin to act within that chord.

Answer 3:

.....it's complicated.....

This is actually a really important point to bring up. Once you get into more complicated chords (like Dominant 7 #9 11 B13, and other extended jazz style chords) things start to fall apart rather quickly. In fact, for many of these kinds of chords it is literally impossible to perfectly tune them to the harmonic series. This is because every frequency interacts with every other frequency, and tuning a note to align with a certain reference pitch can detune it from another.

Simple chords work out just fine because they have a lot of overlap in their harmonic series' and there are a limited number of interactions to potentially conflict. Larger chords introduce more and more points of failure where conflicts can occur. Dom7 #9 11 b3 has a perfect 5th from the 7 to the 11, but when they are both tuned to the root they have a ratio of 9:16 which puts them WAY out of tune with each other (over 2 semitones sharp!) Fix their ratio to make it a perfect 5th again, and now one or both are out of tune with the root.

So basically, equal temperament tuning is absolutely essential to jazz music!


Question 4

What was the logic or process behind the interval ratios that were selected to build major,minor,diminished,augmented,sus2 and sus4 triads? Why were those specific intervals utilized to build each specific triad?

Answer 4:

The intervals were not chosen, they just are. They come from how pitches and frequencies naturally behave. It was more of a discovery than an invention. The frequency ratios have always existed, we just figured out how to reproduce them and gave them names.

  • Tremendous thanks again will. I took your ratio formula and have compressed it into an even simpler formula thanks to your assistance, cheers!!! – Seery Sep 20 at 5:58
  • In regards to question 2.. This question relates to the context of a single chord and not within the context of a chord progression.. To clarify, an interval ratio itself is the numerical translation of how many wave cycles the root must perform to coincide with the crest of the wave of the second note of the interval? If this is true, then a chord ratio is a numerical translation of how many cycles each ratio within the chord has to perform for all of them to coincide? If so, how could the post common factor ratio be explained specifically? – Seery Sep 20 at 5:59
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    @Seery A non-reduced ratio is essentially just multiple cycles of the ratio; the ratio itself does not change. The only thing we care about is what happens in a single cycle, so we always reduce. A minor 10:12:15 cannot be compared to a non-reduced sus2 16:18:24 because the sus goes through 2 full cycles, but the minor only gets to go through 1. Its kinda link saying "You make $10 in an hour, but I make $12 in 2 hours, therefore I make more than you". $6/hour is the same as $12/two hours, but the latter adds needless complexity and gives us no additional information. – WillRoss1 Sep 20 at 16:02
  • Thank you! I completely get it now, great explanation Will! – Seery Sep 22 at 23:46

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