# Accounting for intervals not present within the chord ratio?

Here are a selection of chord ratios..

Minor Chord - 10:12:15

Root (1:1) - Major 3rd (4:5) - Perfect 5th (2:3)

Major 7th Chord - 8:10:12:15

Root (1:1) - Major 3rd (4:5) - Perfect 5th (2:3) - Major 7th (8:15)

Major 9th Chord - 8:10:12:15:18

Root (1:1) - Major 3rd (4:5) - Perfect 5th (2:3) - Major 2nd (8:9) - Major 9th (8:18/4:9)

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If we look at the Major 7th Chord.. according to the chord ratios, it holds the intervals Major 3rd, Perfect 5th and Major 7th.

But yet, there are more intervals within the Major 7th chord than what the chord ratio 8:10:12:15 represents. The intervals are the following..

• Major 3rd - C - E - (4:5)
• Perfect 5th - C - G - (2:3)
• Major 7th - C - B - (8:15)
• Minor 3rd - E - G - (5:6)
• Perfect 5th - E - B (2:3)
• Major 3rd - G - B - (4:5)

With those additional intervals above not being included in the chord ratio, does that make the chord ratio inaccurate given that the Major 7th chord holds a Minor 3rd which is not accounted for in the ratio or if not, where are these additional ratios within the chord ratio equation?

Also, there are duplicates of the Major 3rd and Perfect 5th. Does that make the chord that much more dissonant as opposed to if there were only 1 Major 3rd and Perfect 5th and not duplicates?

Many thanks guys!

• You may be interested in Paul Hindemith's "The Craft of Musical Composition", in which he apparently "classifies chords based on the intervals in the chords and then classifies these categories into different levels of consonance/dissonance (...) to analyze tension/release." See youtube.com/watch?v=bCnBRJRE_lM&t=42s Oct 1 '19 at 2:30
• Thanks Bob, i checked out the video. The guy said he had trouble finishing the book as it was rather complex and so for someone with my level of technical knowledge, i don't know how beneficial it'll be but ill check out the book and see if it proves to be useful. Oct 1 '19 at 2:55
• Same problems with the humble major and minor triads, Major has M3, P5 and m3. Minor has m3, P5 and M3. Which tantalisingly, are the same intervals as each other, just 'not in the right order' (Eric Morecombe). And what about if there was an octave of the root? That then gives an extra interval of P4.
– Tim
Oct 1 '19 at 9:31
• Not really - as my rating of consonance/dissonance is different from that of others.
– Tim
Oct 1 '19 at 10:25
• From listening. The maths doesn't mean a lot to me - and numbers can be manipulated to mean many things! So, purely from a musical angle, which I know doesn't help your cause a lot.
– Tim
Oct 1 '19 at 10:40

## 3 Answers

where are these additional ratios within the chord ratio equation?

They're right there, almost in plain sight - all you have to do is simplify the numbers:

Major 3rd - C - E - (4:5) = 8:10:12:15
Perfect 5th - C - G - (2:3) = 8:10:12:15
Major 7th - C - B - (8:15) = 8:10:12:15
Minor 3rd - E - G - (5:6) = 8:10:12:15
Perfect 5th - E - B (2:3) = 8:10:12:15
Major 3rd - G - B - (4:5) = 8:10:12:15

In the first row, 8:10 simplifies to 4:5; in the second row, 8:12 simplified is 2:3, and so on.

Also, there are duplicates of the Major 3rd and Perfect 5th. Does that make the chord that much more dissonant as opposed to if there were only 1 Major 3rd and Perfect 5th and not duplicates?

You should consider each occurrence of each interval, rather than regard them as 'duplicates'. Nevertheless, that doesn't mean that each occurrence of each interval makes the chord more dissonant; an interval that is less dissonant than the average for the interval relationships in the chord is arguably making the chord more consonant.

• "They're right there, almost in plain sight - all you have to do is simplify the numbers." Thanks for clearing this up. It's just that when i add up all the interval ratios with the formula provided by WillRoss1 to produce a chord ratio, i don't think the chord ratio added up. "You should consider each occurrence of each interval, rather than regard them as 'duplicates'." So basically what you mean is not to look at the interval per say but more so the note to note relationship. (Probably not the best way of explaining my point). Thank you! Oct 1 '19 at 3:03
• Would the consonance/dissonance be different in different temperaments? Or is all this based on 12tet?
– Tim
Oct 1 '19 at 9:33
• @Tim they're in just intonation. In 12TET the ratios are not nice and clean plain numbers, apart from the octave. The major triad in 12TET is approximately 1:1.25992:1.49831. In 12TET the numbers are simple(ish) only when written as fractional powers of 2, for example 1:2^(4/12):2^(7/12). Oct 1 '19 at 11:37

In general, older (like pre 1000AD theorists) counted only intervals from the bass. Thus, a major chord has the ration 4:5:6. This is enough to calculate other ratios though. The musical reason is that intervals against the bass supposedly (I think so too) show up stronger than against other voices. (Thus the dissonance of the fourth against the bass as opposed to the consonance against the alto in the 64 vs 63 chords.)

• Before 1000? As far as I'm aware, triads came about somewhat later, and there is considerable evidence that Pythagorean tuning continued for five centuries or so, in which the major triad is 64:81:96. That is, the major third was a dissonant interval. Oct 1 '19 at 4:27

I'll call these other intervals: imaginary intervals

# Where are these additional ratios within the chord ratio equation?

With the ratios `8:10:12:15`, it does contain the imaginary interval ratios. As every number is a ratio to each other. For example if you had `5 bananas` to `3 apples` to `4 pears`, there would be the ratios `5 bananas : 3 apples` and `5 bananas to 4 pears` while still having the ratio `3 apples to 4 pears`.

Therefore while the aforementioned, `8:10:12:15` has ratios `8:10` (`4:5`), `8:12` (`2:3`) and `8:15`. It also has the ratios between the other non root numbers like `10:12` (`5:6`), `10:15` (`2:3`) and `12:15` (`4:5`).

# Does these imaginary intervals makes the chord more dissonant or consonant?

This question will always have a vague answer. The inter-ratios or imaginary intervals are not really specific to a chord. For example, for the major triad and minor triad have the same intervals when considering the "inbetween intervals" (Maj 3rd, Min 3rd and P5). So if we judge by a chord is as dissonant as its imaginary intervals then the minor triad and major triad would be the same dissonance.

However, potentially a better approach to this is to considered these imaginary intervals as less weighted - but this deals with the issue of subjectivity. You can't approach this and answer this question without making assumptions.

• The first part of your answer was stupendous, cheers. Yes, the Maj and Min chord share the same intervals, so just out of curiosity.. Although the Maj and Min hold the same intervals, is the reason that the Minor chord is more dissonant because the more dissonant of the 2 intervals (Minor 3rd) is "further down" the octave than the Major 3rd interval and lower octave pitches are known to be harmonically heavier? Oct 1 '19 at 3:10
• @Seery Its difficult to say. Lower pitches aren't more dissonance in just intonation (to a certain extent). Generally wider intervals make the actual interval more ambiguous in terms of dissonance (similarily if you played a note very quickly its harder to tell its dissonance), so it doesn't feel as dissonant but also doesn't feel as consonant. Also, with chords the root intervals is generally the strongest interval, that is why I suggested weighted the other imaginary interval less. So Major triad has a strong M3 and P5 with a weak m3 and minor triad has a strong m3 and P5 with a weak M3. Oct 1 '19 at 9:07