Just Intonation > Equal Temperament "Consonance and Dissonance"?

Question

So for months now I'm becoming increasingly frustrated with this Just Intonation versus Equal Temperament, so maybe with this question we can nip it in the bud. If we could refrain from "complex mathematics" which i have trouble comprehending and utilize language and simple math, it would be most efficient.

I will firstly articulate my goal behind asking this question..

I have produced with the help of you good people an interval ranking. The purpose of this interval ranking is solely to tell me what intervals within an octave are most consonant to dissonant from most to least.

The reason i sought this information is so that i can build a chord with the exact desired level of consonance and/or dissonance that i preferred so that when i build a chord progression, i have entire mathematical control over the result. Wrong or right, this is my absolute goal.

The very specific question follows..

If i am using the interval ranking chart for the stated above purpose, does this ranking of intervals from most consonant to dissonant apply to composition in 12tet or when composing in 12tet do the degrees of consonance and dissonance of intervals in just intonation differ in 12tet. I do not mean in their numerical properties but their general sonic properties.

Alternatives..

If this is not the case..

• How can i translate the interval ranking from Just Intonation to 12tet, so that i have a list of intervals from most consonant to dissonant applicable to 12tet?

The desired end result of this question example..

• Unison = 1st most consonant interval in an octave.
• Octave = 2nd most consonant interval in an octave.
• Perfect Fifth = 3rd most consonant interval in an octave.
• Perfect Fourth = 4th most consonant interval in an octave.
• Major Sixth = 5th most consonant interval in an octave.
• Major Third = 6th most consonant interval in an octave.
• Minor Third = 7th most consonant interval in an octave.
• Minor Sixth = 8th most consonant interval in an octave.
• Minor Seventh = 9th most consonant interval in an octave.
• Major Second = 10th most consonant interval in an octave.
• Major Seventh = 11th most consonant interval in an octave.
• Minor Second = 12th most consonant interval in an octave.
• Tritone = 13th most consonant interval in an octave.

I mean this in the most polite manner possible, i am looking for a very straight cut answer. I do not require complex mathematics or incredibly detailed answers. I am just seeking to have a list of intervals within an octave from most consonant to least consonant applicable to 12tet composition. Ideally the desired end result of this question i displayed would be the perfect format to answer this question.

Tremendous thanks to you.

EDIT!!

There seems to be quite a few comments stating that it is impossible to rank 12 intervals in an octave by most consonant to dissonant so let me elaborate.

If a chord progression is developed through tension and release, that tension and release is accounted for by a combination of intervals (a chord) that through each bar go from dissonant (tension) to consonance (rest). If we acknowledge that a chord could be considered to be multiple intervals at work simultaneously, how can we absolutely deny their individual degrees of consonance and dissonance within context of a chord progression? The denial is absolutely baffling to me. It is obvious that the interval of a perfect fifth is more consonant than the interval of a minor second, so evidently the intervals do have their individual degree of consonance/dissonance relative to the root. I will even assume that maybe some of you good folks are jumping in the gun in your answers as opposed to what im specifically asking which revolves around interval to interval relationship and not for example a chord progression. I am not asking how this method would be applicable to a progression, i am very specifically asking what intervals are most consonant to dissonant relative to the root (interval) in 12tet. Nothing more and nothing less.

I also saw a comment stating "what may sound consonant to one listener may be dissonant to another". I think it is undeniable that any listener on planet earth would find an octave more dissonant than a minor second. So evidently a consonance/dissonance ranking does exist and i will paste below my actual ranking which i think some confused my example one for a real ranking.

My question is, is the ranking below applicable to 12tet, not by its numerical values which i understand differ slightly, but the general sound.

I hope this helps.

• Unison = 1:1 Perfect Consonance (1st note of an octave)
• Octave = 2:1 Perfect Consonance (13th note of an octave)
• Perfect Fifth = 3:2 Perfect Consonance (8th note of an octave)
• Perfect Fourth = 4:3 Dissonant when the bass note (6th note of an octave)
• Major Sixth = 5:3 Imperfect Consonance (10th note of an octave)
• Major Third = 5:4 Imperfect Consonance (5th note of an octave)
• Minor Third = 6:5 Impefect Consonance (4th note of an octave)
• Minor Sixth = 8:5 Imperfect Consonance (9th note of an octave)
• Minor Seventh = 9:5 Dissonant (11th note of an octave)
• Major Second = 9:8 Dissonant (3rd note of an octave)
• Major Seventh = 15:8 Dissonant (12th note of an octave)
• Minor Second = 16:15 Dissonant (2nd note of an octave)
• Tritone = 7:5 Dissonant (7th note of an octave)

Answer 1

To keep it simple as you requested, let's just take another look at the well known 12-TET curve from Sethares' site, slightly edited - I've added the curve 'values' at each 12-TET interval.

The values I've added are simply the number of pixels of the line from the top of the image - but that's fine for the purposes of checking the ranking - and for any comparison purposes, because the dissonance 'unit' is arbitrary anyway.

Lets check those numbers against your ranking:

• Unison = 0
• Octave = 0
• Perfect Fifth = 29
• Perfect Fourth = 50
• Major Sixth = 54
• Major Third = 82
• Minor Third = 93
• Minor Sixth = 101
• Minor Seventh = 73
• Major Second = 147
• Major Seventh = 148
• Minor Second = 293
• Tritone = 101

We can see that the numbers there aren't in order regarding the minor seventh and tritone, so we can say that your ranking does not seem to be right according to the definition of consonance by which that graph was created. If you want to know more about that, Sethares' site has a good explanation.

If by 'consonance', you aren't referring to the Plomp/Levelt idea of consonance but something else, you need to be clear what definition you mean.

The reason i sought this information is so that i can build a chord with the exact desired level of consonance and/or dissonance that i preferred so that when i build a chord progression, i have entire mathematical control over the result. Wrong or right, this is my absolute goal.

If you really want to get that right, I would suggest that you don't use a ranking at all, but use the curve values directly. This is because adding the rankings together for the different intervals that would be found between notes in a chord might not give you the same equivalent overall levels of comparative dissonance as if you were to add together the curve values.

If you want to calculate a more precise curve that has values that are better than the 'pixel' values here, again Sethares' site has a lot of information - e.g. the code at http://sethares.engr.wisc.edu/comprog.html

Answer 2

I have produced with the help of you good people an interval ranking. The purpose of this interval ranking is solely to tell me what intervals within an octave are most consonant to dissonant from most to least.

The reason i sought this information is so that i can build a chord with the exact desired level of consonance and/or dissonance that i preferred so that when i build a chord progression, i have entire mathematical control over the result.

Your table doesn't tell you anything about what the intervals sound like. It is just your personal definition of how they are ranked according to some (not very well defined) criterion.

Many people will disagree that such an "absolute" ranking even exists, and those who believe it does exist will probably disagree with you about the exact order.

Wrong or right, this is my absolute goal.

That may be your goal, but it doesn't mean it is possible to do it.

Don't forget the quote attributed (probably incorrectly) to Einstein: "Insanity is repeating the same action over and over again, but hoping that next time you will get a different outcome."

You seem to be trying to construct a "theory of music" which ignores the facts that (1) music is sound that is perceived by humans and (2) the way it is perceived is a learned social construct, not something that is innate at birth.

By my definition, what you are doing might be interesting as an exercise in logic or mathematics (though most of the "mathematics" invented by music theorists is trivial and uninteresting from a mathematician's viewpoint), but whatever it is, it's not "music" by any common-sense definition of the word.

FWIW, after 50+ years of playing and writing music (which actually gets played and listened to by other people apart from myself) I don't even think the terms "consonance" and "dissonance" have any practical use at all in a general "theory of music". The theories that do use those terms are only an approximate description of the sort of music that was made in some limited situations - limited to certain groups of people, and during certain historical periods.

Answer 3

This is too lengthy for a comment, so I'm posting it as an answer. The point, however, is to challenge some assumptions.

First, an actual answer:

How can i translate the interval ranking from Just Intonation to 12tet, so that i have a list of intervals from most consonant to dissonant applicable to 12tet?

The 12-tone intervals are approximations of the just intervals, so their consonance and dissonance is roughly equivalent to that predicted by the corresponding just ratios.

An assumption challenge:

If a chord progression is developed through tension and release, that tension and release is accounted for by a combination of intervals (a chord) that through each bar go from dissonant (tension) to consonance (rest).

To the extent that a chord progression is developed through tension and release, it is not entirely related to the internal harmonic dissonance of each chord. That is, the degree of a chord's dissonance cannot be determined solely from the chord's intervals. To illustrate this, consider the chord progression C major, F major, G major, C major. Each of these chords is a major triad, containing the same intervals in the same frequency ratios. The ratios are certainly the same in equal temperament, and they can also be the same in other temperaments. They can also be entirely just. One of two things must be true about this chord progression: either it is devoid of tension and release, or the tension and release cannot be explained by the intervals comprised in each chord. In the first case, the assumption fails because we have a chord progression that is developed without tension and release. In the second case, the assumption fails because we have tension and release that is not caused by the vertical intervals in each chord.

I also saw a comment stating "what may sound consonant to one listener may be dissonant to another". I think it is undeniable that any listener on planet earth would find an octave more dissonant than a minor second.

That may be true. But will they agree on the relative consonance and dissonance of inversions? Is a minor seventh more or less dissonant than a major second? Why? What about a major third and a minor sixth? What about a perfect fourth and a perfect fifth? In beginning harmony, I was taught that the fourth in the chord G-C-E-G is dissonant, but the fourth in the chord C-E-G-C is consonant. How is that possible?

The answer lies in the idea that certain melodic shapes create certain expectations about what is to follow. This may have little or nothing to do with acoustical tuning. But regardless, it shows that devising chord progressions that "work" requires taking more into account than simply the degree of consonance and dissonance of each chord in isolation.

As a final example, consider this exercise: Take a Bach cantata or other four-part piece, or any harmonic piece really, and a random number generator. For each chord in the piece, generate a random number between -6 and +7. Transpose the entire chord up or down by that number of half steps. If this hypothesis about harmonic music were correct, that it is driven by the internal degree of consonance or dissonance in each chord of the progression, then the resulting music should be just as successful as the original piece.

In fact, that would be true of every possible piece that could be derived this way. That is, instead of a random number generator to generate one piece, use every combination of n values between -6 and +7, where n is the number of chords, to generate 12ⁿ pieces.

It should be obvious that a significant proportion of the resulting pieces, aside from the ones where each chord is transposed by the same number of half steps, or most of them are, would in fact be enjoyable to listen to.

Answer 4

It is not possible to have such a range of single intervals as your concept of dissonance and consonance is not fully congruate with the theory that has been developed in thousands of years not only by mathematical and physical phenomenas but also by the position of 2 tones in a chord with other tones and also in a chord progression.

2 examples:

1. The fourth was considered as consonant in the time of the fourth parallel organum. It is still consonant in a major triad with the octave.

But the fourth can be considered as dissonant in a I46 or V4 chord as sustained 4th resolving in a consonant 3rd.

1. The minor 6th (consonant) can be the enharmonic exchange of an augmented 5th which sounds dissonant and has an enormous tension to be resolved in a 3rd of the fourth.

But you probably want to create a mathematical theory outside of a concept of classical harmony.

Than you have to look up in Hindemith

http://www.hindemith.info/en/life-work/biography/1933-1939/work/principles-and-categories/

https://notingharmonics.wordpress.com/2015/03/11/hindemith-on-intervals/

etc.

or you can listen the intro of my first brass-quartet that operates with major 7th and augmented 5th chords. I think they are all pretty consonant.

Answer 5

It's not always easy to determine the consonance or dissonance of a chord from the properties of the chord's constituent intervals. Based on practice, the fourth in two-part harmony is generally treated as a dissonance. In three-part harmony, the fourth is dissonant against the bass but consonant between upper parts (64 chord vs 63).

Ulehla Ludmilla has a good book which discusses the dissonance level of various chords in different styles.