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I understand that some intervals are considered dissonant (minor second) and some intervals are consonant (perfect fourths).
However whenever I see intervals listed, they are always listed from smallest to largest.
What would the list of intervals look like if it was ordered from most dissonant to most consonant.
It can be tough to define exactly what "dissonance" is (it changes throughout history, and it changes between genres), but Paul Hindemith created his own theory regarding ranked dissonances.
He had two "series": Series 1 was a list of melodic intervals from most consonant to least consonant. Series 2, meanwhile, was a list of harmonic intervals from consonant to least consonant. (By melodic intervals, we mean two pitches sounding at different times; with harmonic intervals, the pitches sound simultaneously.
Here's a quick picture from this website:
Series 1 thus presents the melodic intervals (always from C) in an order from most consonant to least consonant. Thus C to G is the most consonant, C to F is the next most consonant, then C to A, all the way to C to F#. You can generalize this to be in any key by focusing on the intervals in order of increasing dissonance:
P5, P4, M6, M3, m3, m6, M2, m7, m2, M7, TT
Note that the list of intervals changes slightly in Series 2!
Dissonance is an interesting concept in music. As I have learned the idea that dissonance is inherently bad, is a completely outdated concept. For the longest time, classical music has operated under the header of consonance.
That is to say, that is what is easy on the ear was held as good. The name of our intervals reflects that fact. The octave, the unison, the fifth and the fourth are given the title of "perfect" for their inherently consonant sounds.
The second and seventh are relegated to merely being Major or minor, there dissonant quality betraying them. Only trough the work of the modern genre has there been a paradigm shift in regards to dissonance and the means in which music applies it.
So back to the question at hand. The second has a truly uneasy quality about it. It simply is too close to the root. The minor second, in particular, does not sound all that good. It sounds very much like a piano player playing two notes next to each other.
The third on the other hand starts to get better. The Major third is a particularly strong, consonant interval. The minor third also has a different quality but can also work very well in certain context. The quality of the third gives a chord its main character and is the most important part of the chord in a harmonic context
The fourth is a bit awkward. Although it does bear the title of perfect it to me sometimes sits awkwardly between the third and the fifth.
The fifth along with the thirds are the two intervals that form the consonant basis on which chords can be build. The diminished fifth or the devil's interval has a hauntingly beautiful character.
For a fair number of years, this interval was considered to have an evil character and was only used in composition when the devil or things that were considered evil, were composed about.
The Major sixth is another important interval as it has a good sound on stringed instruments. The Seventh is important in the way in which it with the tonic provides the finality on which music can end.
The Major seventh that forms part of the dominant chord of keys in both the harmonic minor and Major keys was the most important development the Baroque era developed and has been the basis of most music the west has produced for nearly five centuries.
The octave is good to provide notes sounding different, do not take away or give anything to the character of the chords. C Major chord still sounds Major regardless how many of the octaves, of the chord notes, are played.
Please have a look at this resource. https://www.sfu.ca/sonic-studio/handbook/Just_Tuning.html
It lists the frequency ratios in ascending order according to interval size. Dissonance and consonance are concepts we've used to understand the effect on our ear that frequency ratios of different complexities have. More complex frequency ratios like 15:8 (Major Seventh) will sound more dissonant, while frequency ratios that are simple like 3:2 (Perfect Fifth) will sound more consonant.
If you order the intervals according to the complexity of their frequency ratios (using Just Intonation system) you'll arrive at...
1:1 (unison); 2:1 (octave); 3:2 (P5); 4:3 (P4); 5:3 (M6); 5:4 (M3); 6:5 (m3); 8:5 (m6); ...
The last few are challenging, because different tuning solutions have been used to get around the dissonance involved in the m7, M2, A4, and D5. I personally finish the series off with:
9:5 (m7); 9:8 (M2), because the spaced-out interval of a m7 is more pleasing than a crunchy M2, and
15:8 (M7); 16:15 (m2); 45:32 (A4), because for simplicity I'd choose the Tritone (three whole-tones above/below the Tonic) which is consistent with the definition of A4 but not D5. It also sounds better
Wikipedia on: Just Intonation https://en.wikipedia.org/wiki/Interval_(music)#Frequency_ratios , and Frequency Ratios https://en.wikipedia.org/wiki/Interval_(music)#Frequency_ratios
I hope this helps. Also, as you can tell my order differs from posted one, in that I place the m7 before M2 in terms of consonance. Another reason to place the m7 first is that the compound m7 (m21) is earlier in the harmonic series than any note close to the compound M2.
From my experience with mathematics, computer science , signal processing and digital music production:
The other answers do well, provided we are considering the 12-tone western system of music.
However, if we are interested in wondering about ALL musical intervals...
Dissonance results when the roots of the sine waves of our complex signal don't have regularly repeating intersection, a.k.a. overlapping commonality, with each other.
Consonance results when there are shared roots between the waves. In other words, you could layer the waves over top of each other and you would see one of them fit into the other one quite harmoniously at recurring intervals.
Perhaps the simple human mind prefers the feeling of simplicity in recurring patterns?
By the way, this goes for sound design, melody design, life, and beyond.
This thread has been very helpful to me. Thanks, all. It's 5 years old (in 2022) and I imagine others are finding value in it as time goes on.
I tried an ordering based on the sum of the absolute values of the monzo primes. There was one tie which I broke with the count of the number of primes. The order for the first eight intervals through m6 agrees with most approaches and maybe offers an interesting approach to the final intervals.
name steps sum of primes monzo primes
abs val
Unison 0 1 1
Octave 12 2 2
Perfect 5th 7 5 3 -2
Perfect 4th 5 7 2 2 -3
Major 6th 9 8 5 -3
Major 3rd 4 9 5 -2 -2
Minor 3rd 3 10 3 2 -5
Minor 6th 8 11 2 2 2 -5
Minor 7th 10 11 3 3 -5
Major 2nd 2 12 3 3 -2 -2 -2
Minor 2nd 1 13 2 2 2 2 -5
Major 7th 11 14 3 5 -2 -2 -2
Tritone 6 21 3 3 5 -2 -2 -2 -2 -2
