# Under what arithmetic conditions do collections of vibrations sound "nice"

I can't recall where but I had read somewhere (different places and different times) the following two pieces of advice:

1. when playing an interval of two notes one after the other at A-hertz and B-hertz that the closer A/B is to a fraction of "small" integers, the nicer that interval will sound.

2. Intervals that sound nice, and in general scales that sound nice give rise to chords that sound nice.

Is there a way to create a rule (1) for chords of 3-or more notes? Like if I have a sequence of notes A_1 hertz, A_2 hertz, A_3 hertz... with the property that A_2/A_1 is close to some small fraction and A_3/A_2 is close to another small fraction etc...

Will the whole thing necessarily sound like a decent chord? Or does the cross note interaction like A_3/A_1 need to also be close to another small fraction.

Can this be stated in a different framework perhaps? (Like if A1, A2, A3 can be rounded with very little change to integers that share a very large Least Common Multiple then they will sound consonant together)

My motivation is exploring creating microtonal chords.

If you want to get into a purely quantitative view, probably skip the term "nice" and use "consonant" instead.

The idea is simpler frequency ratios are more consonant. So unison `1:1` is most consonant, next the octave `2:1`, then the perfect fifth `3:2`, etc.

IMO it doesn't take long before you get to the point where the quantitative doesn't really match up to music aesthetics. The perfect fourth would be next on the list at `4:3`, but in the Classical tradition it was treated as a dissonance. The sense of stability and consonance of thirds and sixths has a lot more to do with their placement in a key than any absolute, quantitative measure by ratio.

The other thing complicating a quantitative approach is the different tuning systems. The ratios above are all "just intonation" but today most everything is tuned to equal temperament where all intervals except octaves are slightly out of tune with "just." For example, in equal temperament perfect fifths aren't really an interval of `3:2` but they are consider to be such by the phenomena just-noticeable difference.

That's a complex question, but I would agree: generally, intervals close to small integer ratios sound more consonant (or perhaps less complex would be a more neutral way of putting it) than those of higher integer ratios. But primeness also plays a role. At least for me, the interval 8/15, the just major seventh, sounds more consonant than 8/11, one of the just "tritones", even though the lowest common denominator is smaller. There's also the factor of which note is at the bottom of an interval or chord. For instance, the just major chord in second inversion, 3/4/5, sounds less consonant (to me anyway) than in root position, 4/5/6, even though the smallest common denominator is the same.

Perhaps this could all be factored into a program that would yield a scale of relative consonance. But I suspect it would have to be rather complex, and it would of course not reflect personal and cultural ways of perceiving consonance.

• 8:11 isn't really a just “tritone” at all; it's right between a perfect fourth and a tritone. And “lowest common denominator” doesn't really make sense there, “common” amongst what? — IMO a 8:15 major seventh by itself sounds actually pretty dissonant, it only becomes consonant in a maj7 chord. Commented Sep 30, 2020 at 4:30
• @leftaroundabout - well, as I said, personal ways of perceiving consonance differ. Commented Sep 30, 2020 at 14:29

You should really provide sources for your "advice". I have never heard such things and I think it is safe to say that the second is plainly incorrect, and I can create chords from the major scale that sound like crap. Also, the closer two notes are the worse they will sound when played together, though I see that you are discussing consecutive notes in item 1. However, you then ask about chords (a little confusing), so I am focusing on that aspect.

On another note the question presumes an objective measure of "sounding nice" and that is subjective. Though on average most people agree with certain conventions regarding consonant and dissonant intervals.

To understand the phenomenon better requires delving a bit into physics and psycho acoustics. Related to the ratio of the frequencies of the two notes is a tower of harmonics and those harmonics can interfere even when the fundamentals are far apart. The ratio of the b5 to the 1 would be smaller than 5 to 1 yet considerably more dissonant, and a minor second is extremely dissonant. The key to understanding this is to look at whether or not the harmonics line up. In the case of the minor second the fundamentals are close and that creates beating in the combined waveform. Larger intervals can exhibit the same beating, for example a major 7th (an inverted minor 2nd) due to the proximity of other frequencies present. In the case of M7 the fundamental of the 7th is a half step away from the harmonic of the Root note in the interval. So, for all intents and purposes there are 1/2 steps in the spectrum and the ear hears it.

Even if you try to kill these harmonics the ear will create them because it is a non-linear system. This phenomenon is called aural harmonics. This essentially makes it impossible to hear a "pure tone". Our bodies are biologically designed to experience the harmonics sequence, f_n = n*f_1.

The intervals of M3 and P5 have many harmonics aligned and those that are not aligned are far enough apart to avoid the beating that contributes to dissonance. These intervals are generally considered consonant.

I would recommend looking at a text like "Physics and the Sound of Music" by Rigden. It covers all these topics and does not assume a science background.

When it comes to building chords there is more that choosing notes, the order of the notes contributes to the overall quality of sound and there are some rules of thumb in this regard. For example it is generally a good idea to have large intervals in the bass and smaller intervals in the upper voices. The reason for this has to do with the fact that the human ear has limited resolution for distinguishing notes and this resolution is a function of average frequency of the interval. It turns out that out judgement of "consonant" and "dissonant" are frequency dependent. In the bass register a 3rd can sound muddy and is sometimes judged as dissonant, in contrast in the soprano register an M2 can sound consonant.

As for micro tonal chords? I have been interested in such things too, more specifically micro tonal resolutions in chord movement. I use 1/4 tones frequently but it is usually in passing. The 12TET system actually does not adhere to the harmonic structure that our ears and brain are tuned to and there are more that 12 distinct tones in an octave that can be derived from harmonics, I've read about Just scales that have 17 notes in them. In such cases one can manipulate the intervals to play with the harmonics and perhaps create novel (or rediscover old) chords. However, if you are looking at micro-tone scales built up from equal temperament, like 24 tone chromatics etc. I think these will always produce more dissonant intervals than the standard scales we use. In the case of the 12TET major scale (i.e. diatonic scale built from tones in the 12TET chromatic scale) the difference is extremely small and most folks cannot tell that the harmonics are slightly off. In my opinion introducing more micro tones hurts in this regard more than it helps but that's just my opinion. Some of us enjoy the beating (pun intended) and dissonance becomes enjoyable. Other than the description of consonance and dissonance in terms of harmonics I have not seen a general rule of thumb for chord building that would generalise to micro tonal scales.

• "fraction of small integers" means something like 3/2 (pure fifth) or 6/5 (minor third), not the 18/17, 17/16 or more realistically almost irrational number that minor second is.
– ojs
Commented Oct 2, 2020 at 18:08
• A just minor second is NOT irrational. On the other hand ALL 12TET intervals are irrational.
– user50691
Commented Oct 2, 2020 at 18:49
• With "realistically" I meant that any tuning with reasonable amount of well tuned larger intervals isn't going to have just minor seconds.
– ojs
Commented Oct 3, 2020 at 9:30

Notes with low integer ratios between them will sound most consonant and pleasing and restful. The notes low in the harmonic series will naturally have this property because they are made by fitting a wave into a particular length of tubing or string an integer number of times. If we look at the key of C we would have the following harmonics

1. C
2. C
3. G
4. C
5. E
6. G
7. B flat
8. C
9. D
10. E
11. F (very sharp)
12. G

Octaves sound great, and always have a 1:2 ratio. Other ratios like 2:3 giving a perfect 5th (C and G) and 3:4 giving a fourth (G and C) and 4:5 giving a major third. A tritone though has a ratio of 5:7

The lowest three distinct notes (ie. Classing octaves as the same note) are C E and G and the most complex ratios are 3:5 and 5:6 between the two Gs and the E, forming a 6th and a 3rd, meaning that major chords will have the least tension of any triad.

A minor chord (in this harmonic series, G minor) requires the use of the 7:9 ratio and 6:7 ratio as well as the 2:3 (6:9) ratio and adds more tension.

The more distinct notes you add to the chord the more different ratios you will have, and the more likely it will be that you have many complex or large ratios, and potentially adding more tension and less consonance.