As I understand, harmony in music can be achieved either through melody / chord — when the frequencies of the sounds are in phase sync or through the rhythmicality (the same rhythmical pattern that repeats itself).
Can we say in this case that harmony is when things repeat themselves, at least from the mathematical perspective?
Can we say in this case that harmony is when things repeat themselves, at least from the mathematical perspective?
No, we cannot say that. It is incorrect:
Harmony - Alan Rich - ENCYCLOPÆDIA BRITANNICA
Harmony, in music, the sound of two or more notes heard simultaneously. In practice, this broad definition can also include some instances of notes sounded one after the other.
If the consecutively sounded notes call to mind the notes of a familiar chord (a group of notes sounded together), the ear creates its own simultaneity in the same way that the eye perceives movement in a motion picture. In such cases the ear perceives the harmony that would result if the notes had sounded together. In a narrower sense, harmony refers to the extensively developed system of chords and the rules that allow or forbid relations between chords that characterizes Western music.
Nothing to do with repetition.
the same rhythmical pattern that repeats itself.
Repetitive rhythmic patterns are not related to harmony in our terminology. It's called rhythm and sometimes meter,depending on the context.
Metre (British Spelling) - ENCYCLOPÆDIA BRITANNICA
Metre, also spelled Meter, in music, rhythmic pattern constituted by the grouping of basic temporal units, called beats, into regular measures, or bars; in Western notation, each measure is set off from those adjoining it by bar lines. A time (or metre) signature, found at the beginning of a piece of music, indicates the number of beats in a measure and the value of the basic beat. For example, 3/4 metre has three quarter-note beats per measure.
Quoting @topoMorto in his excellent answer:
In English, the word 'rhythm' refers to something like a drum beat that repeats over a much longer period than an audio waveform. Our auditory system processes this in a different way, so we don't call this 'harmony'.
The other answers do a good job of outlining why harmony is generally not the same thing as rhythm.
I wonder, though, if your question was provoked by videos such as these:
If you haven't seen them, the presenter (Adam Neely) uses repeating patterns of notes, and speeds them up to produce a tone. If you have multiple patterns repeating at the right ratios, you get harmony.
This works because of how sound works. You can consider any repeating rhythm as having a frequency; that is, the number of times it repeats per second.
If a rhythm is fast enough (say 20 repeats per second, also called 20 Hertz), we start to perceive it as a sound. If you keep speeding it up, you hear it as a higher sound. We can say that it has higher pitch.
Now, if you have two different "rhythms" (or repeating patterns), and their frequencies have some ratio between them, we hear it as a harmony. The type of harmony (or chord) depends on that ratio.
So, if you speed a rhythm (or any repeating pattern) up past the point at which we perceive it as a rhythm, you do get an audible tone. You can use that to produce harmony. But, at that point, it's not really a "rhythm" anymore. At least, not in the conventional sense of the word.
What relation does harmony have to repetition?
A rather tenuous one, I think!
As I understand, harmony in music can be achieved either through melody / chord — when the frequencies of the sounds are in phase sync
Hmmm... that's not 'wrong' exactly... - consonance in music occurs when we put together sounds with simple ratios between their frequencies. You could see this as a kind of phase correspondence, but actually, as mentioned in my answer to this question, when 'looking for' harmony, the ear doesn't listen for phase relationships in the time domain, but pitch relationships in the frequency domain. And it is able to find consonances even when there isn't an exact 'phase lock'. So I don't personally find this the most helpful way to think about it.
or through the rhythmicality (the same rhythmical pattern that repeats itself).
In English, the word 'rhythm' refers to something like a drum beat that repeats over a much longer period than an audio waveform. Our auditory system processes this in a different way, so we don't call this 'harmony'.
Can we say in this case that harmony is when things repeat themselves, at least from the mathematical perspective?
The human ear is cleverer than just looking for repetition - it can do a frequency analysis of a waveform with very little repetition, and find bands of energy that have something close to simple ratio relationships with those in another waveform that itself has very little repetition. Or to put it another way, the ear can find 'harmony' in some really messy collections of sounds, because it can find pitch in very messy waveforms. Here's an example of a single pitched wave that the ear can definitely get a sense of pitch out of - It's brass_acoustic_006-032-075.wav from nsynth-test at https://magenta.tensorflow.org/datasets/nsynth.
This is only the start of the note - it settles down and looks a little more periodic later. And yet even from this messy bit of note (a tenth of a second or so), the ear can pick out a definite sense of pitch, which could have a harmonic relationship with other waves.
But - can you see any repeating pattern in the wave? Maybe you kind of can, but it's very unclear. This goes back to what I said before - the ear works in a cleverer way than just looking for obvious repeating patterns. Then again, you could just say that it's very good at finding repeating patterns at levels that aren't immediately obvious.- It really comes down to the fact that you can look at audio from a time or frequency domain (again, as mentioned in the question What causes consonance in music?).
