Why notes of the harmonic series sound good together
From Bounce Metronome
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If you play two notes of the harmonic series, particularly the ones lower in it, they will go well together. Intervals between notes of the harmonic series sound particularly pure.
To go into a little more detail - when one plays two successive notes of the harmonic series, the ear also picks out the lowest note of the series as the difference of the two pitches. Then the two notes, and the difference tone, all sound together without beats, so you get a particularly sonorous chord, especially for notes lower in the series.
When one plays two non consecutive notes, the same thing happens, only this time the difference tone will be one of the higher notes in the harmonic series. The notes both belong to the harmonic series of their difference tone (if a note belongs to a harmonic series, so also do all the notes in its harmonic series).
And in fact, the whole harmonic series played as a single chord also sounds pretty good:
[help_files/harmonic_series_as_chord.ts harmonic_series_as_chord.ts] (played on pan flutes, getting faster and quieter as it gets higher)
The clip goes up to the 30th harmonic, which is as far as one can go using the 15 simultaneous pitch bends available in MIDI (one for each melodic channel).
New pitch bends are needed for the fundamental, and all the odd numbered harmonics:
1 3 5 7 9, 11 13 15 17 19 , 21 23 25 27 29,
so the 31st harmonic brings the number of simultaneous pitch bends needed to 16, and one of the earlier notes in the harmonic series would need to be switched off at that point.
Just for fun, here are the first five hundred members of the harmonic series, played on pan flutes again, with each note sounding to overlap the next twelve. Notice how they seem to merge together to make almost a single note.
[help_files/harmonic_series_500_as_chord.ts First five hundred harmonics as drifting chord]
First note is C0, or C - the C two octaves below the base clef (one octave below lowest note of double bass).
The blue lines are positioned just above the note played (like the "ties" of tied notes). They show how long the note sounds for.
Towards the end the notes may go beyond the range of your pan flute voice, though they remain below the theoretical highest MIDI note of 127. Highest note in the clip is 41.0645 cents below c''''' (C9, midi note number 120). The difference in pitch between last two notes is 3.4668 cents.