This track is constructed from all chords using the numbers 1 to 6 which meet certain criteria. Using whole numbers to construct harmony is 'Just Intonation'. Both the otonal chord (a, b, c) and the utonal chord (1/c, 1/b, 1/a) were used.

This track is intended to help composers to familiarise themselves with the chords available in 5-limit harmony.

For example, (3, 4, 5) is a (G, C, E') major triad in the second inversion. It is otonal, and its utonal counterpart is (1/5, 1/4, 1/3) = (12, 15, 20) which is (G, B', E') minor triad in the first inversion.

The rules for including a chord (a, b, c) were:
  a < b < c
  gcd(a,b,c) = 1
  4/3 <= c/a <= 4/1
  chord is otonal, which is (abc)^(1/3) < lcm(abc)^(1/2)

The final otonal harmonies found were:
1	2	4
1	3	4
1	2	3
2	5	6
2	3	5
2	4	5
2	3	4
3	5	6
3	4	5
4	5	6

The corresponding utonal harmonies were:
1	2	4
1	3	4
1	2	3
2	5	6
2	3	5
2	4	5
2	3	4
3	5	6
3	4	5
4	5	6
(note that 1 2 4 does not have a utonal counterpart)

Each harmony was used 3 times, by multiplying each chord by 6, 8 and 9, giving versions in keys of C, G and D.

They were then arranged in:
  descending order of MaxRatio = c/a
  ascending order of bass note 'a'
  descending order of middle note 'b'.

I hope that you find this track useful! Dave.

PS apologies for the lack of 11th and 13th harmonics, however those were not part of the construction schema :/

------------------------

See also:
- https://arxiv.org/a/ryan_d_1 (arXiv) for a list of my academic paper pre-prints, including my writing on music theory and Just Intonation
- http://orcid.org/0000-0002-4785-9766 (ORC ID) for my academic profile
- https://www.linkedin.com/in/davidryan59 for my LinkedIn profile

Example track of 3 note harmonies in Just Intonation constructed from the numbers 1 to 6

most of the music is straight from tiktok.  the other music is from a friend

I'm into Just Intonation (JI), which is using harmonics (or undertones) to generate music. Over time, my profile will fill up with wonderful and melodious New Harmony, fresh from the infinite vaults of the original and best source of harmony, Just Intonation.

I’ve written some papers on the music theory of Just Intonation. You can check them out via the arXiv link below. Topics include the mathematical analysis of harmony (in Just Intonation), and a notation system to cover all of Just Intonation.

Enjoy listening!

a little bit of everything  including  Christmas  music 

Example track of 3 note harmonies in Just Intonation constructed from the numbers 1 to 6

David Ryan