11th or 13th harmonic in a progression
I know the dominant seventh chords include a 7th harmonic and there are progressions using them. What about utilizing higher harmonics? Maybe an add6 chord implies the ratio 13/8 for some certain places?
For instance in
i - bIII - IV7 - iv(add6) progression, I suspect the added 6th can serve as the 13th harmonic of the fourth. I lowered the major second of scale to 13/12(at about 138 cents) and it sounds natural to my ears(maybe cause i used to the middle eastern music :) . Also, it seems lowering the minor third degree about 10 cents results a better dominant seventh experience.
Not directly related with the specific progression above but since the topic is in general about the usage of higher harmonics, here is an arrangement of a piece originally written for oud, arranged by professional Turkish classical guitarist Celil Refik Kaya. He says in the description that he transposed the piece to tonic B and the C is bent up about 40 cents. I think that bend is to nail the 13/12 ratio.
I strongly recommend anyone to watch the performance and listen to this beautiful piece.
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Starting from a just iv chord, with a 6:5 minor third and a 3:2 fifth, consider the other intervals that arise when you add the sixth. A just major sixth would be a ratio of 5:3 above the root; a just minor sixth would be a ratio of 8:5; and you are asking about a ratio of 13:8.
The ratio between the 5:3 sixth and the third is 25:18; the ratio with the fifth is 10:9. The first interval is the inversion of the tritone in a just dominant seventh chord (1:1, 5:4, 3:2, 9:5), so it is somewhat dissonant, but should be tolerable. The 10:9 whole step is of course quite common in justly-tuned music.
The ratio between the 8:5 sixth and the third is 4:3, a perfect fourth, so it is far more consonant. The ratio with the fifth in that chord is 16:15. This is dissonant, but it is a common and pleasant dissonance in just intonation.
By contrast, the 13:8 sixth has a ratio of 65:48 with the third and 13:12 with the fifth. Not only is the first ratio quite distant from the land of simple ratios, but it is also 26.8 cents sharper than a perfect fourth. To my ear, it just sounds wrong: it's not far enough from a perfect fourth to sound like a tritone, but it's much too far away to sound consonant.
If I could hear these intervals in the context of middle eastern music, I might change my tune a bit, as it were. I don't have much experience with such music, but what I am familiar with uses interesting tuning as a feature of melodic expression, not so much harmonic. Perhaps the 13:12 ratio is part of its tonal landscape. If that is the case then this tuning might be well suited to such music.
Once you get above the sixth note in the harmonic series things start getting pretty badly out of tune with an even tempered scale, so dominant seventh chords don't really include the 7th note in the harmonic series. Even though textbooks may identify that note as the b7 of the fundamental, they only do that because it's the closest note - the 7th note of the harmonic series is actually a little more than 2/3 of the way from the major 6th to the b7th. It's flat by 31 cents.
That applies to the 11th and 13th as well. The 11th in the harmonic series falls almost exactly halfway between the 4th and the #4 /b5, and the 13th is a little bit more than halfway from the b6 to the 6th.
Having your ears accustomed to a non-Western temperament (like middle Eastern music) means they might not be as jarring to your ears as they would be to a European or American listener... but they're still so far from our Tertian system of harmony that they're not really going to work for building chords.
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