User:TallKite/Overshadowing
Overshadowing happens when a ratio falls too close to a much simpler (lower integer limit) ratio or ratios. An extreme example would be 201/100 being overshadowed by 2/1 and completely losing its identity. A lesser example is 11/9, which is hard to tune by ear, and hard to sing, because it's overshadowed by 6/5 and 5/4.
There follows excerpts from a private convo which one day I will clean up and turn into a xenwiki page.
I would classify 11/9 as dissonant because you've got interference beats from the 5th harmonic of the lower note and the 4th harmonic of the higher note missing by only 45/44 = 39¢, and also 6th of lower vs 5th of higher missing by 55/54 = 32¢. And these clashing harmonics are much louder than the 11th and 9th harmonics.
Basically, 11/9 is both a sharp 6/5 and a flat 5/4. This is a general issue with delta-2 ratios. 13/11 has even worse beating as a 7/6 off by 78/77 = 22¢ and 6/5 off by 66/65 = 26¢.
This also explains why 9/7 is so much rougher than ratios with similar sized numbers in them like 9/5 or 7/4. It's 5/4 off by 36/35 = 49¢ and 4/3 off by 28/27 = 63¢. IMO this is an ambisonance, right on the edge. 63¢ isn't so bad but 49¢ is pushing it.
Whereas 7/5 misses 4/3 by 21/20 = 84¢ and misses 3/2 by 15/14 = 119¢. Those two misses are more like semitones than commas. They are wide enough that you don't really get interference beats. And indeed 7/5 sounds quite consonant to me.
BTW the 2 nearby simpler ratios are the stern-brocot ancestors of the delta-2 ratio.
For delta-3 ratios:
- 8/5 = 3/2 + 16/15 and 5/3 - 25/24, totally fine
- 10/7 = 3/2 - 21/20 and 7/5 + 50/49 = 35¢, pushing it
- 11/8 = 4/3 + 33/32 and 7/5 - 56/55 = 31¢, really pushing it
- 13/10 = 4/3 - 40/39 and 9/7 + 91/90 = 19¢, ouch!
Note that the ancestor with much smaller numbers and hence much louder harmonics is always further away, making its impact less. As we go farther, it gets close enough to totally wreck the ratio. 16/13 = 5/4 minus 65/64 = 27¢!
So by this metric, what are some good 11-limit ratios?
- 11/7 = 3/2 + 22/21 and 8/5 - 56/55 = 31¢, dissonance
- 11/6 = 2/1 - 12/11 and 9/5 + 55/54 = 32¢, dissonance
- 11/5 = 2/1 + 11/10 and 9/4 - 45/44 = 39¢, dissonance
- 11/4 = 3/1 - 12/11 and 8/3 + 33/32 = 53¢, ambisonance
- 11/3 = 4/1 - 12/11 and 7/2 + 22/21, consonance
- 11/2 and 11/1 are consonances of course
So my theory is that if a just ratio has lower, louder harmonics missing by something comma-sized, it's dissonant, and very hard to tune by ear. (Unless you memorize the melodic size, which arguably is an example of an overeducated ear.) If the lower harmonics miss by something quartertone-sized, it's an ambisonance. If it misses by something semitone-sized or greater, it's a consonance and easily tunable by ear. And it just so happens to work out that most of the consonances are 7-limit.
Now this is all very neat and tidy, but music is an art, not a science. Music breaks the rules of math all the time. So I'm not saying that 11-limit music is invalid or anything. But I do think that we can say that 11-limit ratios tend to be pretty hard to tune precisely by ear using interference beats. (No fair using square waves!) Partly because of this analysis I just did, and partly because the 11th harmonic is usually pretty faint. Whereas the 7th harmonic is louder and more obvious. But a 7-limit ratio like 14/9 (3/2 + 63¢, 8/5 - 49¢, 11/7 - 18¢) would be likewise "untuneable". And "tuneability" matters when writing vocal harmonies. Not so much for big choruses with their inherent, um, chorusing. More for say a barbershop quartet or quintet.
Now I'm only analyzing intervals here, not complete chords. And a chord can have one dissonant interval and lots of consonant ones and sound very nice, e.g. 8:10:12:15. But let's look at say 7:9:11, probably the simplest 11-limit triad. What do the 3 intervals miss by? 9/7 by 49¢, 11/9 by 39¢ and 32¢, and 11/7 by 31¢. That's a lot of clashing harmonics! But on the other hand, the difference tones probably help a lot. So who knows?