User:TallKite/Overshadowing

Clashing happens when a ratio falls overly close to an overly simpler (lower integer limit) ratio or ratios. An extreme example would be 201/100 clashing with 2/1 and completely losing its identity. This is known as overshadowing, but the "clash region" of a ratio is generally larger than its shadow. For example 11/9 is hard to tune by ear, and hard to sing, because it's clashing with both 6/5 and 5/4. But it's not overshadowed by either because it retains its identity as a distinct interval.

I would classify 11/9 in isolation as dissonant because there are interference beats from the 5th harmonic of the lower note and the 4th harmonic of the higher note missing by only 5/4 ÷ 11/9 = 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 coinciding 11th and 9th harmonics.

Thus 11/9 is both a sharp 6/5 and a flat 5/4 (the two stern-brocot ancestors of 11/9). This is a general issue with all but the smallest delta-2 ratios. 13/11 has even worse beating as a 7/6 off by 78/77 = 22¢ and a 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¢ is tolerable 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.

Clashing is based on integer limit not odd limit, so voicing matters. For example, 19/16 is a clashing 6/5, but 24/5 is a clashing 19/4.

A delta-2 ratio always has stern-brocot ancestors with about half the integer limit, so the clash is pronounced. A delta-3 ratio has one ancestor with less than half the integer limit and one with more. The two integer limits of the ancestors always add up to the integer limit of the ratio, by definition.

• 8/5 = 3/2 + 16/15 and 5/3 - 25/24 (71¢), fine
• 10/7 = 3/2 - 21/20 and 7/5 + 50/49 (35¢), pushing it
• 11/8 = 4/3 + 33/32 (53¢) and 7/5 - 56/55 (31¢), really pushing it
• 13/10 = 4/3 - 40/39 (44¢) and 9/7 + 91/90 (19¢), dissonant

Note that the ancestor with the smaller integer limit and hence louder harmonics is always further away, making its impact less. As the integer limit increases, this ancestor gets close enough to totally wreck the ratio. 16/13 = 5/4 minus 65/64 (27¢)!

Example delta-4 ratios:

• 9/5 = 7/4 + 28/27 and 2/1 - 10/9, consonant
• 11/7 = 3/2 + 22/2 (81¢) and 8/5 - 56/55 (31¢), small comma but also small difference in integer limit, so only somewhat dissonant
• 13/9 = 10/7 + 91/90 (19¢) and 3/2 - 27/26 (65¢), fairly dissonant
• 15/11 = 4/3 + 45/44 (39¢) and 11/8 - 121/120 (14¢), quite dissonant

Delta-1 ratios tend to not clash because the closer stern-brocot ancestor has an integer limit smaller by only 1. For example, 8/7 = 7/6 - 49/48 (36¢), but the 6th and 7th harmonics are not much louder than the 7th and 8th harmonics, so the 6th-7th clash doesn't obscure the 7th-8th coinciding. However, a small delta-1 ratio like 81/80 clashes with its simpler stern-brocot ancestor, 1/1.

So by this metric, what are some consonant 11-limit ratios?

• 11/9 = 6/5 + 55/54 (32¢) and 5/4 - 45/44 (39¢), double dissonance
• 11/8 = 4/3 + 33/32 (53¢) and 7/5 - 56/55 (31¢), double dissonance
• 11/7 = 3/2 + 22/21 (81¢) 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 (81¢), consonance
• 11/2 and 11/1 are consonances of course

So my theory is that if a just ratio has sufficiently lower/louder harmonics missing by something comma-sized, it's in isolation 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 (e.g. 5/4 is 4/3 - 16/15). 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 11-limit ratios in isolation 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. So we can use a clashy interval in a chord, but only if the other chord notes provide enough "glue" to help the chord hang together.

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?