Quoting David Doty from Just Intonation Primer (highly recommended if you can find a copy):
"7:4, 7:5, and 7:6 can be identified with the flatted seventh, fifth, and third beloved of blues and jazz musicians". He may not be the most well known person to write about music theory but his understanding of the fundamentals of music/tuning/harmony is solid. Anyway, it is fairly self evident that the blues scale would not sound the way it does if it didn't have these three intervals (the others just being the fourth, fifth, and octave, which is about as basic as you can get)
As an aside, the flat third certainly has a different representation in 5-limit tuning (6:5) but both this as 7:6 are approximated by the same interval in twelve tone equal temperament.
In the section 'Approximation using equal temperament' you can see the tritone represented as 7:5.
To be more accurate with my wording, its not that the 'tritone is 7:5', but rather what we call a tritone in 12 tone equal temperament is really an approximation of the interval 7:5. Look at the 7th and 5th notes in a harmonic series, which interval do they most closely resemble?
The most basic and literal form of an interval is always the smallest possible whole integer ratio that can be used to represent that interval, in this case, 7:5.
As you mention, you can arrive at something similar by going up 6 perfect fifths, but the ratio ends up being 729:64 -> 89:64, if you take out the octave. The only way this would have musical significance is if its used as an approximation for 7:5.
Scales that are built by stringing perfect fifths do not produce a clean 'tritone', hence its relative shunning in classical music (and its absence in 5-limit tuning where the interval you mention is more of a byproduct).
It sounds like you are writing from the point of view of twelve tone equal temperament tuning as the standard by which intervals are defined, whereas I'm starting from the concept that the harmonic series is the basis of harmony and (most) tuning systems are just attempts to emulate it.
Hi :-) Thanks. I'm no expert in rational temperaments, I just am objecting when you say things that don't sound true. Like "jazz is said to approximate the scales that could be built off the '7-limit' intervals, which explains its more dissonant nature". (that explains nothing) Or "The only way this would have musical significance.." (only ratios of small integers can 'have musical significance'?)
"look at the 7th and 5th notes in a harmonic series" - I played trombone for years. :-) The 7th is flat, sounds out of tune. (Yes I was mostly playing jazz)
I don't know why you think I'm writing from the point of view of equal temperament as opposed to harmonic series. But it does sound like you're writing from the point of view of a particular rational temperament - maybe because you just read that book. There are many, dozens or hundreds, of different rational tunings/temperaments/intonations, writing today 'from the point of view' of any one of them about music, as if that's what music 'really' is, would be arbitrary, silly.
As an aside, the flat third certainly has a different representation in 5-limit tuning (6:5) but both this as 7:6 are approximated by the same interval in twelve tone equal temperament.
Also: https://en.wikipedia.org/wiki/7-limit_tuning
In the section 'Approximation using equal temperament' you can see the tritone represented as 7:5.
To be more accurate with my wording, its not that the 'tritone is 7:5', but rather what we call a tritone in 12 tone equal temperament is really an approximation of the interval 7:5. Look at the 7th and 5th notes in a harmonic series, which interval do they most closely resemble? The most basic and literal form of an interval is always the smallest possible whole integer ratio that can be used to represent that interval, in this case, 7:5.
As you mention, you can arrive at something similar by going up 6 perfect fifths, but the ratio ends up being 729:64 -> 89:64, if you take out the octave. The only way this would have musical significance is if its used as an approximation for 7:5.
Scales that are built by stringing perfect fifths do not produce a clean 'tritone', hence its relative shunning in classical music (and its absence in 5-limit tuning where the interval you mention is more of a byproduct).
It sounds like you are writing from the point of view of twelve tone equal temperament tuning as the standard by which intervals are defined, whereas I'm starting from the concept that the harmonic series is the basis of harmony and (most) tuning systems are just attempts to emulate it.