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Monday, October 22, 2018

Rule of nine

I learned about this a few months ago.  

The fourth and fifth are the same interval inverted. So a fifth up from C is G, and a fourth down is also G. 

(4+5 = 9).  

The tritone up or down is the same same. You can think of that as four and a half.   

(4.5 + 4.5 = 9).  

So thirds and sixths, and seconds and sevenths, will also be the same. But the minor third, in the other direction, is a major sixth, and vice versa. The same relation holds with major and minor sevenths and seconds. So C up to B is major seventh, and C down a half step to B is a minor second. This makes sense if we think of the major as +.5 and the minor as -.5.   

It's one of those nice little symmetries, like so many in music theory.  

10 comments:

Vance Maverick said...

If you remember any high-school math, it's probably easier to think in terms of modulo arithmetic and the zero. The interval from the first degree of the scale to the same degree is called a "unison", i.e. a "one". But this reflects the way we thought before the zero was introduced. In the terms you learned in math class, the scale degrees (in the modes) are 0, 1, 2, 3, 4, 5, 6, modulo 7. A "fourth" is a difference of three steps, and a "fifth" is a difference of four steps -- 3+4 = 7 = 0 modulo 7. In this light, it's the rule of 7.

Consider that a third plus a third plus a fourth makes an octave. Using the traditional nomenclature, that's 3+3+4, or 10, which breaks the "rule of 9". But if we express this using zeroes, it's 2+2+3, or 7, which still works correctly.

(If you get seriously chromatic, it's all arithmetic modulo 12. A "fourth" is 5 semitones, and a "fifth" is 7 semitones, which add up to 12. A stack of twelve fourths is 60 semitones, or 5 octaves.)

Jonathan said...

That works too, but I am really comfortable with thinking in stacks of thirds right now.

el curioso impertinente said...

The last suggestion makes sense, given the ultimate basis in twelve-tone equal temperament. Exploring just intonation with a keyboard may be more difficult, but who knows what they can do with electronics these days.

Vance Maverick said...

There's no incompatibility between stacks of thirds and modulo-7 thinking. A "third" is a 2 in this system, and everything works out fine. A ninth chord on the tonic has pitches 0-2-4-6-8 (8 being the "ninth" -- modulo 7 it's 1, equivalent to the second degree of the scale).

Another example of the goofiness of the common system -- my daughter came to ask me what "15ma" means as a label on a line above the staff, though she was already familiar with "8va". Wouldn't it be clearer if one "octave" was labeled 7 and two octaves as 14? (In Spanish, do people use "in 15 days" to mean "in two weeks"? I think they do in French.)

As for intonations, many keyboards allow you to set special tunings, expressed as A=435, B=483.3, etc. The problem is that such a tuning can give you "just" intervals only for certain combinations of notes. But this doesn't really have much to do with the arithmetic of steps and half-steps (it's about their realization in frequencies).

Leslie B. said...

This is because the octave is considered 8, right?

Note that 2 weeks is 15 days but 3 weeks is 20. 1 week is 8 days, "de hoy en ocho."

Anyway, 9 is a weird number, there is a way of checking arithmetic, "casting out nines" that seems mystical somehow to me

Jonathan said...

I know it's not incompatible, but what I'm saying is that I am already thinking of each interval in a certain system. It's taken me a long time to know automatically what a sharp 9, is, say. Knowing the relationship of every note to every other note in 12 keys is taking me a while. I can also understand what other musicians or music books are saying when they use the same language as I do.

I'd have to rewrite all the fake books and even the chord software I use on irealpro.

Leslie B. said...

There is something really yogic about knowing this, though, being that in tune. I'd like to have a piano now.

Vance Maverick said...

Right, the octave is named for "eight", because you can look at it as "a span of eight notes". (From Monday to Monday is eight days if you include both ends.)

And yes, the mnemonic value of the conventional terms like #11 would be lost if you switched to this notation. Most of these names, though, reflect a focus on one analysis or etymology of the chord -- is it really fundamental to think of the eleventh in terms of a stack of thirds? Or might it be better conceptualized, sometimes, as an added-note chord, i.e. a dominant of some sort plus a raised fourth? Without seriously advocating a French Republican Calendar-style renaming of the world, I would like to suggest a detached attitude towards the names we have.

Jonathan said...

Maybe that odd-numbered system is what makes it hard to learn in the first place. Hence my reluctance to let it go! The way I remember a sharp 9 is a minor third an octave up, but that's cumbersome in a way, I agree. This comes from my jazz training where every chord extension has an odd number up to 13.

Vance Maverick said...

And confusingly, the number 6 is sometimes used for the same interval (I think for chords that lack dominant flavor, i.e. sevenths).