"Negative" -- that is, directionality -- could be in the system of identifying intervals.  If  you go from Bb to the next upward C#, and then to the next upward Bb, shouldn't the sum of the two intervals be an octave?  Or at least a unison?

To get from context of Cb to context of B# would be 19 perfect fifths.

Cb Gb Db Ab Eb Bb F C G D A E B F# C# G# D# A# E# B#

quote:

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Originally posted by Alex Z

Visually, the first perception is that the starting B and the starting C are right next to each other.

Suppose they are not?  Suppose the starting B is just below middle C and the starting C is an octave above middle C.

Then sharp the B, flat the C, and what is the interval in between?

Or suppose the B is above the C, with middle C the starting note and B the first B above middle C.

Then flat the C, sharp the B, and what is the interval in between?

A sytem for figuring intervals with sharps and flats should be able to work on any two notes.  What that system might be, I don't know.  

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In contemporary nomenclature, any mixing of sharps and flats tells you it's a diminished/augmented interval of some sort (as opposed to minor/major). 

B to C in the octave above is a minor 9th. B to Cb is a diminished 9th. B# to Cb is therefore a double diminished 9th (yes, that is a real term). The inverse of this interval, Cb to B#, is a double augmented 7th. The somewhat confusing thing is that a double diminished 9th sounds like a major 7th, whereas a double augmented seventh sounds like a minor 9th. 

The existing system is actually very comprehensive, it just doesn't account for "negative" intervals (or maybe it does, but you'd never see it in practice). 

Makes sense. 
 

Start with a double augmented 7th and add a double diminished 9th on top, and you get two octaves. 
 

What can be added to a double augmented 7th to get one octave?

Can C to Cb be called diminished octave?

Alex - That's the problem; you can't reduce an interval by adding to it because all intervals are assumed to have a positive value. In a one-octave framework, the inverse of a double augmented 7th is technically a double diminished 2nd. But the latter interval (and it's enharmonic equivalent, the diminished unison) don't actually exist because they have a negative value (i.e., the "upper" tone is below the "lower" tone). All we can do is un-augment the double augmented 7th once or say that we are reducing it by a semitone/m2.

Jan - C to Cb can be a diminished octave (sounds like a major 7th), but it cannot be a diminished unison.

Diminished Unison - that would be a good name for a progressive Bluegrass group!

Only positive intervals.  No negative.  Makes sense.  Thanks, Mr. banjoboyd.

It took several centuries for negative numbers to be accepted and used in mathematics, other than simply to to take away something small from a larger pile.  And that was around the time we call the beginning of the  "common practice period" in music.  Negative numbers allowed you to take something large from a smaller pile

Maybe 500 years from now, maybe in a parallel universe, people will be hearing what a "negative interval" sounds like.  

Thanks.

quote:

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Originally posted by Alex Z

Only positive intervals.  No negative.  Makes sense.  Thanks, Mr. banjoboyd.

It took several centuries for negative numbers to be accepted and used in mathematics, other than simply to to take away something small from a larger pile.  And that was around the time we call the beginning of the  "common practice period" in music.  Negative numbers allowed you to take something large from a smaller pile

Maybe 500 years from now, maybe in a parallel universe, people will be hearing what a "negative interval" sounds like.  

Thanks.

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The introduction of negative numbers was simultaneous with the introduction of debt. One caveman to another,"You owe me two rocks."

Upon further consideration, this would have been a question for Carlton Haney.

There's a large field of mathematics which revolves around imaginary numbers, and which has applications to all sorts of physics and engineering. Just wait until composers and banjos discover imaginary numbers...