transforming the circle of fifths

Yesterday we looked at the circle of fifths (COF) and regarded how some scales and intervals looked when displayed on that circle. But as interesting as the COF is, it isn't the only way to display all 12 pitch classes: another intuitive way to do so would be to arrange the pitches chromatically. Here's how a circle of minor 2nds (COM2) looks:


It might not be obvious at first glance, but there's an interesting relationship between the COM2 and the COF. To start to see it let's look at how the whole tone scale -- or major 2nds -- looks when displayed on the COM2:

And let's recall how the same collection of tones looks on the COF:

They're the same shape! The two circles differ only by the tones not a part of this collection, and they differ in a very systematic way. The easiest way to see the pattern is like this:

that is, the tones not a part of the major 2nd collection simply swap places with their tritone.

Another way -- a more arithmetical way -- to understand this transformation is to use a formula:

1. Convert all the tones to numbers as in 12-tone music (C=0, Db=1, D=2 and so on).
2. Multiply all tones by 7 mod 12 (i.e. multiply by 7 and if the number is 12 or bigger subtract 12: if it is still bigger subtract 12 again, and so on until the number in question is less than 12).

Some examples:

C [0 M7 mod 12 = 0 x 7 = 0 =] C
Db [1 M7 mod 12 = 1 x 7 = 7 =] G
D [2 M7 mod 12 = 2 x 7 = 14 -12 = 2 =] D
A [8 M7 mod 12 = 8 x 7 = 56 (mod12) = 8 =] Eb

This multiplicative operation is a way of extending the basic operations of 12 tone music (retrograde, inversion, inversion retrograde), but here we have a graphic display.

Lastly we can see how some scales look when displayed on the COM2. Here's a major scale:
which is still a symmetric shape, though far different from the one it takes in the COF.

And here's how perfect 5ths look:

This shape is more visible evidence of a strong correspondence between the two circles under question...if this shape is mapped onto a COF one then has a chromatic scale:

circle of fifths and scales

We touched on the circle of fifths before in this post. The circle of fifths is just that: notes seperated by a fifth (C - G -D - A, etc) and arranged in a circle because (in equal tempered tuning, at any rate) the fifths lead back to the starting point after all 12 tones have been accounted for (...Eb - Bb - F - C).

The circle of fifths (henceforth COF) has many uses, but one I've been playing around with lately is examining how scales look -- i.e. what shapes they take when the collections of notes are joined one to another as in the sequence of a scale. E.g. here's a whole tone scale:

And a chromatic scale:

Not surprisingly the 2 scales above take symmetric shapes when graphically displayed (the scales are symmetric in terms of their construction: comprised of 1/2 steps or whole steps). Surprising -- to me, at any rate -- is that when the major scale is displayed it also forms a symmetric shape:

As does the melodic minor.

Some asymmetric shapes: the harmonic minor:

and the neapolitan minor:

You can do these on your own, of course. Some other symmetric scales: the neapolitan major, the major pentatonic, the double harmonic. Some asymmetric ones: the hungarian minor and the harmonic major.

Lastly we can observe intervals -- actually we've already done this with the chromatic and whole-tone scales (min2 and Maj2 respectively). Here in one diagram are min3, Maj3 and P5:

The remaining intervals can all be found simply by going the other way round: C to Ab can be seen as a Maj3 down or as a min6 up. C to F is a P5 down or a P4 up, and so on...

naked women at goodbye blue monday TONIGHT

Yes, TONIGHT a pared-down quintet manifestation of Naked Women return to the stage at Goodbye Blue Monday.

The lineup:

T-Bone Blatt
bass
Matthew Polashek
sax
m'tazz
guitar
Father Todd
'bone + words
Zebra aka "Tiger"
drums

The show is FREE and starts at 9pm.

creepy chords addendum

I was reading through Ponce's fabulous Guitar Sonata no. 3 last night and realized that in measures 39 and 40 (of the first movement) there is the same chord progression as mentioned in the post on creepy chords. Ponce is in Bb, but the chords qualities are the same: Major 7th +5 and a Minor(Major) 7th:



In this context I wouldn't say that the chords are "creepy", though they are tense. More creepy -- or at least menacing -- are the chords immediately preceding the 2 above (actually starting in measure 35: the structure is repeated):


Here we're really only dealing with one chord per measure: Ponce simply "flips" the outer voices (the 7th and root exchange places) giving him a very nice stepwise-ascending bass line.

And lest we think that these are just contemporary harmonic structures, or something born of jazz, keep in mind that this piece was written in 1927 -- and it's probably a safe bet that Ponce didn't invent this all on his own, so one might be able to trace it back at least to Debussy (maybe even Liszt???)...worthy of investigation.