By Bob Crites

The earlier discussion in this blog about the "Walden sound" was intriguing to me, particularly the part about stacking fifths.  I, too, am obsessed with stacks of fifths.  Musicianship class triggered much of this interest , but another early influence for me was the soundtrack of the original "Outer Limits" TV series from 1963-64, by Dominic Frontiere.  That is one of my favorite TV series of all time, and stacks of fifths are featured prominently in its music, including in the configurations that David Drucker referred to as the "Underwood chord".  Funny thing: Underwood was my dorm for a number of summers and I'm sure I did plenty of improvising with those chords -- I still come back to them all the time.  A 3-CD set of the Outer Limits music was released not long ago in case anyone is interested:

http://www.lalalandrecords.com/OuterLimits.html

Along with Frontiere's music, I also really like the music of Bernard Herrmann, who wrote the Twilight Zone theme and a bunch of excellent film scores, from Citizen Kane (1941) to some great Alfred Hitchcock movies [including North by Northwest (1959), one of my favorites, and side note: visiting Walden several years back, some of the students were creating their own soundtracks for this movie as an assignment, which I thought was really cool], to Fahrenheit 451 (1966) and Taxi Driver (1975).

Back to the topic of fifths: a stack of 7 fifths condensed into a scale is the Lydian mode (or the other modes in its family if one changes which note is the root).  At a Walden alumni gathering recently I overheard a student mentioning "pandiatonic clusters", a fancy name for this same idea I presume.
___

Does anyone remember the Pythagorean comma that we learned about in Musicianship  class?  12 stacked fifths is slightly larger than 7 stacked octaves, i.e., B# is slightly higher than C.  Recall that the ratios of frequencies in the overtone series are:
1 => octave => 2 => perfect fifth => 3 => perfect fourth => 4 => major third => 5 => minor third => 6 etc.
So the octave is a 2/1 ratio, perfect fifth is 3/2, etc.
12 stacked fifths is (3/2)^12.
7 stacked octaves is (2/1)^7.

If you do the math, B# is higher than C by a ratio of 531441 / 524288. Well, the Pythagoreans were really into ratios such as those found in the overtone series, and they looked at ratios geometrically using devices such as similar triangles.  Along with mathematics and music, they also had a mystical religion.  What little is known about their religion has mostly been pieced together from cryptic references by some of their contemporaries.  I've been reading a very interesting book with a chapter on this topic recently called Intimations of Christianity Among the Ancient Greeks, by Simone Weil.  She was a French philosopher who had some fascinating insights before her untimely death at age 34 (1909-43).  She analyzes some passages from Plato that shed light on the Pythagoreans.  The number 1 symbolized divinity for them.  Some other numbers had a particular bond with 1: squares, via a proportional mean; e.g. 1/3 = 3/9.  (Here 9 is the square and 3 is the proportional mean).  The Pythagoreans discovered that numbers that aren't perfect squares have irrational proportional means (irrational means they cannot be written as a ratio).  This was quite a shocking discovery for them!

The Pythagoreans had a saying "Justice is a number to the second power". Aristotle cited this obscure quote with disdain.  It does sound pretty wacky.  However, Weil actually makes sense of it:  Plato in the Theaetetus explains that Justice is assimilation in God.  If a number is to the second power, that means it is a square, and it can be related to 1 by its proportional mean.  Weil argues that the proportional mean of the Pythagoreans prefigures Christ's mediation between God and people.  Kind of a stretch, perhaps, but a fascinating one.

Weil points out that the Pythagoreans found that the musical scale does not contain the geometric mean, but is symmetrically disposed around that mean. The same geometric mean exists between a note and its octave as between its 4th and its 5th.  For example, if you take a 6 as the root note, the 4th above the root is 8 (4/3 ratio), the 5th above the root is 9 (3/2 ratio), and the octave above the root is 12 (2/1 ratio).  Then 6*12 = 8*9!  I found Weil's book fascinating because it simultaneously addressed my interests in music, math, theology, and philosophy.
___

One more thought related to the ratios in the overtone series:  When you stop and think about it, isn't it amazing that these ratios among frequencies sound the way they do?  Why should a 2:1 ratio sound to us like an octave, and a 3:2 ratio sound like a perfect fifth?  Each interval has its own characteristic sound -- but why do we experience them with these particular "flavors"?  Philosophers have a name for the raw feelings of experience, like seeing the color red, or hearing a perfect fifth:  Qualia. One age-old question is how similar or different are the qualia we each experience?  It is a hard question to get a handle on, since we are talking about 1st person experiences, and we can't get inside each other's heads. But I have a few questions that shed some light on whether our qualia are similar or not -- and they point in different directions:

(1) We all agree that notes an octave apart sound like the same note, just in a different register, right?  Whereas with the other intervals, this is not true.  Since we all agree on this, that would seem to be an argument in favor of us having similar qualia.
(2) As someone with perfect pitch, I find that all notes have their own recognizable character or qualia (like the various intervals).  However, people without perfect pitch are not able to recognize a note by its "character".  This seems to be an argument the other way: that we have different qualia.

Synesthesia is where people experience several different senses together, such as hearing sounds with colors.  I experience a version of this called "ordinal linguistic personification", where numbers have their own personalities.  I remember talking with another musician who also has this experience.  I wonder if it is particularly common among musicians?

Consciousness is a very perplexing phenomenon.  I enjoyed a book from 1769 on this topic recently: D'Alembert's Dream by Denis Diderot.  Highly recommended!

Photos by Marshall Bessières

Photo by Marshall Bessières 

Photo by Marshall Bessières

Photo by Marshall Bessières