User:David Z Kasler

Originally a music student at Ohio State University, I now hold an MSIS from Strayer University, Washington, DC by disguising a thesis in Music Theory as a DSP in Information Systems. However, the process is indeed an information system that is also simple to program. Therefore, it is both Music Theory AND Information System. The DSP research is based on my earlier work entitled "The Scales: The B.A.D. Catalog."(C) 1984. In my DSP entitled "the Universal Harmony: Computing the Scale (Combination) the Permutation, and the Rhythmezomenon"(C)2004, I expand the original scope of my scale study to be a general view of sequence as applied to music. The focus is on the very foundation of music, without which music as we know it could not exist.

However, the numbers I present here are from my original 1984 work, the assertion of which is that every possible scale relationship within the scope of an octave, can be represented as a little-endian binary exponents (ascending scales). Those binary numbers can be converted to decimal format such that any possible scale is uniquely cataloged by a decimal number that can be reduced to the sequence of pitches - binary exponents - that produce it. However, it is far more than that.

Consistent with the work of early 20th century music theorist Joseph Schillinger, the binary sequence is rhythmic. Schillinger postulates that rhythm is the foundation of all music. (Among Schillinger's students, Wikipedia lists George Gershwin, Earle Brown, Burt Bacharach, Benny Goodman, Glenn Miller, Oscar Levant, Tommy Dorsey, and Henry Cowell. Former Peabody professor Dr. Asher Zlotnik, with whom I studied privately is listed as well. The Berklee College of Music in Boston is Schillinger's legacy.) My work demonstrates that rhythm is the foundation of sequence, especially with regard to subsetting a Universe. For example, if I say a Universe is the set of black and white keys proceeding from some reference note on a piano (a 12 tone equal temperament), then a rhythmic relationship exists within each the subset of black keys, and the subset of white keys. It is based on the proposition that either a note exists within the subset - binary 1 - or it doesn't - binary 0. Converting a binary number to rhythmic notation is direct, and is commonly used in selecting combinations of registers in computer programming.

The Binary Analog Decimal (B.A.D.) Catalog employs the binary sequence as a control structure for selecting objects from a set universe - specifically, the 12 tone scale. In the early 1980's, I came across the idea as I was attempting to avoid needing a physical catalog to contain the 2047 possible scale structures ( $\sum _{i=0}^{12-2}{2^{i}}$ or $\sum _{i=1}^{12-1}{{(12-1)}C_{i}}$ ) in a way that can be easily calculated with pencil and paper. Today, it has the advantage of being simple to program and fast to process, since it relies on the CPU - not an external database. Nevertheless, paper and pencil is still the preferred approach. Computer only executes the calculation fast. Using a computer prevents you from thinking about what you're doing.

Rotating the Binary Analog of a sequence produces a set of logically related sequences. In music, they're called modes. Traditionally, those modes are identified Ionian (the Major scale), Dorian, Phrygian, Lydian, Mixolydian, Aeolean (Melodic minor, descending), and Locrian. Each of those modes is a logically shifted set of the same seven notes: C, D, E, F, G, A, and B - where the original reference note is C. As a rotational sequence, their catalog numbers run 1370, 854, 725, 1386, 858, 726, and 693 respectively. That is the 693 Module - from the notion of modes in traditional music theory. The idea is that starting with 693 - the reference mode - you can easily determine every other mode in the module by logically shifting the binary. That logical shifting results in a consistent listing of the sequence of notes in each successive mode, as well as its catalog number. Modules are listed according to their reference mode, the lowest catalog numbered mode. The logic is that it is the first instance of its module you come across when counting up from zero.

The binary digits do not include the reference pitch of the scale. This is not tonality. There is no "tonic." Tonic anchors tonality. A reference pitch anchors the scale sequence. That's why a scale's reference pitch carries an absolute value of zero. 12 is the "Megathos" (Mu) of the 12 tone equal temperament of the modern piano, and can be thought of as the sample space of sequences conforming to ANY 12 tone temperament. Therefore, Mu = 12. Having said that, notice I account for the reference pitch in the Sigmas above. As exponents of 2, I subtract 1 from Mu to account for the reference pitch, and another 1 to account for exponents starting at 0, then begin the summation at 0. As combinations, I simply simply subtract 1 from Mu and begin the summation at 1. (In this context, 0 is the reference note, and doesn't participate in the Summation, even though it physically exists in every scale combination as a function of the additive identity: n + 0 = n.)

In the example regarding the traditional modes above, the "keynote" of each mode is its reference pitch. Zeroing the reference note turns the equation into a function where a unique input produces a unique output. Where Mu = 12, if we do not zero the reference note, we end up having 12 different numbers pointing at exactly the same scale structure. That is, the same scale carries a different catalog number according to which of the 12 possible notes anchors the sequence. However, expressed as a function, the catalog number of Ionian mode - for example - is always 1370, regardless of its "keynote." That is the significance of removing the reference pitch from the calculation. That is why the catalog numbers of scales within a Module remain stable when logically rotating across the binary analog that defines a Module.

That is the nature of the horizontal - time like - relationship of modes within a module. However, across 12 elements (notes), vertical - space like - relations exist. One of them is the distribution of scales having the (Mu - 1)Cr where r > 0, or what I call the "Hypomegathos" (hMu) relationship, where Mu is 12 - signifying the 12 tones of our temperament. One of them is the module relationship of the reference modes of each Module. For example, the musical structure called the "interval" consists of 2 pitches, so hMu = 2. Therefore, within Mu there is the hMu sequence of Modules such that 1, 2, 4, 8, 16. From that you can easily extrapolate that the absolute hMu sequence where hMu = 2 is "the sequence of doubles," where:

             ref mode:        rotates to:   
                  1               1024
                  2               512
                  4               256
                  8               128
                 16               64, and finally
                 32               32, perfectly symmetrical on Mu = 12.     

However, that sequence of doubles is a model of what happens where hMu = 2. hMu > 2 is a bit more complex, involving the compound summation of the binary exponents. The "magic" is that the "vertical" compound summation of the binary digits of the catalog number results in an index order that mirrors the catalog number. For example, recall our sequence of doubles.

                    Index number: Catalog number: 
                         11            1024
                         10             512
                          9             256
                          8             128
                          7              64
                          6              32
                          5              16
                          4               8
                          3               4 
                          2               2
                          1               1

Note here, the index numbers are not the same as the binary exponents. Index numbers begin at 1, and are the result of summation, not exponentiation. The product is an ordered listing of all hMu = x for any x <= 12, where x = 12 and x < 3 are trivial.

I'll also be listing sequences of numbers within this system that are parallel to the "Pitch-Class Set Complex" devised by Alan Forte in his work "The Structure of Atonal Music." I say parallel to because my work is based on the B.A.D. Catalog process, which focuses not on interval content, but rather on sequences of "note objects." Called the Rho (p) complex (px), its focus is rather on sets and subsets of objects. Forte's approach is based on traditional Pythagorean relationships. The approach I'm using is what I call "Boethio-Guidonian," and derives from the time Boethius gave letter names to Pythagorean derived pitches, and Guido d'Arezzo wrote them down on paper as objects called "notes."

Forks in the music history highway, both they are. Musical events are treated as discrete objects, rather than as mathematical ratios. Schoenberg came within a "hair's breadth" of this approach in the early 20th century, in his focus on sequence. Moreover, his colleague Anton Von Webern was a mathematician!! However, in their focus on the "tone row" or simply "the row" they were interested in creating a system of musical composition that abandoned tonality, and missed - what I consider - the obvious. Their contemporary, Joseph Schillinger came even closer, but got bogged down in the complexities "interference patterns" of wave interaction. That was a result of not abandoning Pythagorean principles for something closer to George Boole. Digital methods are simpler than analog approaches every time. My approach, along with the language I use derive from Aristoxenu, a contemporary of Pythagoras. Aristoxenu's work "Rhythmicon Stoicheon," discusses the simple rhythmic principles of his time. It was a path not taken in Western music, but yet preserved in the art and discipline of poetry. You may know of the work Latinized as "Elementa Rhythmica," or "The Elements of Rhythm." Indeed, in the original Greek, it's meaning is closer to "Elementary Rhythm." The simple rhythmic principles of the time as Aristoxenu explains them will fit almost directly over any sequence of 1's and 0's you should care to provide. To me, the B.A.D Catalog is simple and obvious.

Motivation:

Although the system is easily programmed, I still regard it as a pencil and paper approach. Although an MSIS, I'm a techno-skeptic - not to be confused with a Luddite. I admire technology, but I question its ultimate sustainability. Today, computer systems are easily had by the masses. Will that always be so? My conclusion is that understanding this system as simple arithmetic will provide meaning to the sequences it generates. That way it will always remain a sustainable skill. Besides, it would simply be a cryin' bloody waste of 40 years effort if no one else ever knows of it.

To that end, I intend to write a primer on all this, complete with software to accomplish it. The main purpose of the software is to allow the user to hear these sequences. That's not so important with ordinary 12 TET piano tuning. However, most people have never knowingly heard the vast store of traditional and modern temperaments. For example, very few have ever knowingly heard music composed using the Mu = 43 temperament created by American composer Harry Partch, let alone had the means of fiddling around with it in order to facilitate their ability to play and/or create music using that set of pitches.

My object here is an attempt at presenting the concepts in a disinterested venue where the reader is already familiar with the underlying arithmetic techniques. I need to validate the process by explaining how the numbers come from in order to establish their significance. Only then will I stand a chance of explaining all this to the literate public generally, and musicians specifically.

To that end, I invite any civil discussion of the principles I present here.

Education: MSIS - Strayer University BSIT (SCL) - Strayer University BSCIS (SCL) - Strayer University

AA in Music (highest honors) Howard Community College, Columbia, MD AAS in Computer Support (highest honors) Howard Community College AA in Liberal Arts (highest honors) Howard Community College AA in Social Sciences (highest honors) Howard Community College AA in Education, secondary(highest honors) Howard Community College AA in Education, elementary (highest honors) Howard Community College AA in General Studies (highest honors) Howard Community College