Symmetric Questions

Posted by mbase on Friday - August 3, 2007 · 13 Comments 

WordPress.com

Many times people send me email asking various questions about my music. If I have the time (many times I don’t) I respond in detail. I have decided to post a few of these responses as some may find the discussions interesting. I will keep these posts pretty much in the same informal style as in the original email message.

—–
I am also very much interested in the different melodic approaches that you ‘programmed’ in ‘RAMESES’ (the logic in their own architecture) and how it reacts to what the musicians play.

Another big question! I’ll try to answer. Rameses 2000 is the name I gave to the interactive computer program that I developed as a result of a commission from IRCAM in Paris. I cannot really say the entire program was developed as a result of this commission because I had been working on various parts of it for years, and what I presented at IRCAM was a further development of it, using IRCAM’s resources to add to what I had been doing. In the initial stages of the program I worked on a Commodore 64 computer (back in the mid 1980s) using BASIC and 6502 machine language (really using assembly language). At this time I worked either alone or with the help of musician/programmer Joe Ravo, a guitarist still living and teaching in the New York City area. Later, at the end of the 1980s, I ported the program over to the Atari ST computer (following the advise of trombonist/composer/programmer George Lewis) and started using 68000 assembler and a programming language called FORTH. I had a residency at the Banff Center for Fine Arts where I worked out a lot of the initial real time routines, due to the Atari ST’s faster microprocessor. Finally with the help of Takahiko Suzuki and Sukandar Kartadinata, I ported the majority of the program over to the Macintosh Power PC environment and the Max/MSP programming environment. It was also in this form that I worked on the commission at IRCAM. Its really too much too explain the entire operation of this program in words as it is an ongoing (when I have the time) project that covers a period of 20+ years and it keeps changing form. However I basically see the computer as a tool, a laboratory of sorts, to help to work out various conceptual ideas. There have been a few times where I have used it in performance and this concert in 1999 at IRCAM was one of those times. Normally the computer is a behind the scenes tool for me.

——————
Regarding symmetry, I can speak on this a little. The idea of symmetry is pretty deep as basically I believe that the structure of man’s ideas is symmetrical by nature. Certainly all of the musical systems that I have seen from the various cultures around the world and the examples from ancient times are symmetrical in principle (the basic idea behind them) even if not in actual execution.

As you may know many musicians use symmetrical ideas from Bach to Bartok, from Palestrina to Bartok, from Charlie Parker to John Coltrane, so I just have my way of using this stuff and I’ve gotten my ideas from meditation on principles (inside yourself is always a good source). The external sources are the following:

Von Freeman, Art Tatum, Johann Sabastian Bach, Bela Bartok,

Ancient and Medieval musicians and writers
Plato
Aristoxenus of
Ptolemy (Claudius Ptolemaeus)
Aristides Quintilianus

The entire ancient Greek Harmonia system (similar to what we call modes today) is symmetrical if looked at as a whole. There are several Genera of these Harmonia but I will only deal with the Diatonic Genus here. The Harmonia in the Diatonic Genus are formed from what the ancient Greeks called Tetrachords (meaning four strings).

Lydian Tetrachord	C D E F	(semitone on the top)
Phrygian Tetrachord	C D Eb F	(semitone in the middle)
Dorian Tetrachord	C Db Eb F	(semitone on the bottom)

Although all of these structures are physically possible it was the Dorian Tetrachord that the Greeks used when thinking functionally in their music. I am very much into this Tetrachord thing as well as various rhythmic modes but I don’t want to go to deeply into it here.

From what I can figure out the ancient Greeks thought of this tetrachord as the main structure, combining two of them with an interval of separation (called a disjunction) between them. This interval was called ‘Mese’ (meaning ‘middle’) and the note at the bottom of this interval was also called ‘Mese’. So there are two ways to look at these Harmoniai (plural for Harmonia), from a structural point view as Lydian, Phrygian or Dorian Tetrachords separated by a tone…

Dorian Harmonia	C Db Eb F G Ab Bb C	(two disjunct Dorian Tetrachords separated by a tone)
Phrygian Harmonia	C D Eb F G A Bb C	(two disjunct Phrygian Tetrachords separated by a tone)
Lydian Harmonia	C D E F G A B C	(two disjunct Lydian Tetrachords separated by a tone)

or from a functional point view as only Dorian Tetrachords separated by
‘Mese’…

Dorian Harmonia	C Db Eb F G Ab Bb C	(two disjunct Dorian Tetrachords separated by a tone, i.e. [C Db Eb F] and [ G Ab Bb C], F = Mese = 4th degree)
Phrygian Harmonia	C D Eb F G A Bb C	(two disjunct Dorian Tetrachords separated by a tone, i.e. [D Eb F G] and [A Bb D D], G = Mese = 5th degree)
Lydian Harmonia	C D E F G A B C	(two disjunct Dorian Tetrachords separated by a tone), i.e. [E F G A] and [ B C D E], A = Mese = 6th degree)

Notice that ‘Mese’ is successively higher in each of the above Harmonia, and this was very important to the Ethos (character) that the Harmonia was supposed to help express. This subject is too much to go into here but it is enough to say that the Dorian Harmonia was considered the most manly and noble position as it’s mean or middle position contained Mese. The Harmonia with Mesai (plural for Mese) higher than the 4th degree were considered more effeminate and pathetic while the Harmonia with Mesai lower than the 4th degree were considered to be of a gregarious, happy, relaxed and fun loving nature (like drinking songs for example). The ones with Mese near the mean or middle were considered noble and heroic (in other words the ideal or mean).

So these modes are formed by combining the relevant tetrachords with a disjunction of a tone between them (called disjunct tetrachords). More Harmonia could be formed by forming conjunct tetrachords with the tone either at the bottom or the top of the two tetrachords. The terms Hypo (below) or Relaxed and Hyper (above) or Tense were generally used with these other Harmonia.

Hypodorian Harmonia	C D Eb F G Ab Bb C	(a tone followed by two conjunct Dorian Tetrachords)
Hypophrygian Harmonia	C D E F G A Bb C	(a tone followed by two conjunct Phrygian Tetrachords)
Hypolydian Harmonia	C D E F# G A B C	(a tone followed by two conjunct Lydian Tetrachords)

Hyperdorian Harmonia	C Db Eb F Gb Ab Bb C	(two conjunct Dorian Tetrachords followed by a tone)
Hyperphrygian Harmonia	C D Eb F G Ab Bb C	(two conjunct Phrygian Tetrachords followed by a tone)
Hyperlydian Harmonia	C D E F G A Bb C	(two conjunct Lydian Tetrachords followed by a tone)

It appears to me that the ‘Hyper’ terms were not used that much. As you can see there is some duplication here – Hyperphrygian is the same as Hypodorian, Hyperlydian is the same as Hypophrygian. However the Hyperdorian Harmonia is unique from the others. The Hyperdorian Harmonia was normally called the ‘Mixed Lydian’ Harmonia by the Greeks, what we call Mixolydian. Looking at the structure of the remaining unique Harmonia (i.e. Mixolydian, Hypolydian Hypophrygian and Hypodorian) from the functional point of view we have…

Mixolydian Harmonia	C Db Eb F Gb Ab Bb C	(two disjunct Dorian Tetrachords separated by a tone), i.e. [F Gb Ab Bb] and [C Db Eb F], Bb = Mese = 7th degree)
Hypolydian Harmonia	C D E F# G A B C	(two disjunct Dorian Tetrachords separated by a tone), i.e. [B C D E] and [F# G A B], E = Mese = 3rd degree)
Hypophrygian Harmonia	C D E F G A Bb C	(two disjunct Dorian Tetrachords separated by a tone), i.e. [A Bb C D] and [E F G A], D = Mese = 2nd degree)
Hypodorian Harmonia	C D Eb F G Ab Bb C	(two disjunct Dorian Tetrachords separated by a tone), i.e. [G Ab Bb C] and [D Eb F G], C = Mese = 1st degree)

I believe that the Hypolydian was also called ‘Slack Lydian’ by Plato
and ‘Relaxed Lydian’ by Plutarch.  This mode was considered the
‘opposite’ of Mixolydian and using symmetry we can see why by this
demonstration where ‘1’ = semitone and ‘2’ = tone…

Ascending Mixolydian structure	1 2 2 1 2 2 2
Descending Hypolydian structure	1 2 2 1 2 2 2

Plato called the Lydian Harmonia ‘Tense Lydian’ because of its high pitch.

This concept of the ‘Mean’ continued into the middle ages and later and eventually got transformed into the modern triad concept. The ancients thought in Geometric terms. For example the ancient Egyptians were great Geometers (people skilled in Geometry) and the Greeks received a lot of their education from the Egyptians. Proportion was very important to these older traditions with even the concept of the major and minor triad being developed out of this. However this concept of proportion was not only geometrical but also philosophical, metaphysical and theological, dealing with such concepts as the Father, Son and Holy Spirit. For example in the major triad C-E-G, the pitch ‘E’ was seen as the Harmonic Mean between ‘C’ and ‘G’ (from the perspective of string proportions). In the minor triad C-Eb-G ‘Eb’ was seen as the Arithmetic Mean (or reverse Harmonic Mean musically speaking) between ‘C’ and ‘G’. The Harmonic Mean is the reciprocal of the Arithmetic Mean of the reciprocals of a finite set of numbers.

For example the Arithmetic Mean of 6 and 12 is 9 (i.e. (6+12)/2 = 18/2 = 9).
The reciprocals of 6 and 12 are 1/6 and 1/12.
The Arithmetic Mean of 1/6 and 1/12 is 1/8
The reciprocal of 1/8 is 8.

So…
The Arithmetic Mean of 6 and 12 is 9
The Harmonic Mean of 6 and 12 is 8

All of the above is using whole numbers as in representing pitches in the language of hertz (i.e. vibrations per second – or the measure of frequencies). But if we are thinking of things like the ancients then we have to do our proportion calculations using parts of the lengths of string (for example using the monochord). This produces what at first sight looks like the reverse of the results we get above.

The Harmonic Mean of 1/6 and 1/12 is 1/9
The Arithmetic Mean of 1/6 and 1/12 is 1/8

So using this way of thinking we take a string that is equal to length 1 (that is 1 anything, for example 1 Meter (almost the same as 1 American Yard, 1.09 Yards to be more precise or a little more than 3 Feet). If we take the tone produced by vibrating the entire length of the string as the pitch C4 then the pitch produced by vibrating 1/2 the length of the same string (that is by vibrating 1 out of two parts of the same string) will be C5, i.e. the pitch an octave above C4. So we have…

1 = C4
1/2 = C5

Now we use pitches of the Octave as our extremes between which we find the Harmonic and Arithmetic Means.

The Harmonic Mean of 1 and 1/2 is 2/3
The Arithmetic Mean of 1 and 1/2 is 3/4

So if 2 out of 3 parts of our string is vibrating then the pitch produced will be G4, and if 3 out of 4 parts of the string is vibrating then the pitch produced will be F4. So now we have…

1 = C4	Generator
3/4 = F4	Perfect 4th	Arithmetic Mean (AM)
2/3 = G4	Perfect 5th	Harmonic Mean (HM)
1/2 = C5	Octave

We can take this even further by finding the HM and AM inside of the
interval of a perfect 5th, so now our extremes are 1 and 2/3.

1 = C4	Generator
5/6 = Eb4	Minor 3rd	Arithmetic Mean (AM)
4/5 = E4	Major 3rd	Harmonic Mean (HM)
1/2 = C5	Octave

We could continue and find the HM and the AM between every new interval that results. For example the HM between 1/2 and 2/3 (that is between the fifth and the octave) is 4/7

2/3 = G4 Perfect 5th
4/7 = Bb4 Flat Seventh (HM)
1/2 = C5 Octave

I would call 4/7 the Natural Seventh as it occurs naturally with these small number proportions and also it is the ratio that occurs in the Harmonic Series. In other words 4/7 is the octave reduced form of 1/7).

1/1 1/2 1/3 1/4 1/5 1/6 1/7 1/8 1/9 etc…

This series, when representing the lengths of strings, is the same as what we call today the Overtone Series (some still say the Harmonic Series). However by representing the Natural Seventh as 4/7 I am reducing it to within an octave, that is to a ratio between 1 and 1/2.

Harmonic Series Interval	Octave Reduced Interval
1/1 1/2 1/3 1/4 1/5 1/6 1/7 1/8 1/9	1/1 1/2 2/3 1/2 4/5 2/3 4/7 1/2 8/9

So this concept of the Mean is very old and it is in fact the basis of our present day music systems. The ancient musician/philosophers such as Pythagoras, Socrates, Plato, Aristotle, Aristoxenus, Ptolemy, Aristides etc. knew this. The composers and music theorists of the Middle Ages, Renaissance and Classical periods continue in this same tradition, I am speaking of people such Guido of Arezzo, Marchetto of Padua, Gioseffo Zarlino, Glareanus, Giovanni Pierluigi of Palestrina, Claudio Monteverdi, Nicola Vicentino, Johannes Lippius, Johann Sebastian Bach, Rameau, Chopin etc., and also the more modern composers such as Arnold Schoenberg, Bela Bartok, Paul Hindemith, Ernst Levy and Olivier Messiaen.

All of this symmetry is still working in the music of Duke Ellington, Charlie Parker, Von Freeman, John Coltrane, etc., right up to what we are doing today. There is a page on my website where I explain the basic principles behind some of my symmetrical ideas.

This is not something new, as I state in this email these ideas go back to ancient times. Many thinkers in more modern times like Moritz Hauptmann, Hugo Riemann, Heiner Ruland, Ernst Levy and Ernest McClain talked about Polarity Theory in harmony in more direct terms. Levy especially talks about these ideas in functional terms, but this is only the modern expression of an entire proportionate musical system that existed before among the ancients (in the structures of tones and rhythms) and in the tonal modes of the Middle Ages and Renaissance (Authentic and Plagal modes for example) and the rhythmic modes from the Middle Ages and early Renaissance.

In particular I am interested in the people who take these musical concerns connected with physical and metaphysical ideas. All of the earlier composers follow this way of thinking to various degrees, and also people like Guido, Lippius, Bach, Levy, Ellington, Messiaen, Coltrane, McClain, Ruland, and Charles Muses (he also called himself Musaios). These people were concerned with using music to expand consciousness. Of course the musicians who were not writers simply created their expression though the actual music. Others who were writers expressed some of their ideas in the books they produced. Lippius, Ruland and Musaios in particular, using slightly different approaches, wrote some very interesting things.

Ruland wrote a book called ‘Expanding Tonal Awareness’ where he expands on the ideas of Hans Lauer, who comes out of a Germanic school of musical mystics. Ruland describes cycles of intervals formed from the prime numbers. He uses the ratios series 8:8 (unity) 9:8 (maj2nd – extended prime 3) 10:8 (maj3rd – prime 5) 11:8 (tritone – prime 11) 12:8 (P5th – prime 3) 13:8 (min6th – prime 13) 14:8 (minor or natural 7th – prime 7)

According to Ruland Human Consciousness has been making a steady progression towards the outer planets of our solar system with the Sun acting as a mediator which facilitates the harmonization of the inner and outer expressions of consciousness. The tunings dealing with the primes 7 (i.e. 14:8) and 13 (13:8) belong to a time when civilizations generally looked outward, experiencing the world through the Cosmos, Universe and Nature. Their inner experience was molded by their contact with this ‘outer’ world. Beginning with the experience of the 5th (cycle of prime 3) there was a mixture, a bridge and the beginnings of a look inward as well as outward.

In order to deal with these concepts practically Ruland proposes using a 24tEQ system in which the quarter tones are treated in a ‘flexible’ manner, in order to simulate the intervals of what Paul Hindemith calls ‘The Holy Domain’, i.e. 14:8, 13:8, 11:8 (primes 7, 13 and 11). So Ruland says that you should start with a quarter tone and then adjust it according to which of these primes you are dealing with.

14:8 (7:4) = 969 cents, tempered to 960 cents
13:8 = 840.5 cents, tempered to 840 cents
11:8 = = 551 cents, tempered to 550 cents

Here are the eras in history and the corresponding consciousness that Ruland associated with each of the prime numbers and their respective ratios:

14:8 Cycle of 7 Movement/Spirit	Outward	Egypt	Prime 7
13:8 Cycle of 13 Dimension	Outward	Egypt/Mesopotamia/India	Prime 13
12:8 Cycle of 3 Spirit	Outward	above + China/Greece	Prime 3
11:8 Radial/Cycle of 11 Mind	Inward/Outward	Egypt/India (not sure)	Prime 11
10:8 Radiality of 5 Soul	Inward	above + Modern times	Prime 5
9:8 Radiality of 3 Movement/Soul	Inward	near Future	Prime 3 (extended)
8:8 Prime Consciousness (1)	Inward	Future	Prime 1

These ideas are interesting to me because I link this kind of consciousness expansion with the spontaneous compositions of people like Von Freeman (and the tradition that he comes out of). I am thinking of these things in terms of more specifics musically (i.e. in terms of musical movement and functions) but the basic idea is the same. Also most of these modern philosophers ignore rhythm completely, for this kind of information I find only the writings of some African philosophers (such as Professor Willie Anku) and ancient writers (i.e. Aristoxenus, Aristides, etc.) where they suggest some specific poetics involved with the movement of rhythmic forms (the writings of Messiaen and Schillinger would be some examples of some of the exceptions, and there are others).

Similar to Musaios, Ruland suggests some planetary correspondences to his ratios. Ruland also suggest some planetary correspondences to these ratios but his considerations only seem slightly Astrological (these considerations are explained on page 172 through 176 in ‘Expanding Tonal Awareness’). Here are Ruland’s correspondences.

Saturn = 8:8
Jupiter = 9:8
Mars = 10:8
Sun = 11:8
Venus = 12:8
Mercury = 13:8
Moon = 14:8

Musaios is similar to Schwaller de Lubicz in that he directly studied ancient Egyptian texts, but Musaios was a scientist and mathematician who is able to go deeper. He developed some ideas pertaining to using sound to help in the raising of consciousness which he expresses in a book called The Lion Path. Musaios created tapes with musical progressions on them to help open up the pathways to expand consciousness. His tapes contain a 5-Limit Just Intonation tonal system based on 22 tones to the octave. Musaios system is not constructed from 22 equal intervals, they are based on small interval ratios that do not use any prime number greater than 5. However his system is completely symmetrical! The symmetry does not require that the intervals between pitches or durations between rhythmic onset times are equal, just that there is some kind of system of balances at work where elements are equal around a particular axis. In the case of Musaios’ tonal ratios these distances are proportionately equal around a tritone base. Since the tritone is really the Geometric Mean within the Octave (for the mathematical definition of the Geometric Mean see below under my discussion of the Golden Mean) then these 22 intervals fall on either side of the Geometric Mean, 11 intervals on each side.

And here are Musaios’ planetary correspondences.

Planet(?)	Scale position	Ratio	Frequency	% of people drawn to this world	absolute pitch
Plu	1	1:1	151.70		Eb
Pan	2	25:24	158.02		E
Vulcan	3	16:15	161.81		E
Horus	4	10:9	168.56		F
Ven-Sothic	5	9:8	170.66	0.33	F
Sat-Sothic	6	75:64	177.77	0.20	Gb
Sun-Sothic	7	6:5	182.04	0.13	Gb
Moon-Sothic	8	5:4	189.63	0.07	G
Mar-Sothic	9	32:25	194.18		G
Mer-Sothic	10	4:3	202.27	0.07	Ab
Jup-Sothic	11	27:20	204.80	0.07	Ab
Ura-Sothic	12	45:32	213.33 (64:45 215.75)	0.07	A
Nep-Sothic	13	40:27	224.74		B
Plu-Sothic	14	3:2	227.55		Bb
Pan-Sothic	15	25:16	237.03		B
Vulcan-Sothic	16	8:5	242.72		B
Horus-Sothic	17	5:3	252.83		C
Sothis-Sothis	18	128:75	258.90		C
Pre-Pleromic Moon	19	16:9	269.69		Db
Pre-Pleromic Sun	20	9:5	273.06		Db
Pre-Pleromic Sat	21	15:8	284.44		D
Pre-Pleromic Ven	22	48:25	291.26		D
Pre-Pleromic Jup		2:1	303.40		Eb

I consider both of these approaches philosophical, although both Ruland (in his book) and Musaios (cryptically explained in his book and practically applied on his Lion Path tapes) try to express these ideas musically. In particular the Lion Path tapes sound more like sonic mantras than music.

It is interesting to me that the philosopher’s approach to this almost always involves working with tonality through manipulating the structures of tuning systems, whereas the musician’s approach is mostly to manipulate the accepted musical sounds of the then current era through the actual movements (i.e. rhythms) of rhythmic/melodic/harmonic structures. Therefore the ideas behind placement, duration and combinations of pitches and what they symbolize seem to be important to the musician and much of the time the form that the sonic symbols take are various kinds of musical motion.

This is the approach that I favor also. Movement is at the basis of everything that we know, and so if we are to express concepts of balance, it is to the movements of these balances that we must look for our symbols – in other words Dynamic Balance is possibly the key. Although there is no doubt that the structures of tuning systems plays a role in all of this, I would say that this role is much less important than the movements. Humans and other living creatures, as with all things created by Nature, have a tolerance that allows us to recognize principles as they approach the ideal mean. For example a perfect 5th does not have to be ‘perfect’ for us to recognize the quality of the fifth and respond to it psychologically. It is the idea of the 5th that we respond to, not its perfection. In fact every fifth that we hear in a piece of music is different if it is performed by humans. Just like in Astrology there is an orb of influence, an area which when approached triggers certain reactions within us. That area varies in size from person to person, depending on their personality, genetics, background, training, etc., but these orbs are remarkably consistent across the human spectrum. We must look for evidence across the human spectrum, in all cultures and all eras, if we are to find clues to what are the musical Universals, the primal musical elements that we all possibly respond to. And for this we must go back in time and help to define and clarify our most basic musical concepts…

What is…

music
musical rhythm
musical tone
musical melody
musical harmony (by harmony I mean agreement, which is harmony in its broad sense, not the major-minor tonal system which is harmony in its narrow sense)
musical form
musical poetry (also called musical rhetoric, i.e. expression of ideas, feelings, beauty and emotions by use of the elements of music)

For the ancients proportion played a major role in all of these areas.

About symmetry: you say you are interested in the ‘function’ of the musical idea: how would you define the function of ‘symmetry’ then? What is the ‘Harmonic Mean’ and the ‘Arithmetic Mean’ ?

The Harmonic Mean and the Arithmetic Mean (along with the Geometric Mean and the Golden Mean) I explain a little above, and anyway a lot of this information is available on the Internet or in books, it is not that complicated to figure out.

By Function I mean how something works in relation to the things around it. Since Music can be considered a science of movement then Function in music generally means how something moves in relationship to other things around it. So I could use symmetry in the following way and say that these two things are symmetrical around an axis of ‘C’…

G C
E Ab
C F

because C to E to G are the same intervals ascending as C to Ab to F are descending. But there is not really any obvious function implied here, it is just a statement that the structure of these two triads are symmetrical around ‘C’. Now if I say that the second triad is ‘moving’ or progressing to the first triad, now I am talking about Function because I am stating that the Fmin triad progresses to the Cmaj triad and I can talk about the movement between these two sounds as one entity, as well as refer to the relationship of these two triads. So I can speak of how the Fmin triad Functions in relation to the Cmaj triad, I can talk about how the Cmaj triad functions in relation to the Fmin, or I can talk about the function of the movement itself – a third and invisible relationship that involves the ‘sound of the movement’ itself. I can speak of this particular function and movement as being centripetal motion, of the Fmin being ‘pulled’ toward the Cmaj, and of the symbolism that this motion expresses.

In the above case Fmin functions as the negative dominant of Cmaj (but more naturally of Cmin). First let us imagine two parallel keys, Cmaj and Cmin. These two root triads are symmetrical around and axis of Eb/E. That is the relationship of Cmaj to Eb is the same as the relationship of Cmin to E in the opposite direction. And the axis of Eb/E could be described as a Sum 7 axis, because Eb (3) and E (4) sum to 7 (3+4=7). This I discuss elsewhere on my website and maybe on my blog.

G G
E Eb
C C

If you meditate on this and you will find that it is true but you need to be able to think in a descending as well as ascending mode.

C up to Eb (min3rd)	G down to E (min3rd)
E down to Eb (semitone)	Eb up to E (semitone)
G down to Eb (maj3rd)	C up to E(maj3rd)

But the tonic of both these triad is C, even though the systems are being generated in opposite directions symmetrically. Now if you extend these two triad in the direction of tonal relationships, we say that traditionally Gmaj is seen as the dominant of Cmaj. This is because the Gmaj triad is constructed upwards from the dominant degree of Cmaj. When in the key of Cmaj we are thinking as if C is the tone that is generating this particular tonality.

The triads of Cmaj – in the diagram below the red triad (Gmaj) is the dominant in the key of Cmaj and the blue triad (Fmaj) is the subdominant in the key of Cmaj. Here I am thinking in a positive and ascending manner and I am thinking about how the triads Gmaj and Fmaj Function within the gravitational sphere of Cmaj.

G A B C D E F

E F G A B C D

C D E F G A B

So the following are dominant and subdominant progressions respectively.

D E C C

B C A G

G G F E

The triads of -Gmaj – when in the key of -Gmaj we are thinking as if G
is the tone that is generating this particular tonality negatively
(that is to say thinking in a descending manner). In the diagram
below the blue triad (-Cmaj or Fmin) is the dominant in the key of
-Gmaj (Cmin) and the blue triad (-Dmaj or Gmin) is the subdominant in
the key of -Gmaj. Here I am thinking in a negative and descending
manner and I am thinking about how the triads -Cmaj and -Gmaj Function
within the gravitational sphere of -Gmaj (or Cmin).

G F Eb D C Bb Ab

Eb D C Bb Ab G F

C Bb Ab G F Eb D

So the following are negative dominant and negative subdominant
progressions respectively.

C C D Eb

Ab G Bb C

F Eb G G

However since Gmaj is also considered the dominant of Cmin (or -Gmaj)
then -Cmaj (or Fmin) can be considered the negative dominant of Cmaj.

D Eb C C

B C Ab G

G G F E

If you look at these progression you will see that they are all symmetrical in movement (i.e. in function) and in terms of the relationships of the intervals (not just symmetrical in structure around a particular axis). What I have described here with Cmaj and -Gmaj are two tonal systems that are symmetrical around the Eb/E and A/Bb axes. The A/Bb axis is a Sum 7 interval as well, because A (9) + Bb (10) = 19 – 12 = 7 (minus 12 because our musical system is really a mod 12 or base 12 system).

So this is what I mean by function, and usually I give names for functions just so that I and the people who I communicate with know what I am talking about. But the name is fluid, and the functions can change, however the principles are immutable.

It is true that Messiaen’s ideas and theory is not new if you see it as part of the whole. Messiaen himself usually refers to the Nature when he explains the use of symmetry (non-retrogradable rhythms and their harmonic equivalent, the modes of limited transpositions ), he actually only refers to his religious belief when he explains how he used the ‘theory’ in his music. What appeals to me is the musical consistence he has reached by the extreme use of a few basic ideas and by the potential of their combinations and additions.

Yes, I like this about Messiaen’s ideas also. They are great ideas – even if I feel that the principles are the same as have been expressed many many times in the past (similar to my own ideas which are also not new) but with Messiaen’s own particular language and insights.

But so I would like very much to know about the way you define ‘function’ of a musical idea, and what it refers to in the globality of your musical thoughts. What are the paths from ‘function’ to their structural and (sonic)-symbolical applications?

I gave some examples of function above. Perhaps what I should say is that function is for me ultimately defined by Nature. Nature is the final arbiter of what works and what does not work, and this is the Global or Universal aspect of what I am dealing with. The path from function to structure is really only based on experience and trial and error. Not only your own trial and error but also the experience of those contemporary, older or past musicians whom you may respect or trust. Still I feel that a musician must base their own work on what they themselves have experienced. So if I get a certain feeling or experience from the concept of a circle or triangle this can give me more confidence when I use this symbol or shape in my work, as a means of projecting a certain feeling, emotion or vibration. Then I must meditate on the idea behind this function to arrive at the musical method or musical tool that I can use that will best (for the moment) express this function through sound. So I would say that the real path is experience and contemplation and meditation on that experience. The experience is outsight (that which is brought in by the senses) but the meditation is insight (that which is contemplated by the mind).

Regarding the example of symmetry you gave in the last mail, the ‘chord of resonance’ being the conjunction of the C Maj ascending partials and D Maj descending partials: from a theoretical point of view the conjunction of both overtone ( C ) and undertone ( D ) series gives the 3rd of the modes of limited transpositions ( in its 1st transposition : mode 3 of C : C D Eb E F# G G# A# B ). And actually in each sum of the partials ( C or D ), there is only one note missing to obtain the entire mode 3 : in the sum of the C overtone series it is Eb that is ‘missing’ and in the sum of the D undertone series it is F#. I find this very interesting because it probably means that the other modes of limited transpositions could also be seen as the conjunctions of partials. I’ll try to find out!

Hey, let me know what you find out, I would be very interested to know. I noticed as soon as I looked as Messiaen’s modes that he was really dealing with the tempered overtone and undertone series in his own way. There is a great tradition going all the way back to the Egyptians and Babylonians of this idea, called by very different names in different eras. I just call it Musical Reciprocity.

Sorry but I don’t understand when you say that D negative is Gmin and that G is the root of negative D maj

As I explain above, negative Dmaj is the same as Gmin. For example a Gmaj chord is composed of an ascending major third (G to B) and an ascending minor third (B to D). Well negative Dmaj is the same concept but starting on D and descending. We begin with a descending major third (D to Bb) and then a descending minor third (Bb to G). However we all know that when this triad is sounded we will hear this as Gmin (this part is learned) and we will hear the root as G (this part is as the result of our structure as humans). This is because the strongest interval here is the fifth (D down to G) and with any fifth the initiated musician hears the bottom note as the tonic or main supporting note. So here are the two situations with a positive and negative major triad:

G B D (positive ascending major triad, tonic = G, which is the generator of this overtone series, i.e. 1st partial = G, 3rd partial = D, 5th partial = B, these partials are positive partials of an overtone or positive series)

D Bb G (negative descending major triad, tonic = G, however D is the generator of this undertone series, i.e 1st partial = D, 3rd partial = G, 5th partial = Bb, these partials are negative partials of an undertone or negative series)

You MUST be able to adapt your thinking to understand this! You must be able to imagine in the opposite direction of what you are used to. Most musicians think only in ascending form, you must learn to think in a descending manner, first structurally – then functionally.

Peace,

Steve

Filed under Theory and Concepts · Tagged with computer music application, greek music, mbase, medieval, music theory, negative theory, steve coleman, symmetry