GPS and LPS Greek and then Medieval Theoretical Musical Structures

In this email I am going to use the terminology for both the ancient Greek Harmoniai, the later ancient Greek Tonoi (and Tropoi) along with the Medieval Tonus (which are theoretical constructions that were attempts to organize and ‘explain’ the ecclesiastic plainchants) and later became the Medieval Modes. When I am talking about ‘Mese’ and using Greek terms then I am using the ancient Greek names as the Greek writers used them. However when I go into terminology such as “Species of the Fifth”, “Authentic” and “Plagal” then I am using Medieval terminology for the modes. Some words such as the prefix ‘Hypo’ are applicable to both ways of thinking but this prefix is used differently in Medieval times than it was in ancient Greek times.

Ancient Greek System
——————–

GPS (Greater Perfect System)
(from http://sonic-arts.org/dict/gps.htm )

-A NETE HYPERBOLAION
G Paranete Hyperbolaion
F Trite Hyperbolaion
-E NETE DIEZEUGMENON
D Paranete Diezeugmenon
C Trite Diezeugmenon
-B PARAMESE
-A MESE
G Lichanos Meson
F Parhypate Meson
-E HYPATE MESON
D Lichanos Hypaton
C Parhypate Hypaton
-B HYPATE HYPATON
-A PROSLAMBANOMENOS

LPS (Lesser Perfect System)
(from http://sonic-arts.org/dict/lps.htm )

-D NETE SYNEMMENON
C Paranete
B Trite Synemmenon
-A MESE Synemmenon
G Lichanos Meson
F Parhypate Meson
-E HYPATE MESON
D Lichanos Hypaton
C Parhypate Hypaton
-B HYPATE HYPATON
-A PROSLAMBANOMENOS

Tetrachord Names

Hyperbolaion = Exceeding, going beyond, at zenith
Diezeugmenon = Disjoined
Meson = Middle (see Mese below)
Hypaton = Highest, farthest. (See Hypate below)

Synnemmenon = Conjoined

Tone Names
(These names refer to the position on the lyre, not to pitch height as the Greeks only rarely characterized pitch as “high” or “low”)

Nete = Lowest, nearest
Paranete = Beside Nête
Trite = Third (from top of Tetrachord)

Paramese = Beside Mese
Mese = Middle
Lichanos = Licking (finger), i.e. forefinger
Hypate = Highest, farthest

Proslambanomenos = Taken in addition, (i.e. the “added note)

Remember that ‘Para’ means ‘Beside’ or sometimes ‘Beyond’ as in…

Paranormal (beside normal)
Paralegal (beside legal or beside a lawyer)
Paranoid (from the Greek term for madness which is ‘Paranoos’ (demented), i.e. Para (in this case ‘Beyond’ and the Greek ‘Nous’ or ‘Noos’ (mind) and so literally ‘Beyond Mind’.

Remember Species is different from Genera (plural for Genus) as follows:

Species is the cyclic reordering of intervals within a given pitch space boundary (for example within an octave, fifth or fourth). For example take the ancient Greek Harmoniai (diatonic genus) or the Medieval modes, many see these species as occurring within the span of an octave. Genera were the interval structure in a given pitch space, this was called a tetrachord when the boundary is the interval of a perfect fourth.

For an example of the latter say that 1/2 = a quartertone, 1 = semitone, 2 = tone, 3 = three semitones and 4 = a ditone. So what we really have here is the number referring to how many semitones (i.e. 1/2, 1, 2, 3 or 4). So for the ‘Genera’ of the tetrachord we have (from bottom to top)…

1 2 2 Diatonic Genus
1 1 3 Chromatic Genus
1/2 1/2 4 Enharmonic Genus

Since in ancient times the exact tuning of these approximate quartertones, semitones and tones varied, then we can see that the Genera had more to do with attunement (i.e. tuning), and in fact everything in ancient times had more to do with attunement.

Now the various ‘Species’ within for the Diatonic Genus is as follows (I will call these Tetrachord Species X, Y and Z, this type of thing was perfected in Medieval times)…

1 2 2 Diatonic Genus (Species X) ex. C Db Eb F
2 1 2 Diatonic Genus (Species Y) ex. C D Eb F
2 2 1 Diatonic Genus (Species Z) ex. C D E F

And the same thing for the other Genera of the tetrachord, that is each would have two other Species. The concept of ‘Genus’ was more or less discarded but the concept of ‘Species’ we retain today in the way we look at the modern Modes.

Now in this sense the ancient Greek Harmoniai can be thought of as ‘Species of the Octave’. As an example of the former suppose that 1 = semitone and 2 = tone, and I am only going to use the diatonic genus for the Greek Harmoniai…

Greek Medieval Interval
Harmoniai Modes Structure type function
—————————————————-
Dorian Phrygian 1 2 2 2 1 2 2 B- (- tonic, more or less)
Phrygian Dorian 2 1 2 2 2 1 2 A+- - dominant
Lydian Hypolydian 2 2 1 2 2 2 1 B+ + tonic
Mixolydian Hypophrygian 1 2 2 1 2 2 2 C- (diminished, more or less)
Hypodorian Hypodorian 2 1 2 2 1 2 2 D- - tonic
Hypoprygian Mixolydian 2 2 1 2 2 1 2 D+ + dominant
Hypolydian Lydian 2 2 2 1 2 2 1 C+ (+ tonic, more or less)
Hypomixolydian (really the same structure as Greek Dorian)
Hypomixolydian (really the same structure as Medieval Dorian)

In the table above the ancient Greek Harmoniai can be thought of as ‘descending’ from Dorian and the Medieval Modes can be thought of as ‘ascending’ from Dorian (i.e. Descending or ascending from Dorian to Hypomixolydian).

Notice that the prefix ‘Hypo’ means ‘Below’ (for example ‘hypodermic’ means below the skin, the Greek ‘Derma’ means ‘Skin’). The prefix ‘Hyper’ means ‘Above’ (as in hypertension or hyperactive).

Between ancient Greek and Medieval times the interval that is being referred to by the ‘Hypo’ did not change but the interpretation of that interval changed!

In ancient Greek times Hypodorian was ‘below Dorian’ but was thought of as being a ‘Perfect Fourth below Dorian’, this even though the actual Harmonia is a Perfect Fifth below the Dorian Harmonia. To explain this you have to understand the concept of Tonoi in ancient Greece.

If you try to represent the Harmoniai on the piano then you can see that Dorian would run E-E on the white keys and Hypodorian would run A-A, so it would appear that the Hypodorian Harmonia is a fifth below the Dorian Harmonia, however this is not how things were conceived in ancient times as the ‘Perfect Fourth’ (i.e. the downward ‘Perfect Fifth’) represented the main interval structure of their melodic thinking (possibly after the octave).

In this more developed method, the Greek Harmoniai were conceived were as a representation of the GPS (Greater Perfect System) where the interval ‘Mese’ was thought of as moving up and down within a fixed octave. This concept evolved in late ancient Greek thought and was usually referred to as the Greek ‘Tonoi’ (or called ‘Tropoi by some of the later writers). So you could look at the various ‘Mesai’ (plural for ‘Mese’) as being related to each other by a concord (4th, 5th or Octave) between a fixed octave boundary. So the change in the pitch-relations of the ‘Tonoi’ was the change of the position of ‘Mese’ and what we end up with is a certain range of the GPS rotating between the same two pitches (i.e. the limites of an octave) with the Dorian Tonos representing the original position of the GPS. If we use E-E as our octave then we have the following…

Tonoi Tonoi Position
Structure Name Of Mese (from bottom)
———————————————
E F G A Bb C D E Mixolydian 7 Hypate H. to Paramese
E F# G# A B C# D# E Lydian 6 Parhypate H. to Trite D.
E F# G A B C# D E Phrygian 5 Lichanos M. to Paranete D.
E F G A B C D E Dorian 4 Hypate M. to Nete D.
E F# G# A# B C# D# E Hypolydian 3 Parhypate M. to Trite H.
E F# G# A B C# D E Hypophrygian 2 Lichanos M. to Trite H.
E F# G A B C D E Hypodorian 1 Mese to Nete H.

The names of these ranges come from the names in the GPS (Greater Perfect System) above. Notice that, in terms of the interval structure, the range of Hypodorian from Mese to Nete Hyperbolaion is the same as saying it is from Proslambanmenos to Mese. However the former terminology puts Mese on the bottom of the structure, which is I believe how the Greeks thought of it.

When looked at in this way then it can be seen that the Greeks thought of the ‘Hypo’ Tonoi as being ‘ a fourth below’ (i.e. ‘Hypo’) their named counterparts, this because ‘Mese’ is a fourth below in each of the ‘Hypo’ Tonoi. So we have ‘Mese’ in the middle representing the structure of the music of the ‘Doric’ Greek race, then the Tonoi with ‘Mese’ in a lower position than the Dorian Tonoi have the ‘Hypo’ prefix and the Tonoi with ‘Mese’ in a higher position than the Dorian Tonoi do not have the ‘Hypo’ prefix.

However in order to think of each ‘Mese’ as ‘modulating’ up a Fifth then you would need to begin on the Mixolydian Tonos and the order would be as follows…

Tonos Tonos Position
Structure Name Of Mese (from bottom)
———————————————-
E F G A Bb C D E Mixolydian 7 Hypate H. to Paramese

E F G A B C D E Dorian 4 Hypate M. to Nete D.
E F# G A B C D E Hypodorian 1 Mese to Nete H.

E F# G A B C# D E Phrygian 5 Lichanos M. to Paranete D.
E F# G# A B C# D E Hypophrygian 2 Lichanos M. to Trite H.

E F# G# A B C# D# E Lydian 6 Parhypate H. to Trite D.
E F# G# A# B C# D# E Hypolydian 3 Parhypate M. to Trite H.

Here you have ‘Mese’ falling by three degrees within each Tonoi. This corresponds to the modulation of ‘Mese’ ‘up’ by Fifths, which means the structure of the Tonos is modulating ‘down’ by Fifths). And this also implies that there is a theoretical eight ‘HyperMixolydian’ structure as follows…

Eb F G A Bb C D Eb HyperMixolydian 3 Parhypate M. to Trite H.

Or the same structure as Hypolydian if placed in the octave of E-E.

In this way the Greek Harmoniai can be thought of as ‘Species of the Octave with the Tetrachord structure being the fixed ‘micro structure’ and the GPS (or LPS) being the fixed ‘macro structure’. There were many other esoteric associations in ancient times for the Harmoniai and Tonoi but we can save that for another day.

The Transition between the Greek and Medieval Systems
———————————————-

In Medieval times Hypodorian was also ‘below Dorian’ and was thought of as being a ‘Perfect Fourth below Dorian’, however they were then referring to the entire mode being a perfect fourth below. I think this was a result of an early misunderstanding of the ancient Greek writings, and there are other indications that this misunderstanding occurred, for example the Greek names referring to different melodic configurations, even though the modes occurred in the same sequence as the ancient Harmoniai.

I believe that the Medieval ‘Hypo’ concept morphed into the ‘Plagal’ concept because the relationships are the same, and I will say more about what I believe are the origins of this way of thinking below. Around the time the word Plagal began to be used was also the same time the ‘Hypo’ prefix got dropped (of course this is an oversimplification of what actually went down).

The Medieval Modes can be thought of as ‘Species of the Octave’ with the Pentachord and Tetrachord structures being the fixed ‘micro structures’ and the concept of complimentary modes as being a kind of ‘macro structure’. This eventually led to our present day concept of ‘Key’ and ‘Major-Minor Tonality’ as being the ‘macro structure’.

Medieval System
—————

Originally in Medieval times there were eight modes numbered (and later given ancient Greek names) as follows…

# Mode Element Polarity
————————
1 Dorian Water Male
2 Hypodorian Water Female
3 Phygian Fire Male
4 Hypophrygian Fire Female
5 Lydian Air Male
6 Hypolydian Air Female
7 Mixolydian Earth Male
8 Hypomixolydian Earth Female

The above modes were mainly used in the Church and in some secret circles carried some esoteric associations. Then the theory further developed, and with the addition of more material the acceptable modes became twelve and were numbered and named as follows…

1 Dorian
2 Hypodorian
3 Phygian
4 Hypophrygian
5 Lydian
6 Hypolydian
7 Mixolydian
8 Hypomixolydian
9 Aeolian
10 Hypoaeolian
11 Ionian
12 HypoIonian

For the above modes the odd numbered modes were eventually called ‘Authentic’ and the even number modes (i.e. the ones with the ‘Hypo’ prefix) were eventually called ‘Plagal’ (see my theory on Plagal modes below).

Now remember our discussion of the Species of the Fourth as follows…

1 2 2 Diatonic Genus (Species X) ex. C Db Eb F
2 1 2 Diatonic Genus (Species Y) ex. C D Eb F
2 2 1 Diatonic Genus (Species Z) ex. C D E F

In Medieval times the Species of the Fifth were also common (I will call these Pentachord Species A, B, C and D,…

1 2 2 2 (Species A) ex. C Db Eb F G
2 1 2 2 (Species B) ex. C D Eb F G
2 2 1 2 (Species C) ex. C D E F G
2 2 2 1 (Species D) ex. C D E F# G

In Medieval times (after the Tonus period) the modes were originally thought of as Species of the Octave, but later as Species of the Fifth and Fourth. An Authentic Mode was one which had a species of the Fifth below a species of the Fourth. A Plagal Mode was one that had a species of the Fifth above a species of the Fourth.

At that time Locrian was not used because there is no Fifth above the final (i.e. there is no species of the Fifth on the bottom). The Final of all of these modes, Authentic or Plagal, always has a tone an interval of a Fifth above it. The reason why the Hypophrygian mode can be used is that it was considered as having a species of the Fourth on the bottom and a final a fourth higher than the starting tone. For example here is the Plagal mode ‘E Hypophrygian’ (i.e. octave species between E-E)…

E F G A Bb C D E

This mode’s final is ‘A’, that is the melodies written in this mode would tend to end on ‘A’, and this tone has an ‘E’ a fifth above it. Its complimentary mode is the Authentic mode ‘A Phrygian’. This was considered different from what would happen with ‘E Locrian’ which would cadence on ‘E’ (which has no Fifth above it).

Notice that all of the Authentic modes (i.e. with species of the Fifth below the species of the Fourth) have odd numbers and the Plagal (or Hypo) modes (i.e. with species of the Fifth above of the Fourth) have even numbers.

So the Authentic modes have the structure (the numbers below the letters are the Medieval mode numbers)…

X Y Z X Y Z X Y Z X Y Z
A A A B B B C C C D D D
———————
3 0 0 9 1 0 0 7 11 0 0 5

For these Authentic modes all the numbers here are odd (not considering 0 which just signifies unacceptable here). Notice that the pattern above is two modes acceptable, two unacceptable (i.e. 5-3, 0-0, 9-1, 0-0, 7-11, 0-0). Also notice that with the above pattern of Authentic modes it is impossible to have a Locrian mode since the very definition of an Authentic mode is that it contains a species of the Fifth on the bottom (and Locrian does not have a Fifth on the bottom).

And the Plagal modes have the structure…

A B C D A B C D A B C D
X X X X Y Y Y Y Z Z Z Z
———————
4 10 0 0 0 2 8 0 0 0 12 6

For these Plagal modes all the numbers here are even (except 0 of course). Notice that the pattern above is four modes acceptable, three unacceptable, two modes acceptable, three unacceptable (i.e. 12-6-4-10, 0-0-0, 2-8, 0-0-0). Also notice that with these Plagal modes there is always six numbers between each pair of Plagal mode numbers (i.e. 4-10, 2-8, 12-6). The Authentic modes share a similar characteristic, except there are four numbers between each pair of Authentic mode numbers (i.e. 9-1, 7-11) where the pairs are both within the same species of the fifth. In the cases where the Authentic modes have a unique species of the fifth this is not the case (i.e. with modes 3 and 5).

As you can see, not all of these 24 modes were not used, twelve were rejected because they give rise to successions of steps not found in a diatonic scale (the modes that I designated above with a 0). Well, that is how we would phrase it today, basically they were considered non-melodic in nature or another way of putting it is that they were melodically ‘non-consonant species’. The melodic elements that led to a particular mode being considered a ‘non-consonant species’ were either of the following: four or five consecutive whole tones in succession, two semitones in succession, a single whole tone between two semitones.

For example if we place tetrachord species ‘Y’ over pentachord species ‘A’ then you would get the following ascending mode (expressed within the octave E-E)…

E F G A B C# D E

And here you have four consecutive whole tones between F and C#, so this was not allowed in Medieval times, although I personally like it. You also have a tone between two semitones, because of the tone D-E between the semitones C#-D and E-F (remember to think cyclically).

Another example, we place tetrachord species ‘X’ over pentachord species ‘D’ then you get…

E F# G# A# B C D E

And here you have two consecutive semitones (between A# and C), so this was also not allowed in Medieval times, but again I can dig it!

For our last example if we place pentachord species ‘D’ above tetrachord species ‘Y’ then you would get…

E F# G A B C# D# E

And here you have four consecutive whole tones between G and D#, again I dig i, but theoretically this was not supposed to be allowed in Medieval times. However, even though this was not an ‘official’ mode, it did appear in actual use as a substitution for Dorian (and later Aeolian), especially in ascending melodic passages. In fact today we call this he ascending melodic minor, and it is used in both ascending and sometimes descending melodic passages. So we must always keep in mind that these modes were very general structures that helped to organize melodic materials, but should not be thought of as laws written in stone (the same with chord structures today).

Also notice that the structure of the Plagal modes above is just the reverse of the Authentic modes, and any two modes represented by the same letter terminology are ‘compliments’ of each other…

1 Dorian Y/B 2 Hypodorian B/Y
3 Phygian X/A 4 Hypophrygian A/X
5 Lydian Z/D 6 Hypolydian D/Z
7 Mixolydian Y/C 8 Hypomixolydian C/Y
9 Aeolian X/B 10 Hypoaeolian B/X
11 Ionian A/C 12 HypoIonian C/Z

So, of the modes that are left what you end up having is six Authentic and six Plagal modes. Eventually the distinction between Authentic and Plagal was dropped and there developed a method of organizing the modes outside of the Authentic and Plagal species of fifths and fourths. The Authentic and Plagal concepts eventually merged into one idea with only the cadences (i.e. Authentic and Plagal) being carried over into the modern way of thinking. Also instead of thinking of the modes as containing a species of Fifth below (Authentic) and above (Plagal) a species of Fourth there seemed to be a transition where the species merged into each other!

This method was based on the way the modes ‘functioned’ and was developed by several theorist but one of the best explanations comes from an Alsatian theologian and music theorist, a cat named Johannes Lippius (1585-1612) who thought of using the Third tone of each species of the Fifth as giving this Species a certain ‘polarity’ (this is the word that I am using today). Lippius used the Third of the bottom Fifth in each mode as the main organizing feature of his theory, but I still detect the influence of the older ‘Plagal’ way of thinking in the way all of this is organized.

Even though it looks like only the six Authentic modes were used I believe that the concept of the Plagal modes is still being expressed in terms of a ‘downward’ Fifth from a tonic tone (however this was given secondary priority after Lippius, who died very young before he could further develop his ideas). If you look at the work of Lippius and some of the writings of Gioseffe Zarlino (1517-1590) then you will see the beginnings or triadic thinking. You can see that they begin to organize the modes according to the Thirds and the Sixths. Well a Sixth is nothing but an upside down (or negative) Third so organizing according to Thirds and Sixths is like organizing positive Fifths (i.e. ascending) according to positive Thirds and negative Fifths (i.e. descending) according to negative Thirds. For example a new way or organizing the modes followed the approach of Zarlino in organizing modes according to the quality of the Third and Sixth above the Final (an older concept of what we call the tonic) and the quality of the Third and Sixth above the Fifth. Since the Sixth above the Fifth is the same as the Third above the Final then this is redundant. What we really have is modes organized according to the quality of Thirds, Sixths and Sevenths (i.e. the Third above Fifth) as follows…

Number, Mode, Third (above Final), Sixth (above Final), Third (above Fifth) Sixth, (above Fifth), sex
———————————-
3 Phygian minor minor minor minor female
9 Aeolian minor minor minor minor female
1 Dorian minor major minor minor female
7 Mixolydian major major minor major male
11 Ionian major major major major male
5 Lydian major major major major male

What this really represents is the symmetrical organization of Thirds around the three most important tones, the Final, the Dominant and to a lesser extent the Subdominant…

Major or Minor Third above the Final (Third)
Major or Minor Third below the Final (Sixth above Final)
Major or Minor Third above the Dominant (Seventh above Final)
Major or Minor Third below the Dominant (same as the Third above Final)

To this can be added…
Major or Minor Third above the Subdominant (same as the Sixth above Final)
Major or Minor Third below the Subdominant (Second above Final)

So, taking out the redundant intervals, the end result is…

Major or Minor Third above the Final (Third)
Major or Minor Third below the Final (Sixth above Final)
Major or Minor Third above the Dominant (Seventh above Final)
Major or Minor Third below the Subdominant (Second above Final)

So what we have here are Thirds above and below the Final, Thirds above the Dominant and Thirds Below the Subdominant. We could say that these are the modern equivalents and expansions of the concepts of ‘fixed’ and ‘moveable’ tones from ancient Greek music theory. However now the ‘fixed’ tones are the Tonic, Dominant and Subdominant and the ‘moveable’ tones (i.e. the ones that take on a major or minor identity) are the Third, Sixth, Seventh and Second.

Zarlino, (and later on Lippuis in much more detail) went on to say that the Third above the Final was the most important of these ‘moveable’ configurations and that determined the “variety of the harmony”. Zarlino notices that something other than a composite of intervals is involved when there are more than two voices sounded together and this is the beginnings of a recognition of the power of the triad…

“The variety of the harmony in such situations does not consist solely in the variety of the consonances that are found between two voices, but also in the variety of the harmonies - which [variety] is determined by the position of the tone that makes a third or tenth above the lowest voice of the composition. Either these [intervals] are minor, and the harmony that arises is determined by or corresponds to the arithmetical proportion or division, or they are major, and such a harmony is determined by or corresponds to the harmonic mean. On this variety depends all the diversity and perfection of harmonies… For as I have said elsewhere, when the major third is below, the harmony is cheerful, and when it is placed above, the harmony is sad.”

The arithmetic and harmonic means mentioned above are based on the string length perspective (as opposed the the frequency perspective which is the reciprocal perspective) and this was the normal way of looking at these proportions during this time.

Thinking in string lengths when an octave is divided by the arithmetic mean then you get an interval of a perfect Fourth on the bottom and a perfect Fifth on top, and this corresponds to the division of the Plagal modes in Medieval times. When an octave is divided by the harmonic mean then you get an interval of a perfect Fifth on the bottom and a perfect Fourth on top, and this corresponds to the division of the Authentic modes in Medieval times.

Correspondingly when a Fifth is divided by the arithmetic mean then you get an interval of a minor Third on the bottom and a major Third on top, and this corresponds to the division of the minor triad. When a Fifth is divided by the harmonic mean then you get an interval of a major Third on the bottom and a minor Third on top, and this corresponds to the division of the major triad. This is what Zarlino is referring to.

Back to our new organization of the modes…

Number, Mode, Third (above Final), Sixth (above Final), Third (above Fifth) Sixth, (above Fifth), sex
———————————–
3 Phygian minor minor minor minor female
9 Aeolian minor minor minor minor female
1 Dorian minor major minor minor female
7 Mixolydian major major minor major male
11 Ionian major major major major male
5 Lydian major major major major male

I have organized the modes here in terms of increasing ‘brightness’. Of course if Locrian had been included it would be the ‘darkest’ mode, however it seems that the male dominated culture could not deal with the ‘extremely feminine nature’ of the Locrian mode.

Also notice that only Dorian and Mixolydian have structures that are not completely minor or major. This is very important for considering the ‘function’ of the modes! Both Dorian and Mixolydian have modulatory functions, Dorian most naturally expressing a negative dominant function and Mixolydian expressing a positive dominant function.

If only the thirds are used as the organizing criteria then we have the minor-major female-male associations above and the beginnings of the major-minor tonality system. This is a step beyond the completely modal theories of Heinreich Glareanus (1488 1563) and his contemporaries.

Johannes Lippius went further and actually described the triad, partly by using analogies that were based on his detailed knowledge of Christian theology as models for his musical ideas, and presented a unified theory of a harmonic approach. Lippius recognized all inversion relationships between intervals, including those between Thirds and Sixths and Seconds and Sevenths, he suggested that music be composed from the bass (not from the tenor which was the norm at that time), he coined a term for the triad calling it ‘trias harmonica’, he differentiated the modes by the major and minor tonic triad, and he replaced the study of counterpoint with a study of harmony based on the triad. After defining music as a mathematical science Lippius turns to a systematic treatment of harmonic structures based on one, two and three pitches, i.e. Monands, Dyads and Triads (by the way this approach is very similar to what Von Freeman told me long ago, but I only found out about Lippius rather recently). As far as the modes, in the end the quality of the third degree above the final was the most important aspect for Lippius, not species (i.e. Interval structures). In this way Lippius was a bridge between modal and major-minor Harmony just as Botheius, many Arab theorist, Guido and Glareanus were bridges from ancient Greek music theory to Medieval modality. I ultimately believe that the Greeks themselves were a bridge from the much older music systems of the ancient Egyptian and Sumerian\Babylonian civilizations. So we may be going full circle here.

One final thought, I have meditated for a long time about where this ‘Plagal’ mode concept (i.e. the Medieval ‘Hypo’ concept) is coming from functionally. Finally I believe now that the Plagal modes could be thought of as negative (upside down in structure) Authentic modes that have a final on their Fifth degree! This is because the root of negative structures is always the negative Fifth! In this sense, in the example above of the Plagal mode ‘E Hypophrygian’ the actual negative mode (descending) is ‘negative E Lydian’, which has a final (i.e. Tonic) of ‘A’…

E D C Bb A G F E

We would ‘hear’ this in sound as ‘A Phrygian’ but with a melodic range between E-E. So I believe the the Plagal modes are the feminine polar counterparts to the male Authentic modes, but they functioned according to ‘Telluric’ Gravity principles. This means that they were the same structures as the Authentic modes but reversed (i.e. descending instead of ascending), however the finals (and later the tonics) of these Plagal structures was the Fifth of the negative structure. This because humans will always hear the bottom tone of a perfect Fifth interval as the primary gravitational point (i.e. Final, Tonic or Root).

Please feel free to respond to any of the points here.

Peace,

Steve

Negative Rhythms?

Hey Mr.Coleman,
Thank you very much for the class yesterday at SIM. It was very intense, and the most engaging masterclass we have had so far. I thought it was very well organized as far as playing vs. explaining, and I had a great time clapping and singing through everything. I am writing because you told us outside the SIM building to email you with any questions we might have. I have one: what did you mean by negative rhythm? Did you mean taking a certain pattern and playing it backwards? or some sort of palindrome cycle? What are some examples of negative rhythm?

The term negative is just a polarity term for positive, and it may not be the best word since it has ‘bad’ connotations for most people. What I am really into could be best described as correlative thought (associative thinking) and polarity.

The negative rhythm thing is a matter of balancing. The temporal things that I talked about in the workshop, (3 2 2 3, etc.) these are forms, not rhythms. Forms are the imaginary structures that the actual rhythms can reside in. Just because you have a 3 2 2 3 form (which is just a kind of 10, or 5 in cut time, but with the weights distributed in another way than the 5 you normally hear) does not say anything about the actual rhythms involved.

A discussion about 4/4, 5/5, etc, is not a discussion about rhythm really, no more than a discussion about scales or modes is a discussion about melody. However in former times (ancient Greece, Middle Ages) discussing modes was closer to talking about melody because a mode was not just the assemblage of pitches in the structure but mostly a ‘way’ of playing a particular group of pitches and the relationship these pitches had to each other. This way of thinking still holds in the raga system on the Indian subcontinent. In ancient times a tetrachord was not just a group of four pitches, these pitches has dynamic functional relationships to each other, that is there was a system of movement tendencies associated with the group of pitches also. And since tetrachords (and in the Middle Ages pentachords also) were the cellular structures that formed the modes, then these dynamic pitch relationships got transferred to the modes also.

On to rhythm. What I call rhythmic modes (i.e. 3 2 2 3, 3 2 3 3 2 2, etc.) are kinds of forms that you can use as a way or organizing time. Using one analogy you could think of the rhythmic modes as the body and the rhythms as the soul. Or we could say that the rhythms are the meat, the actual sounded musical material, whereas the rhythmic modes are simply imaginary structures in your mind that are used as organizational devices used to help keep your place, tools that you use to negotiate time in the same way that we use a watch as a tool to negotiate time.

You see, the human sense of time is based on reality, that time is relative. So our sense of time is relative and the musician relies mostly on a relative time base which is flexible. Not much of an attempt is made, especially in spontaneously composed music, to use absolute time in music. In composed music you see absolute markings that relate to seconds or minutes, but unless you are using a computer, the human relative sense of time takes over once the music starts and the time is variable from that point onward. Of course even so-called absolute time is based ultimately on cycles in Nature. The most obvious of these cycles are the rotation of the earth about its axis (the day), the lunar phases resulting from a combination of the revolution of the moon around the earth and the revolution of the earth around the sun, all creating the angular relationship of the earth-moon-sun system (the month) and the revolution of the earth around the sun (the year). You can see that the day and year are simple relationships that are basically one cycle each, but the month is a more complex relationship resulting from the combination of two cycles.

Our own sense of time is relative, based on how long we have been alive and other factors. Children have not been living so long so that everything is relative to their current lifespan. So one month in the life of a child is a relatively longer period of time than one month in the life of an adult (you frequently hear older people talking about how ‘time flies’)

However we use internal references also such as the beating of our heart, which generally beats slower as we get older. One of the reasons an older person, for example saxophonist Von Freeman, are better at playing ballades is that

In all of my teaching one of the main things I notice is that young people (who make up most of the class when you are teaching) tend to rush when playing music. Young people have less patience, and the tendency to want to push the beat is greater. So you have to make a conscious effort to relax and lay back. This tendency is counteracted in some cultures, especially in the African Diaspora. This may be because initially in these cultures it is frequent for much older people to play alongside younger people and the ‘way’ of playing may more easily be transferred to the younger musicians, but I’m just speculating here. My own experience is that I picked this up from playing with much older musicians. I remember when I first joined the Thad Jones-Mel Lewis Orchestra that I was always ahead of Thad in terms of where I felt the time (this was true when I played alongside Von Freeman also), so I had to consciously slow down – and after some time this became a habit.

What I was referring to when I mentioned negative rhythm was motion in the actual rhythms (not the rhythmic mode forms), that is the manner in which the rhythm figures move. For example take this rhythm where the lower pitch is a bass drum and the higher is a snare…



In my opinion the rhythm goes negative in motion for a brief moment in the second measure, because of the forward motion that was set up in the first measure. In the beginning all of the sounds drive towards (or get pulled towards) the downbeat, in terms of feeling. However in the middle of the second measure this feeling shifts to getting pulled ‘away’ from the downbeat (or towards the upbeat). Now this is a very simple example, but from this kind of directional thinking, toward and away from a particular nodal point (not necessarily forwards and backwards because we don’t move backwards in time, in this dimension anyway) and also another balancing idea that deals with masculine and feminine rhythms (another subject for another day) you can get a kind a analogous feeling towards a rhythmic symbolism that accomplishes the same kind of functions as its tonal counterparts, or even universal counterparts.

Now for a more complex example, take the bass and drum part to a composition of mine called ‘Black Hole’ recorded on the Dave Holland CD called ‘Extensions’…



You can hear a little of the composition here.

This is just one of many possible ways to approximately notate this. There are several things happening with this composition symbolically, but I will just discuss the rhythmic elements that pertain to this conversation now. The musical effect that I was going for here was that of energy being pulled back into a ‘Black Hole’, so I was trying to generate a perpetual forward and backward motion. The backwards motion starts because of the shift in the relationship of the drums and the bass at the end of the first measure. Now without the help of a first class drummer (in this case the incomparable Marvin “Smitty” Smith) this effect would not come of as sounding very natural. “Smitty” gives the drum chant a very natural feel enabling it to breathe and flow. I just want to stress that for me it is the relationship between the two parts that gives the feeling of the backward or negative rhythm here. You could say that the drums ‘sit’ on the opposite side of the pulse in relation to the bass. However this effect is complicated in ‘Black Hole’ because the pulse gets flipped because of the odd number of elements in the S S S L S S rhythmic mode (i.e. 2 2 2 3 2 2). So we are talking about a rhythm sitting on the other side of a group of pulses (expressed in the drum part), where the thrust (forward motion) of the pulses themselves (expressed in the bass part) could be perceived as changing in polarity. Now I developed a theoretical language to speak about this kind of motion but it is too much to go into this here. However I can say that it is a kind of rhythmic counterpoint that I developed from years of listening to mostly West African drum music.

Another question: the cycles i have heard from your cd’s and the one we did in class dealt with combinations of 2 and 3 note groupings. i know that those two groupings can be used to express all other subdivisions, but there is definitely a change in feeling when one is playing over a beat subdivided in 5 equal notes, as opposed to 3+2, or 2+3 (for example). Do you deal with larger beat subdivisions which are not broken down into arrangements of 2 or 3? I’d like to go in that direction, and have cycles made up of sparser, farther apart beats than 2 and 3 can allow. Any thoughts?

Sure, you could do anything that you want to do, whatever you feel like doing, go for it. I have reasons for using 3 and 2 that are symbolic of something else, and its just expressed as music. Of course I have thought about using other divisions and it does pop up occasionally. There are a few things we do that use 1, 2, 3, 4, 5 etc., but I guess you can see that what makes a beat 2, 3, 5, etc. is the proportion of it to something else. In other words if all your beats were 5 then you could just as well call the beats 2. And of course any beat can be subdivided any way you want it to be. So in the end all of this is just illusion, its just the way you organize the time in your mind. Because in the end these things don’t really exist, they are figments of our imaginations. But if they help you to get close to whatever it is you are trying to express then its all good.

Now when I say “these things don’t really exist” I am referring to these organizational tools, the time signatures, rhythmic modes, etc. The actual sounded material (which could be interpreted in any number of ways) was what I was referring to in the class at SIM. For me there is a big difference between the sounded material and the mental constructs that we use to organize this material. I keep repeating this because I feel that the actual sound is not stressed enough and we get hung up in things like the various means to organize the music (which changes based on training, culture, etc.) and even more secondary concepts like transcription and notation. Basically I choose various musical tools because these are the tools that I need to develop and express certain principles. I try to focus on the principles, then I select the appropriate tools, so that the principles are guiding which tools I use.

In conclusion, I have always had a fascination with temporal matters and for me this cannot be divorced from tonal concerns. There is a very interesting article somewhat related to this where the different kinds of metaphors used to address time are discussed.

Anyway, thank you again for your class. It really gave me a lot to think about and excited me to see I am on the right track (as far as rhythm is concerned) in my playing of rhythms like yours.

Well, I cannot say that I am on the ‘right track’, and I certainly cannot say if you are or not. I’m not sure there is a right track, but I try to follow that voice from within as much as possible.

Peace,

Steve

Symmetric Questions

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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.

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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: and 13 (13: 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