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
                    |
        tetrachord  |  G  paranete hyperbolaion
       hyperbolaion |
                    |  F  trite hyperbolaion
                     - E  nete diezeugmenon
                    |
        tetrachord  |  D  paranete diezeugmenon
       diezeugmenon |
                    |  C  trite diezeugmenon
                     - B  paramese
tone of disjunction
                     - A  mese
                    |
        tetrachord  |  G  lichanos meson
          meson     |
                    |  F  parhypate meson
                     - E  hypate meson
                    |
         tetrachord |  D  lichanos hypaton
          hypaton   |
                    |  C  parhypate hypaton
                     - B  hypate hypaton
tone of disjunction
                       A  Proslambanomenos

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

              -  D  nete synemmenon
             |
 tetrachord  |   C  paranete synemmenon
 synemmenon  |
             |   Bb  trite synemmenon
              -  A  mese
             |
  tetrachord |   G  lichanos meson
    meson    |
             |   F  parhypate meson
              -  E  hypate meson
             |
  tetrachord |   D  lichanos hypaton
   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
---------------------------------------------------------------------------
Doristi         Phrygian      1 2 2 2 1 2 2   B-   (- tonic, more or less)
Phrygisti       Dorian        2 1 2 2 2 1 2   A+-   - dominant
Lydisti         Hypolydian    2 2 1 2 2 2 1   B+    + tonic
Mixolydisti     Hypophrygian  1 2 2 1 2 2 2   C-  (diminished, more or less)
Hypoddoristi    Hypodorian    2 1 2 2 1 2 2   D-    - tonic
Hypophrygisti   Mixolydian    2 2 1 2 2 1 2   D+    + dominant
Hypolydisti     Lydian        2 2 2 1 2 2 1   C+   (+ tonic, more or less)
Hypomixolydisti            (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.

You can always calculate which pitch is Mese, by looking at the relative position of these two tetrachords that determines which Harmonia you are looking at. If you think of these two tetrachords as being separated by a tone (what the Greeks called a ‘disjunction’) then Mese is always the top pitch of the tetrachord just under the disjunction. Its easiest to see this with Doristi. I will identify the tetrachords by putting them between these symbols.

Lets say that the tetrachords use these symbols around their pitches (the initial in the parenthesis is the first letter of the tetrachord name):

|| || = Hyperbolaion (Hy.)
 ( ) = Diezeugmenon (D.)
 [ ] = Meson (M.)
 { } = Hyperbolaion (H.)

If we use E-E as our octave then we have the following…

7   E F G} [A Bb C D] (E        Mixolydisti     (Mese = D)
6   E F#} [G# A B C#] (D# E     Lydisti         (Mese = C#)
5   E} [F# G A B] (C# D E       Phrygisti       (Mese = B)
4   [E F G A] (B C D E)         Doristi         (Mese = A)
3   E F# G#] (A# B C# D#) ||E   Hypolydisti     (Mese = G#)
2   E F#] (G# A B C#) ||D E     Hypophrygisti   (Mese = F#)
1   E] (F# G A B) ||C D E       Hypodoristi     (Mese = E)

Tonoi                       Tonoi           Position
Structure                   Name            Of Mese (from bottom)
---------------------------------------------------------------------------
E  F  G  A  Bb C  D  E      Mixolydisti     7   Hypate H. to Paramese (D.)
E  F# G# A  B  C# D# E      Lydisti         6   Parhypate H. to Trite D.
E  F# G  A  B  C# D  E      Phrygisti       5   Lichanos H. to Paranete D.
E  F  G  A  B  C  D  E      Doristi         4   Hypate M. to Nete D.
E  F# G# A# B  C# D# E      Hypolydisti     3   Parhypate M. to Trite Hy.
E  F# G# A  B  C# D  E      Hypophrygisti   2   Lichanos M. to Paranete Hy.
E  F# G  A  B  C  D  E      Hypodoristi     1   Mese (M.) to Nete Hy.

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, and now we can see the symmetry in the names…

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

E  F  G  A  B  C  D  E      Doristi         4   Hypate M. to Nete D.
E  F# G  A  B  C  D  E      Hypodoristi     1   Mese (M.) to Nete Hy.

E  F# G  A  B  C# D  E      Phrygisti       5   Lichanos M. to Paranete D.
E  F# G# A  B  C# D  E      Hypophrygisti   2   Lichanos M. to Paranete Hy.

E  F# G# A  B  C# D# E      Lydisti         6   Parhypate H. to Trite D.
E  F# G# A# B  C# D# E      Hypolydisti     3   Parhypate M. to Trite Hy.

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…

#   Mode        Third   Sixth       Third   Sixth
                (above Final)       (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…

#   Mode        Third   Sixth       Third   Sixth
                (above Final)       (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

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