The Parallage of John Plousiadinos

[brucewayne]

Could anyone explain how the diagram of John Plousiadinos works? The one with four large circles connected by smaller circles.

I have tried to read the English Wikipedia page on it, but it is a mess. Or perhaps someone has a better reference or article in English, Greek or Russian?

I think the basic idea is that any sound can be considered to be any other sound. For example if you start at πλ δ and go up α β γ then you can find nana in any large circle and jump off from there. For example, you could treat γ as πλ δ and perform the same progression again which gives you a triphonic scale like G A B C D E F. This is indicated by starting at the large bottom right circle and traveling up and right twice.

But you could instead switch to the large bottom left circle and treat γ as β and travel up and left as γ δ α giving G A B C Db Eb F. I am not sure the usefulness of such a progression, but that’s how I am interpreting the diagram. Is this correct? Maybe such a modulation was common in medieval chant?

What is the diagram’s relevance to chant today and what was its relevance to medieval chant?

 

[tudorvolcano]

Could anyone explain how the diagram of John Plousiadinos works?

It's a diagram of connected tetrachords, that shows you how you modulate when you have glued tetrachords, and not disjunct tetrachords, like in the case of the Wheel system.

When you have glued tetrachords, you destroy (φθοράω) the standard scale system of adjacent fifths (τετραφωνία), reason for which we call those modulators phtoras/destroyers.

The one with four large circles connected by smaller circles

The four large circles are the phthoras that join tetrachords together.

For example if you start at πλ δ and go up α β γ then you can find nana in any large circle and jump off from there.

No, Nana is Plagios tou Tetartou upwards, but below the base is Tritos: so you combine the Plagios tou Tetartou tetrachord with the Tritos tetrachord, like this:

If you were in the standard wheel system, below the base, you should have modulated in Tetartos, like this:

You have this conjunction of tetrachords at this part of the parallage:

But you could instead switch to the large bottom left circle and treat γ as β and travel up

That is a misunderstanding of the Papadike's Protheoria;
Authentic modes (Kyrioi Echoi) do not go above the base; they always go below it and then again upwards.
If you want to go up and down, you are modulating in a plagal mode.
Authentic: up-down-up; Plagal: down-up-down : Look at the apechemata for that, and you'll understand why they are designed in that matter.

The diagram is still relevant for today:

1) The first combination (Pthora Nenano) shows you the mixed chromatic-diatonic scale of Plagios tou Deuterou and Echos/Phthora Nenano:

The second combination shows you Echos Protos as it is sung today, with Tetartos below Pa:


The third one is Barys with Phthora Naos on Ga (Bestenigâr Makamı):

The fourth combination is Barys Diatonikos:

The fifth combination seems to be Plagios tou Protou with Phthora Naos on Ga (Sabâ Makamı):

The sixth combination seems to be Barys Diatonikos Heptaphonos:

The seventh combination is Deuteroprotos:

The eighth combination, you saw it, is Phthora Nana:

What is the diagram’s relevance to chant today and what was its relevance to medieval chant?

The diagram is still relevant today, it was pretty much the same thing in medieval chant; it's just that there were different theorists across time seeing things from different perspectives; more modern theorists used the Oriental interpretation, by comparing Byzantine Echoi with the Oriental Makam repertoire, while medieval chanters and theorists did not use Arabic-Persian Music Theory, but rather the Ancient Greek music Tonoi and Music Theory.

 

[Pappous43]

Intervals 13.87, 12.23, 9.04, 3.78 above, are correctly natural (pythagorean)
ratios : 8/7, 9/8, 12/11, 28/27

but what ratios are 8.609849 and 5.413508 ?

 

[tudorvolcano]

Intervals 13.87, 12.23, 9.04, 3.78 above, are correctly natural (pythagorean)
ratios : 8/7, 9/8, 12/11, 28/27

but what ratios are 8.609849 and 5.413508 ?

8.609849 : 88/81 (Al-Farabi's minor neutral tone / undecimal subtone)
5.413508: 256/243 (Pythagorean Limma)

 

[Pappous43]

Yes, I was expecting something like that, 256/243. Too big integers ......

 

[brucewayne]

Thanks, this is a lot of information and I'm still digesting it. Some questions I have so far:
1. What is the meaning of the tones that are written outside the small circles?

2. Am I understanding correctly that you "skipped" over the large circles? Could one also include the larger circles in the progression (making disjunct tetrachords)?

3 Can the circles only be traversed "straight"? Could one not make a curve? Or even jump around, going up and then jumping to a different larger circle?

4. Where did the intervals come from? In the context of medieval chant, the latest research that I know of is inconclusive regarding the intervals that were used, but it's assumed that they were probably diatonic or close to diatonic. Primarily based on Strunk's conclusions by analyzing melodic leaps. Are the intervals you are using an interpretation based on modern intervals?

 

[tudorvolcano]

Thanks, this is a lot of information and I'm still digesting it. Some questions I have so far:
1. What is the meaning of the tones that are written outside the small circles?View attachment 127542

2. Am I understanding correctly that you "skipped" over the large circles? Could one also include the larger circles in the progression (making disjunct tetrachords)?

3 Can the circles only be traversed "straight"? Could one not make a curve? Or even jump around, going up and then jumping to a different larger circle?View attachment 127543

4. Where did the intervals come from? In the context of medieval chant, the latest research that I know of is inconclusive regarding the intervals that were used, but it's assumed that they were probably diatonic or close to diatonic. Primarily based on Strunk's conclusions by analyzing melodic leaps. Are the intervals you are using an interpretation based on modern intervals?

1) The small circles are just progressions by perfect fifths:

2) The purpose of Plousiadinos Parallage is to show how you deal with conjunct tetrachords; the large circles are conjunction points.
Otherwise, you can just take Saint Ioannes Koukouzeles' on the left, and progress on the standard scale by progressions of perfect fifths.

3) When you make a curve, you're just progressing the standard scale, just like in the Trochos system.
So nothing interesting there.

4) The intervals are from Ancient Greek Music treatises and from Al-Farabi (whose intervals are used by Chrysanth of Madytos).
The scales with ”quarter tones” are mentioned as early as Aristoxenus's Elementa Harmonica (look here for τρεῖς διέσεις).

but it's assumed that they were probably diatonic or close to diatonic

What exactly do you mean by "diatonic"? For Byzantine and Ancient Greek Music, "diatonic" carries very different connotations when compared to Western Music; diatonic by etymology means "through tones", meaning that it would indicate a scale made up of only tones.
In the perspective of the diatonic scale being made of "tones", the intervals were always diatonic as the default in Byzantine Music, whether Medieval or Modern, we still call our standard scale "diatonic".

Primarily based on Strunk's conclusions by analyzing melodic leaps

Echos Deuteros would be diatonic too in some ways, according to the Ancient definition.
Because you have 88/81 * 9/8 * 12/11 = 4/3, you get a perfect fourth between Plagios tou Deuterou and Protos, when Protos is destroyed and becomes Nenano, by going down by a quarter tone.
Pretty much, the same leaps are consonant in Modern Byzantine Music, just as in Medieval Byzantine Music.

Are the intervals you are using an interpretation based on modern intervals?

No, not really; it's a simplified view based on Chrysanth's theory.
The actual vocal interpretation of the intervals in Modern Byzantine Music is a whole subject on its own, with many school differences, regional differences etc. (no two psaltes would use an identical set of scales).

 

[brucewayne]

For 1, I may have been misunderstood. Inside each circle is a tone, but written outside some of the circles are alternate tones. What are these alternate tones for? I thought maybe they were supposed to be "equivalent tones", and that if you swap them out with the tones in the circle you would get the same progression, but that appears not to always be the case.

If that were the case then, the first tone and the third tone would be equivalent, which makes no sense.

So I'm not sure what these extra tones are for. Are they just alternate scales? As in, I could erase the tones in the circles and write those in as alternate tones to generate even more scales?

By diatonic I meant the diatonic genus in which the largest interval in the tetrachord is close to a major second (12 out of 72 moria). To ask a particular question, the sixth combination which you identified as Barys Diatonikos Heptaphonos: to me a simple reading of the tones would give something close to a whole tone scale. That is, if I sing "agia, ananes, neanes", I have travelled 12+12+12 moria (or 12 + 12 + 10 if you would like). If I descend down, I sing "aneanes, neagie, aanes" and I have travelled down 12 + 12 +12 moria. So the scale is one of 12 12 12 12 12 12. But you have a very different scale, one which contains a very small interval. Where is my misunderstanding here?

 

[brucewayne]

Assigning numbers to each node from left to right, top to bottom, these are all the combinations I see.

I'm still not sure whether the additional notes outside the circles mean anything. For now I have ignored them and I assume they are just a decoration showing the notes of the wheel.

I'm also not sure why the fourth house (nana) contains the second tone (neanes) or whether it affects anything. It seems like it would only make sense if there were a third branch going up that starts on the third tone.

First House - Plagal First

6 - 4 - 1
D E F G A B C (Plagal First = First + Fourth)

6 - 4 - 2
D E F G A Bb C (Fourth + First)
Transposed: A B C D E F G

7 - 4 - 1
E F G A B C# D (Nenano + Fourth)
Transposed: D Eb F G A B C

or, assuming Nenano was chromatic:

E F G# A B C# D (Chromatic Nenano + Fourth)
Transposed: D Eb F# G A B C

7 - 4 - 2
E F G A B C D (Nenano)
Transposed: D Eb F G A Bb C

or, assuming Nenano was chromatic:

E F G# A B C D (Chromatic Nenano)
Transposed: D Eb F# G A Bb C

Second House - Plagal Second

7 - 5 - 2
E F G A B C D (Plagal Second = Second + First)

7 - 5 - 3
E F G A Bb C D (Mesos Fourth; Modern Diatonic Varys)
Transposed: B C D E F G A

8 - 5 - 2
F G A B C# D E (Varys + First)
Transposed: Bb C D E F# G A

8 - 5 - 3
F G A B C D E (Varys = Third + Second)

Third House - Varys
The fourth note can be either Second or Third and both options are considered below.

11 - 9 - 6 (mixed with Second)
F G A B C D E (Narrow Legetos)
Transposed: Bb C D E F G A

11 - 9 - 6 (mixed with Third)
F G A Bb Cb Db Eb (Plagal Fourth + Second)

11 - 9 - 7 (mixed with Second)
F G A B C# D# E# (Varys + Third)

11 - 9 - 7 (mixed with Third)
F G A Bb C D E (Wide Legetos - Plagal Fourth + Third)
Transposed: Bb C D Eb F G A

12 - 9 - 6 (mixed with Second)
C D E F# G A B (Varys)
Transposed: F G A B C D E

12 - 9 - 6 (mixed with Third)
C D E F Gb Ab Bb (Plagal Fourth + Second)

12 - 9 - 7 (mixed with Second)
C D E F# G# A# B# (Varys + Varys)

12 - 9 - 7 (mixed with Third)
C D E F G A B (Plagal Fourth = Fourth + Third)

Fourth House - Plagal Fourth
The fourth note can be either Third or Fourth and both options are considered below.

12 - 10 - 7 (mixed with Third)
C D E F G A B (Plagal Fourth)

12 - 10 - 7 (mixed with Plagal Fourth)
C D E F# G# A# B# (Varys + Varys)

12 - 10 - 8 (mixed with Third)
C D E F G A Bb (Nana)

12 - 10 - 8 (mixed with Plagal Fourth)
C D E F# G# A# B (Varys + Plagal Fourth)
Transposed: F G A B C# D# E

13 - 10 - 7 (Mixed with Plagal Fourth)
D E F G A B C# (First + Varys)

13 - 10 - 7 (Mixed with Third)
D E F Gb Ab Bb C (?)

13 - 10 - 8 (Mixed with Plagal Fourth)
D E F G A B C (Plagal First)

13 - 10 - 8 (Mixed with Third)
D E F Gb Ab Bb Cb (?)

If the mixture with the second mode is allowed, then we have three additional hypothetical circles with tones three, four, and one, and the following are obtained
12 - 10 - (hypothetical node)
C D E F# G A B (Varys)

13 - 10 - (hypothetical node)
D E F G Ab Bb C (First + Second)

 

[tudorvolcano]

For 1, I may have been misunderstood. Inside each circle is a tone, but written outside some of the circles are alternate tones. What are these alternate tones for? I thought maybe they were supposed to be "equivalent tones", and that if you swap them out with the tones in the circle you would get the same progression, but that appears not to always be the case.View attachment 127597

If that were the case then, the first tone and the third tone would be equivalent, which makes no sense.
View attachment 127598
So I'm not sure what these extra tones are for. Are they just alternate scales? As in, I could erase the tones in the circles and write those in as alternate tones to generate even more scales?

By diatonic I meant the diatonic genus in which the largest interval in the tetrachord is close to a major second (12 out of 72 moria). To ask a particular question, the sixth combination which you identified as Barys Diatonikos Heptaphonos: to me a simple reading of the tones would give something close to a whole tone scale. That is, if I sing "agia, ananes, neanes", I have travelled 12+12+12 moria (or 12 + 12 + 10 if you would like). If I descend down, I sing "aneanes, neagie, aanes" and I have travelled down 12 + 12 +12 moria. So the scale is one of 12 12 12 12 12 12. But you have a very different scale, one which contains a very small interval. Where is my misunderstanding here?

They are indeed alternate tones.

If that were the case then, the first tone and the third tone would be equivalent, which makes no sense.

It does; Varys is the Mesos of Protos, and Plagios tou Protou is the Mesos of Tritos.
They can be used interchangeably, you can actually see the phthora of Pa on Ga in the Exegesis of Saint Ioannes Koukouzelis' Mega Ison.
Just like Tetartos and Deuteros are equivalents, because Plagios tou Deuterou (Legetos) is the Mesos of Tetartos, and Plagios tou Tetartou is the Mesos of Deuteros, reason for which Deuteros is placed on Di instead of Zo' in modern notation, it makes parallage a lot easier, due to commonalities.

By diatonic I meant the diatonic genus in which the largest interval in the tetrachord is close to a major second (12 out of 72 moria)

Deuteros in that regard is diatonic, as the largest interval is a major second; the 14 moria interval is because of an acoustic effect; probably, the major second would be closer to 13 moria rather than 12, when actually heard, at least in Byzantine Music (diatonic Ni-Pa, Di-Ke, Ga-Di etc.).
On instruments, you can use the rational interval of 9/8, which is around 12,235 moria.

to me a simple reading of the tones would give something close to a whole tone scale

That is not present in Byzantine Music; when you have 3 successive major seconds, you create an extremely dissonant tritone; then, an augmented fifth, sharper by a semitone, instead of a perfect fifth, or at worst wolf fifth etc.

"aneanes, neagie, aanes"

Aneanes: you go up by a minor third, then down twice, by two successive neutral thirds; where is the whole tone here?

I'm still not sure whether the additional notes outside the circles mean anything. For now I have ignored them and I assume they are just a decoration showing the notes of the wheel.

They are not for decoration only; they actually show you the equivalent modulations: diapason vs successive fifths, the interventions of the mesoi etc.; I wanted to write a longer reply, but I do not have the time to do it for the moment;

I'm also not sure why the fourth house (nana) contains the second tone (neanes) or whether it affects anything. It seems like it would only make sense if there were a third branch going up that starts on the third tone.

Maybe try making a diagram; also, the letter notes of Western notation seem rather confusing.

 

[brucewayne]

Aneanes: you go up by a minor third, then down twice, by two successive neutral thirds; where is the whole tone here?

Here I was referring to singing the wheel. In which case, you go down a whole tone, then up a minor third and down twice.
Could you explain where your intervals for that scale came from? In particular one of those intervals looks like a quarter tone. Where does that come from?

It does; Varys is the Mesos of Protos, and Plagios tou Protou is the Mesos of Tritos.
They can be used interchangeably, you can actually see the phthora of Pa on Ga in the Exegesis of Saint Ioannes Koukouzelis' Mega Ison.
Just like Tetartos and Deuteros are equivalents, because Plagios tou Deuterou (Legetos) is the Mesos of Tetartos, and Plagios tou Tetartou is the Mesos of Deuteros, reason for which Deuteros is placed on Di instead of Zo' in modern notation, it makes parallage a lot easier, due to commonalities.

I don't think I disagree with anything here, but I think we are using the same words to mean two different things. I see the relationship between the first and third mode. But the intervals are definitely different. If I place a PA fthora on GA = F, I will sing F G Ab. If they were equivalent, then I would instead sing F G A.

If what the diagram means is that instead of C D E F G A B, I can sing C D E F G Ab Bb, I guess that's okay. This is a mixture of Plagal Fourth and First. But I'm not sure that that is what it's saying.



Regarding the extra markings, I've looked at thee different versions of the diagram and they all place the inner markings in different places. For example, in the version you posted above, the markings are not even next to the circles. See for example the markings circled in red: