# What is going on here? Prokofiev Analysis



## Samuel Kristopher

After falling in love with the piece, I've been puzzling over Prokofiev's Cantata for the 20th Anniversary of the October Revolution. (VERY) unfortunately, I cannot find a score for it anywhere, and I've planned a trip to the National Library here in St. Pete to see if I can find it, but at the moment I'm stuck with analysing it by ear.

Of the whole work, there's one section that boggles my musical mind, so I decided to try and score it on Sibelius and come to you guys for a bit of musical analysis.

If you haven't heard the piece, I highly recommend it - here is a recording by Jarvi (I prefer Mark Elder with BBC Symphony Orchestra best but it's not on YouTube that I can see). The theme in question starts at 3:37 and runs until 4:20.

Here is my very stripped-down score of the chords and melody. Please be aware: I'm a self-taught amateur at music theory. I like to think I have a somewhat good ear for pitch and music but these chords & spellings may not be accurate. In hindsight, I feel like the C# should have been written as D-flat, but I'll gladly take some instruction on those points. Anyway, the melody, I'm 99% sure, is pretty much faithful.









Now for my amateur attempt at analysing this: So the progression starts off quite conventionally it seems, moving through I-V-vii-IV, assuming I've noted the chords correctly. Then, all of a sudden, we go from F major (IV) to C#/D-flat. This is a very satisfying leap (aesthetically), which surprised me because D-flat is so far away from the C major scale. I _do_ notice that the melody hangs on F for that quarter-beat before the shift, and obviously F is the only common note connecting F major and D-flat. So one of my questions is, is this common-tone modulation? Is it even modulation at all?

I feel like we get a bit of a cadence (albeit short-lived) on F minor, so I suppose it's plausible to say that Prokofiev used F major's shared tone with D-flat (the VI chord in F minor) to get to F minor. Please correct me if this isn't the case.

What happens next is a bit dazzling, when that momentary cadence leaps to D minor, back towards the C major scale?? There's a bridge chord at the end of measure 11 that enables it but I don't feel I was able to notate it well, and probably that bridge chord provides the context for shifting to D minor.

Then I confess I lose the plot altogether. From D minor to B-flat diminished?? Then to A-flat minor, to some diminished chord (maybe a seventh, maybe not, my brain's fried already), then conclusively back to C major. And after all that compositional wizardry, not once did I feel jarred by the leaps - contrarily, this is now one of my favourite moments in classical music, especially with all the harmonies included.

If anyone could listen to the aforementioned section and take a look at my attempt to score it, and tell me if my assessment was either dribble or on the right track, I'd be very glad indeed  I don't know anyone to ask in real life about this sort of thing!

Cheers


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## JosefinaHW

I am self-teaching myself too and I like to follow instruction and examples of analysis, so I look forward to seeing how people respond to this post. (I might be able to make a small contribution to your project. There is a scanned copy of the orchestral score to this piece on Amazon. I purchased a Russian text from this source a month or so ago; it was legal because the text was out of print for so long; I don't know if it is legal in this case, but with shipping it will cost you approx. $20. They did not rip me off, so I think it isn't a terrible risk. Don't be rushed by the only one copy available thing, it's printed on demand.

http://www.amazon.com/gp/offer-listing/B00JDE6BSO/ref=dp_olp_new_mbc?ie=UTF8&condition=new

Wrong Cantata, Samuel, apologies.


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## Samuel Kristopher

Thanks anyway Josefina  I also hope someone can shed some light, as I love these chord changes so much and it would be great to be able to use them myself somehow in my own compositions.


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## TwoPhotons

Well done with the transcription. Before trying to answer your analysis question, let me point out some notational errors in your score:

1. As you say, I would put Db at measure 9. But for C# major, the F needs to be an E# (it is the third degree, C#->D#->E#)
2. Measure 12, Ab minor is Ab, Cb, Eb, and I would do Abdim7 rather than G#dim7

Now, I think there are some inaccuracies in the harmony in measures 10-12. The way I hear it is like this: the last two beats of measure 10 are Bb minor and A minor. Measure 11 is *D minor*, with E minor on the last beat. Measure 12 is an augmented triad: C# F A. The rest is OK I think. 

Now, I think the best way to analyze this piece (and this kind of music in general) is in terms of voice leading rather than labelling chord progressions and Roman numerals, which is more suited to pieces which _rely_ on "functional" harmony. In this excerpt, Prokofiev goes through a wide range of chord changes to get from C back to C, but those changes are mainly driven by neighbouring tones in the voices, rather than "established" chord progressions like V - I, IV - I, I - vi etc...

So, for example, from measure 8, we go from F major to Db major to Bb minor to A minor to D minor. In Roman numerals, this would be: I - VIb - iv - iii - vi, which frankly is a mess and doesn't tell us anything useful. But when you look at this progression in terms of voice-leading, it looks more logical.

F major to Db major involves "expanding" A and C outwards to Ab and Db, while F is kept stationary (this is why I would use Db major here, because F is part of Db major but not C# major). 
Db major to Bb minor involves shifting the Ab to a Bb, keeping everything else stationary. 
Bb minor to A minor is simply planing the chord down a semitone. 
A minor to D minor involves shifting C and E to D and F (in fact, this is probably better understood as a bog-standard v - i progression).

So Prokofiev is simply shifting voices up/down to create a new chord (more generally it's a sort of counterpoint, but when the harmony is block-like as in this excerpt, you can think of it as shifting voices between chords), which is closely related to the previous chord, but not necessarily to be understood "functionally". It just sounds aesthetically pleasing because of the voice shifts themselves, rather than because we can tell where the music is going, i.e. how the chords function. In fact, not knowing where the music is going is part of what gives that piece its particularly modern sound.

Now, I guess the question one might ask is, how do we know what sounds "aesthetically pleasing"? How do we know where to shift the voices to get a certain sound? IMO this involves a kind of intuition, good ears, and a knowledge of what these shifts sound like. If you listen to and study more and more pieces which use harmony in this way, you will amass a mental "bank" of these chord changes and how they are made.

For instance, consider The Imperial March from Star Wars which everybody is familiar with - that chord change at the beginning is merely the outer notes of a minor triad going out a semi-tone, i.e. G minor to Eb minor, G Bb D -> Gb Bb Eb. So if you know that sound, you've got that particular "shift" in your mental bank, and you will know how to get that sort of sound (or avoid it!) in future compositions. And it is easier to think of it as a fundamental shift of scale degrees than "G minor to Eb minor" or "i - vib", which means very little.

Anyway...back to the Prokofiev.  Don't worry so much about losing the plot, because there is not much of a plot to begin with. Each chord change has its own peculiar quality, and Prokofiev is simply exploiting those qualities to create an interesting harmonisation of a melody (and/or a melody out of an interesting harmonic "progression").

From measure 11:

D minor -> E minor: A planing chord
E minor -> C# augmented: This is a kind of shifting chord also. Note, in the last two chord changes, the voices go like this:

F -> G -> A
D -> E -> F
A -> B -> C#

Each voices climbs up one pitch every time, not due to a traditional cadence or chord progression or anything, but because this has a particular "sound" and Prokofiev was using that sound. Carrying on...

C# augmented -> Bbdim: Here we have the A and F going "inwards" to Bb and E, while the C# is kept constant. 
Bbdim -> Ab minor: This can be thought of as shifting down as well, but it's also a kind of V7b9 -> i progression (the Bb dim can be thought of as Eb major 7th with added Fb). Does it _feel_ like a dominant -> tonic chord progression though? Do we feel an arrival with Ab minor? I don't...
Ab minor -> Abdim7: You take the Eb and split it into D and F
Abdim7 -> C major: This is lovely! Ab goes down to G, B to C, D and F collapse down to E. So each tone in Abdim7 acts as a leading tone to C major. Note, because of this it is a fairly common chord progression, you can think of it as V -> I, but with the root of V shifted up a semitone, to make it lead down to the 5th degree of I. (e.g. Ab -> G)

So, I think it's worth making sense of this excerpt more in terms of voice-leading rather than Roman numerals. Wagner showed with his famous Tristan progression that sometimes you simply don't need a resolution, or a sense of tonic center, to make good music. You can label the progression with Roman numerals, but it won't tell you much, because you don't know where the tonic is. In those cases, you have to go back to the fundamentals, as it were, by analyzing the voice leading.

By the way, this voice-leading approach to analysis is basically what _Neo-Riemannian theory _sets out to do. So if you want a less hand-wavy, less "note-moves-up-note-moves-down" method like mine of analysing an excerpt like the one you provided, take a look at NR theory which sets the process out in a more formal language.

Just as a "further exercise": Take a look at the vocal score of the beginning of R. Strauss' opera "Salome" (on IMSLP). Try to analyse it with functional harmony...within 6-8 measures, you will be scratching your head, because it's darn near impossible without losing the big picture. Strauss has based his chord changes on voice leading, just like Prokofiev does in that excerpt.


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## Samuel Kristopher

Thank you so much for that TwoPhotons, this really opened my mind to a whole new world of possibilities! 

I guess that's the flaw of having no one to guide one's learning - up until recently I thought modulation was the only way to get out of the main key, but I've always suspected it wasn't the whole picture. I thought voice-leading was just a way of describing modulation, but now I see that modulation is just one manifestation of voice-leading. A total "duh" moment 

I'll definitely get hold of that Strauss piece. Sounds like an interesting study 

Thanks again - this has done more for my music study than months of aimless web-surfing!!!


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## JosefinaHW

Samuel Kristopher said:


> Thank you so much for that TwoPhotons, this really opened my mind to a whole new world of possibilities!
> 
> :TwoPhotons: A million thanks for this!


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## EdwardBast

Taking into account some of the corrections TwoPhotons has made to the transcription (however, I don't think we have it right yet), I will offer a simpler explanation for what is going on. The first 8 measures are in C major with nothing terribly out of the ordinary. The progression F-Db-Am-Fm (mm. 8-11) is using a hexatonic system* to elaborate the subdominant harmony. Each hexatonic system comprises 6 notes and 6 triads that can be formed from those notes. In this case the notes are F-Ab-A-C-Db-E, and, using capital letters for major and lower case for minor, the triads that can be formed from them are F-f-A-a-Db-db. The harmonies of mm. 8-11 come from this hexatonic system and they elaborate the IV chord. The Ab minor chord in m. 14 is from the hexatonic system which includes the tonic chord C major (C-c-E-e-Ab-ab are the available triads). 

So the gist is, one can elaborate the principal triads of a key by using triads in the hexatonic system to which they belong. These hexatonic systems are standard vocabulary for all Russian composers after Rimsky-Korsakoff. Check out the opening of the latter's Antar to see this in action using minor chords. By the way, the notes of a hexatonic system are also sometimes referred to as the "Bartok scale." Prominent examples of its use can be found in the third movement of the Concerto for Orchestra and elsewhere.

*See Richard Cohn, "Maximally Smooth Cycles, Hexatonic Systems, and the Analysis of Late Romantic Triadic Progressions" (Music Analysis 15, 1996), pp. 9-40.


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## Samuel Kristopher

Thanks to you as well Edward! I don't know much at all about hexatonic systems so I'll aim my self-study in that direction


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## TwoPhotons

EdwardBast said:


> The progression F-Db-Am-Fm (mm. 8-11) is using a hexatonic system* to elaborate the subdominant harmony.


Only that's not the progression that occurs in the piece - it definitely lands on Dm not Fm. I think the confusion in Samuel Kristopher's transcription arose because it's not clear whether the women are singing the 1st degree or the 3rd degree of the harmony, so it's easy to muddle between Fm and Dm, but you can for example hear the tuba playing D in the bass...the overall harmony is definitely Dm. So Prokofiev abandons that hexatonic system pretty quickly, and because the following E minor does not belong to either the "F system" or "D system" I think using the hexatonic scale as an explanation is overall unfounded in this case. I suppose you could use it to explain the link between F and Db, but IMO seeing it as a shift outwards of the 3rd and 5th degree is closer to the spirit of that chord change.


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## EdwardBast

TwoPhotons said:


> Only that's not the progression that occurs in the piece - it definitely lands on Dm not Fm. I think the confusion in Samuel Kristopher's transcription arose because it's not clear whether the women are singing the 1st degree or the 3rd degree of the harmony, so it's easy to muddle between Fm and Dm, but you can for example hear the tuba playing D in the bass...the overall harmony is definitely Dm. So Prokofiev abandons that hexatonic system pretty quickly, and because the following E minor does not belong to either the "F system" or "D system" I think using the hexatonic scale as an explanation is overall unfounded in this case. *I suppose you could use it to explain the link between F and Db, but IMO seeing it as a shift outwards of the 3rd and 5th degree is closer to the spirit of that chord change.*


It is the same thing. Those shifts and expansions by half steps are how hexatonic progressions work. The Star Wars example you cite, G minor to Eb minor is another instance, as is the A-flat minor in C major or F major to D-flat major in the Prokofiev. That is why Cohn's subtitle includes the words "maximally smooth cycles." All of the moves involve half-step, that is maximally smooth, voice-leading. The word cycles refers to the fact that one can have progressions like F-Db-A-F where each major chord is reached by just the kind of expansion or outward shift you describe, eventually cycling back to the start.

Yes, you are correct about the D minor and about your other fixes to the transcription.


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## JosefinaHW

So the gist is, one can elaborate the principal triads of a key by using triads in the hexatonic system to which they belong. These hexatonic systems are standard vocabulary for all Russian composers after Rimsky-Korsakoff. Check out the opening of the latter's Antar to see this in action using minor chords. By the way, the notes of a hexatonic system are also sometimes referred to as the "Bartok scale." Prominent examples of its use can be found in the third movement of the Concerto for Orchestra and elsewhere.

*See Richard Cohn, "Maximally Smooth Cycles, Hexatonic Systems, and the Analysis of Late Romantic Triadic Progressions" (Music Analysis 15, 1996), pp. 9-40.[/QUOTE]

:EdwardBast: Thank you very much.


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## JosefinaHW

_Using Hexatonic Systems to Understand Russian Composers and Bartok _ Is anyone interested in this topic and also be willing to guide some of us through the article referenced below? I have a PDF of the article to send you if you are interested. Please send me a PM.

So the gist is, one can elaborate the principal triads of a key by using triads in the hexatonic system to which they belong. These hexatonic systems are standard vocabulary for all Russian composers after Rimsky-Korsakoff. Check out the opening of the latter's Antar to see this in action using minor chords. By the way, the notes of a hexatonic system are also sometimes referred to as the "Bartok scale." Prominent examples of its use can be found in the third movement of the Concerto for Orchestra and elsewhere.

*See Richard Cohn, "Maximally Smooth Cycles, Hexatonic Systems, and the Analysis of Late Romantic Triadic Progressions" (Music Analysis 15, 1996), pp. 9-40.[/QUOTE]


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## EdwardBast

Just a little note to this discussion. Two Photons did a fine job of taking the Prokofiev passage apart and correcting the transcription and I should have acknowledged the detail and care he gave to it before abstracting out one aspect of the harmonic language as if that was the only or most important thing. It is a fault of my perspective that I try to boil things down to simple principles, even when they aren't so simple, as Two Photon's generous contribution demonstrates.


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## Samuel Kristopher

I appreciated your input Edward, it put me on to Neo-Riemannian theory which has been a very interesting read so far, as especially as I both live in Russia and adore Russian composers of any era, this sounds like an important line of inquiry 

Having both perspectives has been very helpful!


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## millionrainbows

You can get a concise explanation of Neo-Riemann theory in WIK. Basically, it's a way of relating chords when there is no tonic, and keeping smooth voice leading and producing harmonic results which make musical sense to the ear, even if there is no "function" to the chords as we know it.

This is one of those areas in music which is post-tonal, modern, but still not "atonal" or non-harmonic. It applies modern ideas to octave division, and ideas of symmetry, as opposed to CP tonality's division of the fifth and fourth.

In this sense, it is a way of writing "tonal" music (an inclusive term), or "harmonic" music without a single tonal center as in CP tonality. This does not mean it is "atonal" music, since that term is an exclusionary term, and does not describe the varying shades of "tonality" or harmonic music that can be created using this Riemann system. This creates a form of tonality.

The WIK explanation is interesting, as it shows how the Riemann system can be pictured on a 2-D grid or lattice. In Schoenberg's "Structural Functions of Harmony" he shows a similar grid which he uses to show key areas in "normal" functioning tonality, and this leads me to believe that he was aware of such ideas.


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## TwoPhotons

The text sung for that passage is quite striking (taken from the Chandos booklet of that recording here):

_Filosofy lish' razlichnym obrazom
obyisnyali mir.
No dyelo zaklyuchayetsa v tom
shtob izmyenit' yevo._

which translates from Russian to English as:
_
The Philosophers, in their different
ways, simply used to explain the world,
but changing it
is what it's all about._

(Karl Marx)

So perhaps these jarring chord changes are Prokofiev's way of representing that idea of "changing is what it's all about". Because as soon as the choir begins to sing "No dyelo zaklyuchayetsa v tom", which roughly means "But the point is...", that's when we're dropped into Db major (measure 9) which is the first harmony that feels really different to what we've heard so far. It's as if Prokofiev is trying to catch our attention. The words "shtob izmyenit'" (roughly "in order to change") are in measures 13-14, which I think is the most startling section of that verse, because the harmony feels entirely unrelated to everything that was previous, despite being tonal.

Anyway, this isn't really related to music theory per se, but I thought it was an interesting point to bring up as perhaps a way of explaining Prokofiev's motivation for using such free, constantly changing harmony!


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## JosefinaHW

millionrainbows said:


> You can get a concise explanation of Neo-Riemann theory in WIK. Basically, it's a way of relating chords when there is no tonic, and keeping smooth voice leading and producing harmonic results which make musical sense to the ear, even if there is no "function" to the chords as we know it.
> 
> The *WIK explanation is interesting, as it shows how the Riemann system can be pictured on a 2-D grid or lattice*. In Schoenberg's "Structural Functions of Harmony" he shows a similar grid which he uses to show key areas in "normal" functioning tonality, and this leads me to believe that he was aware of such ideas.


MillionRainbows: I am holding on tight to what Two Photons said in the post that follows yours, but one has to become a mathematician in order to understand the construction of such music? (Those diagrams in WIK and then the detailed "instructions" in that article look 1000 times more daunting than P-Chem!!!) How is one supposed to put into action an analysis of the dynamic toroidal view?--I mean where do you even enter that type of constantly moving, 3-D analysis??


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## Samuel Kristopher

I may be completely wrong but I don't think the graphical representations or the toroid are necessary to understand it, but just alternative ways of viewing the mechanics. I find it's much simpler, but again I may be mistaken, on the six types of transformations as detailed thus:



> The three transformations move one of the three notes of the triad to produce a different triad:
> The P transformation exchanges a triad for its Parallel. In a Major Triad move the third down a semitone (C major to C minor), in a Minor Triad move the third up a semitone (C minor to C major)
> The R transformation exchanges a triad for its Relative. In a Major Triad move the fifth up a tone (C major to A minor), in a Minor Triad move the root down a tone (A minor to C major)
> The L transformation exchanges a triad for its Leading-Tone Exchange. In a Major Triad the root moves down by a semitone (C major to E minor), in a Minor Triad the fifth moves up by a semitone (A minor to F major)
> 
> Secondary operations can be constructed by combining these basic operations:
> The N (or Nebenverwandt) relation exchanges a major triad for its minor subdominant, and a minor triad for its major dominant (C major and F minor). The "N" transformation can be obtained by applying R, L, and P successively.
> The S (or Slide) relation exchanges two triads that share a third (C major and C♯ minor); it can be obtained by applying L, P, and R successively in that order.
> The H relation (LPL) exchanges a triad for its hexatonic pole (C major and A♭ minor)


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## JosefinaHW

:Samuel Kristopher: I have printed your post (TYVM) and also ordered what seems to be a user-friendly text on the topics.

http://www.amazon.com/Oxford-Handbo...&redirect=true&ref_=oh_aui_detailpage_o03_s00

I'll see how it goes.


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## JosefinaHW

:millionrainbows: Thank you for this: several of the pieces of the puzzle are grouping together. Also helped me to scan WIK re/ Riemannian mathematics and the calculus of moving surfaces. I should have know that one would use calculus for any constantly changing system.....


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## EdwardBast

millionrainbows said:


> You can get a concise explanation of Neo-Riemann theory in WIK. Basically, it's a way of relating chords *when there is no tonic,* and keeping smooth voice leading and producing harmonic results which make musical sense to the ear, even *if there is no "function"* to the chords as we know it.


This isn't quite right because these kind of transformations apply whether or not there is a clear tonic. In the Prokofiev, for example, there is a clear and firmly established tonic, but there are also harmonies far outside of the key that are difficult to account for in the terms of traditional tonal grammar. In such cases the Riemannian transformations often provide a reasonable explanation of their genesis and a framework in which to understand patterns in their use.

The question of function looks complicated to me. Chords related by H transformations, for example, often sound like they have a kind of reciprocal tonic-dominant relationship, as in the C major and A-flat minor chords of the Prokofiev example. Each contains the other's leading tone and another strong tendency tone, the flat-6th degree. So in C major, using enharmonic spellings: …

B - C 
Ab - G
D# - E

… the B and Ab function as they would in a V7 chord with a flatted 9th (G-*B*-D-F-*Ab*), which makes the C major triad sound like tonic and the Ab minor triad sound like it has a quasi-dominant function. Conversely, if we conceive the same two triads functioning in A-flat minor: …

G - Ab
Fb - Eb
C - Cb

… then the G becomes the leading tone, and the 3rd of the chord, the E respelled as Fb, becomes the 6th degree. The C major triad thus contains the same tendency tones, the leading tone and the flat-6th, functioning, once again, more or less as they would in a V7b9 chord (Eb-*G*-Bb-Db-*Fb*).

The upshot is that H related triads can take on strong quasi-tonal functions, although they don't have to. And more generally, the altered mediant and submediant relationships described in neo-Riemannian theory made it possible for composers like Prokofiev and Shostakovich to reach and hold distant tonal terrain without abandoning the context of a central key - almost like using a system of magic doors.


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## EdwardBast

Here is a pure, early example of a maximally smooth cycle from 1868, the opening statement of Rimsky-Korsakoff's "Antar" Symphonic Suite, made from the minor triads of a hexatonic system (the horn is written at concert pitch):









The voice-leading is just like the example TwoPhotons cited above from the film music of John Williams (only a half-step lower):



TwoPhotons said:


> For instance, consider The Imperial March from Star Wars which everybody is familiar with - that chord change at the beginning is merely the outer notes of a minor triad going out a semi-tone, i.e. G minor to Eb minor, G Bb D -> Gb Bb Eb.


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## JosefinaHW

In alphabetical order:

:EdwardBast: :Million Rainbows: :Samuel Kristopher: :TwoPhotons:---- Tremendous thanks to all of you!

In the Prokofiev, for example, there is a clear and firmly established tonic, but there are also harmonies far outside of the key that are difficult to account for in the terms of traditional tonal grammar. In such cases the *Riemannian transformations often provide a reasonable explanation of their genesis and a framework in which to understand patterns in their use*.

The upshot is that H related triads can take on strong quasi-tonal functions, although they don't have to. And more generally, the altered mediant and submediant relationships described in neo-Riemannian theory made it possible for composers like Prokofiev and Shostakovich to reach and hold distant tonal terrain without abandoning the context of a central key - *almost like using a system of magic doors*.

Then Ankar example..... I don't get it all yet (and that's OK with me), but I've got enough to see that this is so cool, exciting, and just down-right fun!


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## EdwardBast

JosefinaHW said:


> Then Ankar example..... I don't get it all yet (and that's OK with me), but I've got enough to see that this is so cool, exciting, and just down-right fun!


The short version is this: The chord sequence …

F#m - Dm - Bbm - F#m

… is a circular progression (a maximally smooth cycle in Cohn's terms). It starts and ends on the same triad and in between the root of each successive triad drops by a major third. When one triad moves to the next, all of the individual voices move by the smallest increment possible - nothing but half steps (which is where "maximally smooth" comes from).


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## JosefinaHW

EdwardBast said:


> The short version is this: The chord sequence …
> 
> F#m - Dm - Bbm - F#m
> 
> … is a circular progression (a maximally smooth cycle in Cohn's terms). It starts and ends on the same triad and in between the root of each successive triad drops by a major third. When one triad moves to the next, all of the individual voices move by the smallest increment possible - nothing but half steps (which is where "maximally smooth" comes from).


Yes, I have it right!!! :lol:


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## JosefinaHW

:EdwardBast: Why are you making a distinction between a short and a long version, though?


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## Il_Penseroso

What is the origin of Hexatonic system? It's a big question on my mind. A simple primitive pattern of Hexatonic could be found, perhaps for the first time (I'm not quite sure), in Glinka's Ruslan and Lyudmila (Chernomor's scene) back to 1837-1842 the time he composed it. And yes, as EdwardBast wisely mentioned, one could follow up considerable examples in Rimsky-Korsakov, Stravinsky, Prokofiev and other Russian composers. But as far as I know there's no evidence of such system in Russian folk songs or rural dances. Besides, Liszt and Bartok also used Hexatonic system frequently (Liszt knew Glinka and his Ruslan), so, does it belong to the East European musical tradition by any chance? Or is it a built-up artificial system which composers like Glinka and Rimsky-Korsakov formed with (well, let's say) no trace of it before?


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## millionrainbows

EdwardBast said:


> This isn't quite right because these kind of transformations apply whether or not there is a clear tonic.


Your response is not quite right either, as my statement was not exclusionary ("...when there is no tonic...); "or when there is" is implied, but not excluded. 
As you say, the Riemann transforms whether or not there is a tonic. I did not specify that there could be no tonic.



> In the Prokofiev, for example, there is a clear and firmly established tonic, but there are also harmonies far outside of the key that are difficult to account for in the terms of traditional tonal grammar. In such cases the Riemannian transformations often provide a reasonable explanation of their genesis and a framework in which to understand patterns in their use.
> 
> The question of function looks complicated to me. Chords related by H transformations, for example, often sound like they have a kind of reciprocal tonic-dominant relationship, as in the C major and A-flat minor chords of the Prokofiev example. Each contains the other's leading tone and another strong tendency tone, the flat-6th degree. So in C major, using enharmonic spellings: …
> 
> B - C
> Ab - G
> D# - E
> 
> … the B and Ab function as they would in a V7 chord with a flatted 9th (G-*B*-D-F-*Ab*), which makes the C major triad sound like tonic and the Ab minor triad sound like it has a quasi-dominant function. Conversely, if we conceive the same two triads functioning in A-flat minor: …
> 
> G - Ab
> Fb - Eb
> C - Cb
> 
> … then the G becomes the leading tone, and the 3rd of the chord, the E respelled as Fb, becomes the 6th degree. The C major triad thus contains the same tendency tones, the leading tone and the flat-6th, functioning, once again, more or less as they would in a V7b9 chord (Eb-*G*-Bb-Db-*Fb*).


As in tri-tone substitution, and in late Beethoven, a chromatic resolution takes the place of a V-I. This is because the interval of a fifth, when projected or 'stacked', eventually cycles back to close the cycle (C-G-D-A-E-B-F#-C#-G#-D#-A#(Gb)-E#(F)-C, yielding all 12 chromatic notes. The only other interval that yields all 12 notes is the minor second (C-C#-D-D#-E-F-F#-G-G#-A-A#-B-C). This is most definitely a function of CP tonality, used frequently by Beethoven in his late quartets and elsewhere, transforming a diminished seventh into a flat-nine dominant by placing a new tonic under it.



> The upshot is that H related triads can take on strong quasi-tonal functions, although they don't have to. And more generally, the altered mediant and submediant relationships described in neo-Riemannian theory made it possible for composers like Prokofiev and Shostakovich to reach and hold distant tonal terrain without abandoning the context of a central key - almost like using a system of magic doors.


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## millionrainbows

Il_Penseroso said:


> What is the origin of Hexatonic system? It's a big question on my mind. A simple primitive pattern of Hexatonic could be found, perhaps for the first time (I'm not quite sure), in Glinka's Ruslan and Lyudmila (Chernomor's scene) back to 1837-1842 the time he composed it. And yes, as EdwardBast wisely mentioned, one could follow up considerable examples in Rimsky-Korsakov, Stravinsky, Prokofiev and other Russian composers. But as far as I know there's no evidence of such system in Russian folk songs or rural dances. Besides, Liszt and Bartok also used Hexatonic system frequently (Liszt knew Glinka and his Ruslan), so, does it belong to the East European musical tradition by any chance? Or is it a built-up artificial system which composers like Glinka and Rimsky-Korsakov formed with (well, let's say) no trace of it before?


That's a good area for speculation, so here goes. A six-note scale is half of the total 12 chromatic. If you divide the octave geometrically at the tritone (six semitones wide), instead of at 7 and 5 (fifth and fourth) as in CP tonality, then we have a flatted fifth, or unstable situation. This is manifest in the diminished scale and chords, and also in the whole-tone scale. These are both mathematical projections of intervals (minor third=diminished chord/scale, major second = augmented chord/whole tone scale) which are inherently 'vagrant' or unstable, restless, and in search of resolution, and also chromatic.

The 7 and 5 divisions (V and IV) are divisions based ostensibly on harmonic considerations, being inversions of each other, and also the only two intervals which cycle outside the octave when projected. The fifth, 7 semitones, is 7 x 12 = 84, and the fourth is 5 semitones, 5 x 12 = 60. All the other intervals have 12 as their denominator; minor second/1 x 12 = 12, major second/2 x 6 = 12, minor third/3 x 4 = 12, major third/4 x 3 = 12. In other words, these smaller intervals cycle back "inside" the 12-note octave, while 7 and 5 cycle much further outside the octave before coinciding and closing the circle.
To me, this means that the larger harmonically-based CP intervals 5 & 4 encourage 'travel' or modulation to other key areas which cover larger 12-note areas (using the 7-note scale and its alterations), while the smaller intervals 1,2 3, and 4 encourage more localized areas of control, as in Bartok. This makes them a 'micro-tonality' which is more localized and thus more chromatic, while 7 & 5 want to cover entirely new 12-notew areas of tonality.

Really, I see the 'hexatonic' system as a logical outcome of the 12-note octave and how it is divided and approached. There is evidence of folk-use of six-note scales, which may simply be the result of dividing a flute or string into two geometrically equal parts, resulting in a tritone and two sets of six notes.


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## EdwardBast

Il_Penseroso said:


> *What is the origin of Hexatonic system?* It's a big question on my mind. A simple primitive pattern of Hexatonic could be found, perhaps for the first time (I'm not quite sure), in Glinka's Ruslan and Lyudmila (Chernomor's scene) back to 1837-1842 the time he composed it. And yes, as EdwardBast wisely mentioned, one could follow up considerable examples in Rimsky-Korsakov, Stravinsky, Prokofiev and other Russian composers. But as far as I know there's no evidence of such system in Russian folk songs or rural dances. Besides, Liszt and Bartok also used Hexatonic system frequently (Liszt knew Glinka and his Ruslan), so, does it belong to the East European musical tradition by any chance? Or is it a built-up artificial system which composers like Glinka and Rimsky-Korsakov formed with (well, let's say) no trace of it before?


I prefer to look for motivations and incremental stylistic developments rather than origins because I don't think things in general tend to have origins.  With that in mind and in the spirit of extemporaneous speculation:

I imagine the growing prevalence of hexatonic and similar systems from the mid 19thc forward has a number of roots, motivations and paths of development, including extensions of the Romantic Era practice of modal mixture, a generally increasing interest in symmetrical divisions of the octave, and a preference for maintaining a triadic basis in exploring the possibilities of extended tonality.

Modal mixture - By this I mean the practice of borrowing triads from the parallel major or parallel minor mode. It is easy to see how such borrowing could lead toward the exploration of hexatonic systems. Taking C major as an example, the commonplace practices of borrowing the submediant and tonic triads from the parallel minor, that is Ab and c respectively, gives us four out of the six triads of a hexatonic system (Throughout I will use upper case for major triads and lower for minor): C-c-e-Ab. To get the other two all we need do is alter the mediant and borrowed submediant triads, that is, raise the third of the e triad and drop the third of the Ab triad, thus yielding the full complement (C-c-E-e-Ab-ab). Now it might seem that I have pulled these altered mediant and borrowed submediant relationships from a hat, but they have significant precedents in the Classical and Romantic styles in the choice of keys for the internal movements of multimovement works and in the choice of secondary keys within movements. Beethoven uses E major for the second theme of the Sonata in C major, Op. 53/i; Schubert uses f# minor as a secondary key area in the Sonata in Bb, D. 960/i, to cite just a couple of well-known examples among many. Perhaps this interest in altered mediant and submediant keys led to or inspired the experimentation with altered mediant and submediant triads on the smaller scale, that is in phrase length harmonic progressions?

Symmetrical divisions of the octave: - Symmetrical collections like whole-tone, octatonic and hexatonic scales became increasingly popular in the relevant period. Alexander Dargomizhsky wrote nearly a full scene of his opera The Stone Guest (1866-69) in a whole-tone scale, Liszt and others used octatonic collections over diminished chords and elsewhere, as did Rimsky-Korsakoff. Stravinsky used them extensively. And of course altered mediant and submediant chords of all kinds implied symmetrical divisions of the octave by major thirds and minor thirds. Such symmetrical schemes began to govern key relationships within movements, as in Tchaikovsky's 4th Symphony/i (f-ab-B-d-f), Rachmaninoff's Isle of the Dead (a-c-Eb-f#-a), and Prokofiev's Violin Sonata in F minor/i (f-b-f).

Triadic basis for extended tonality - There are different paths the expansion or (alleged) "breakdown" of tonality is seen to have taken. The more widely acknowledged is through extended tertian (or quartal in the case of Scriabin) accretions to harmony, proliferation of non-harmonic tones, emphasis on unstable tritone based configurations, and the washing out of tonal functions. Many composers, however, chose to preserve a triadic system, but one that was more fluid and compatible with symmetrical scales or modal materials, that eschewed lush extended tertian textures and resolutions, and which facilitated rapid shifts, veiling, or temporary suspensions of tonal centers. Hexatonic systems were just one resource for accomplishing these goals.


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## EdwardBast

Duplicate post ........


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