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First Claim: Harmony and Counterpoint Constrain One Another

7K views 51 replies 8 participants last post by  millionrainbows 
#1 · (Edited)
From A Geometry of Music, Dmitri Tymoczko:

"Any two major chords can be connected by stepwise voice leading in which no voice moves by more than two semitones. This means you can write a harmonic progression without worrying about melody; that is, for any sequence of major chords, there is always some way to connect the notes so as to form stepwise melodies.
What about the chromatic cluster B, C, Db followed by E, F, Gb (its transposition by ascending fourth)? Here, none of the notes of the first 'chord' are within two semitones of any note in the second, and hence there is no way to combine a sequence of these chords so as to produce conjunct melodies. At the same time, however, the chromatic cluster can do things that the C major chord can't. It is possible to write contrapuntal music in which individual melodic lines move short distances within a single, unchanging harmony. This is possible only because the chord's notes are all clustered together, ensuring that there is always a short path between any two of them."
 
#2 ·
From A Geometry of Music, Dmitri Tymoczko:

Any two major chords can be connected by stepwise voice leading in which no voice moves by more than two semitones. This means you can write a harmonic progression without worrying about melody; that is, for any sequence of major chords, there is always some way to connect the notes so as to form stepwise melodies.
What about the chromatic cluster B, C, Db followed by E, F, Gb (its transposition by ascending fourth)? Here, none of the notes of the first 'chord' are within two semitones of any note in the second, and hence there is no way to combine a sequence of these chords so as to produce conjunct melodies. At the same time, however, the chromatic cluster can do things that the C major chord can't. It is possible to write contrapuntal music in which individual melodic lines move short distances within a single, unchanging harmony. This is possible only because the chord's notes are all clustered together, ensuring that there is always a short path between any two of them.
All of the above is a quotation of Tymoczko's words? If not, please use quotation marks to indicate which words are his.
 
#3 · (Edited)
So, I sound like him, or he sounds like me? Either way, I'll take that as an insult. :lol:

That's just one of four claims, Edward. You can look forward to a nice slash-fest. Sharpen up your knife!
 
#6 · (Edited)
It isn't quite, but it's pretty close to what he wrote on pp. 13-14. So, what interests you about this passage? Why are you quoting it? Is there a point you are trying to demonstrate or are you just trying to sell more copies of Tymozcko's book? ;)

You do understand that one should indicate where one's quotation deviates from a text one is quoting, right? If you need any help with how to do this I'd be glad to help.
 
#12 · (Edited)
It isn't quite, but it's pretty close to what he wrote on pp. 13-14. So, what interests you about this passage? Why are you quoting it? Is there a point you are trying to demonstrate or are you just trying to sell more copies of Tymozcko's book? ;)

You do understand that one should indicate where one's quotation deviates from a text one is quoting, right? If you need any help with how to do this I'd be glad to help.
EdwardBast: I consider your questioning of the citation as insulting. The rest of your posts are insulting, as well, and are condescending, designed to abort this thread from its outset. I have reported this to the mods, but they have apparently decided not to remove them.
I really don't understand how you are able to get away with these sorts of actions, unless you hold special favor as an 'inside' advisor to the mods.
 
#7 · (Edited)
error...........
 
#8 ·
Seriously though, it sounds like a very interesting book and perspective and I hope it generates some good discussion.
 
#9 ·
Uh-huh..............Me too.
 
#11 · (Edited)
I've abandoned this thread, Torkelburger. See the other one, with the "corrrected" quote, done for EdwardBast's benefit.

I'll paste your reply over there & reply to it.
 
#13 · (Edited)
I don't get the claim that harmony and counterpoint constrain each other, based on that quote in the original post. The conclusion reached by Dmitri is that you can write counterpoint within unchanging harmony, but that doesn't pose a limitation on the harmony that can be reached with counterpoint. I'll make a counter claim that there are unlimited possibilities in harmony that can be reached with counterpoint.
 
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#17 · (Edited)
I don't get the claim that harmony and counterpoint constrain each other, based on that quote in the original post.
I think if you re-read the thread, and keep following the premise, the extreme simplicity of the idea will finally strike you.

The conclusion reached by Dmitri is that you can write counterpoint within unchanging harmony, but that doesn't pose a limitation on the harmony that can be reached with counterpoint.
That would depend on what notes your counterpoint defines as its harmonic scaffolding, wouldn't it? If the notes you use make up a scale of five or more notes, then yes, almost anything is possible, depending on how large you can tolerate your melodic leaps (i.e. voice leading).

If you want conjunct melodies, then larger spacing begins to prohibit this.

It looks like we'd better present some definitions: https://en.wikipedia.org/wiki/Steps_and_skips

Conjunct: In music, a step, or conjunct motion, is the difference in pitch between two consecutive notes of a musical scale. In other words, it is the interval between two consecutive scale degrees. Any larger interval is called a skip (also called a leap), or disjunct motion.

Melodic motion in which the interval between any two consecutive pitches is no more than a step, or, less strictly, where skips are rare, is called stepwise or conjunct melodic motion, as opposed to skipwise or disjunct melodic motion, characterized by frequent skips.

I'll make a counter claim that there are unlimited possibilities in harmony that can be reached with counterpoint.
But why make such a counterclaim? All this exposition is doing is explaining the nature of tonality, and its harmony and melodic aspects, and how they affect each other reciprocally. What is there to challenge?

Anyway, to say "there are unlimited possibilities in harmony that can be reached with counterpoint" is fraught with inconsistency.
Tonality is not about "unlimited possibilities," is it? It's about how you divide up the octave. If your counterpoint consists of scale notes, then that's your harmony.

You only have 12 notes; what does "unlimited" mean? Dodecaphony? Now we are out of the realm of tonal harmony.
But, yes, the "clusters" in the example are a form of harmony which generates melodic lines (counterpoint) which "burble and bubble" within a limited range of a chromatic cluster. Or you could say it's the counterpoint that's creating the clusters. It's a "chicken or egg" problem.
 
#14 ·
What the OP shows us is that CP classical music is a self-fulfilling system which is virtually 'automatic' in nature; a no-brainer for composers like Mozart, Handel, Vivaldi, and Haydn. The diatonic scale, which divides the octave up so evenly, and fits together in such a closely-related cookie cutter fashion, is an easy environment in which to do counterpoint and compose conjunct melodies for; they practically compose themselves. you can look in any direction and find a closely related chord or voice which is a member of a chord; do you want harmony A, B, or C? A harmonic buffet; whatever suits your fancy.
 
#15 ·
If that's the case, then it should be no problem for you at all to compose say, a double period or so of piano music equally as good or even better than Beethoven Sonatas 14, 17, 21, 23, 29. Since the composition is automatic and the music composes itself, this should take you a matter of minutes. Post your composition that proves your theory to the world one hour from now and let us be the judge.
 
#18 ·
I don't care how easy it is to compose a melody in the diatonic system. That has nothing to do with how good a piece is or how hard it was to compose. Beethoven's fifth symphony entire first movement is based more on extremely simple motifs than any "melody" and it is one of the greatest pieces in the history of music. It's what you do with it that matters. There's more to composition than just connecting dots and painting by numbers. That's not what composition is.
 
#19 · (Edited)
I don't care how easy it is to compose a melody in the diatonic system. That has nothing to do with how good a piece is or how hard it was to compose.
Okay, I agree, but that's not the point of this thread, and I don't think this thread "contradicts" what you are saying. All it is saying is that the diatonic system facilitates the construction of conjunct melodies.

Beethoven's fifth symphony entire first movement is based more on extremely simple motifs than any "melody" and it is one of the greatest pieces in the history of music. It's what you do with it that matters. There's more to composition than just connecting dots and painting by numbers. That's not what composition is.
I don't disagree with that. But the purpose of this thread is not to make value judgements, or to present ideas which contradict that. It's a simple exposition about the nature of tonality.
 
#20 ·
That's the nature of all music, though. Atonality is not any different. Someone could just as easily say that 12-tone music is a self-fulfilling system which is automatic in nature, a no-brainer for composers (and that the chromatic scale divides the octave evenly) and that 12-tone music composes itself. All you have to do is follow the row and plug in the numbers for melodies and harmonies. The row tells you what notes to write. It’s a paint by numbers system.

Would they be correct? Wouldn’t Schoenberg, Berg, Webern, Boulez, Babbitt, Sessions, Stravinsky et al 12-tone music all sound the same then?

Does Bach, Mozart, and Beethoven sound the same? If the music composed itself and is automatic (meaning the composer has no control) then it would, but it doesn’t.

Same goes for composing with pitch-class sets or most any other post tonal technique.
 
#25 · (Edited)
That's the nature of all music, though. Atonality is not any different. Someone could just as easily say that 12-tone music is a self-fulfilling system which is automatic in nature, a no-brainer for composers (and that the chromatic scale divides the octave evenly) and that 12-tone music composes itself. All you have to do is follow the row and plug in the numbers for melodies and harmonies. The row tells you what notes to write. It's a paint by numbers system.
What the book is saying (and this is only the first of four claims about the nature of tonality) is that if you have wide spaces in your harmonic components, you will have wide leaps in the counterpoint or melodic figurations you construct. It's a very simple idea. Since most tonal music uses scales, and if the scales divide the octave fairly evenly, constructing conjunct melodies or smooth voice leading is enabled.

You are on the wrong track. This is not about "how easy" it is to create good music; it simply tells us what features will facilitate conjunct vs. disjunct melody or counterpoint.
 
#21 ·
CP harmony is a fairly straightforward process because it is enabled by the diatonic scale and the triads laid out on its scale degrees.
Again, CP is not alone this regard. It is a system just as any other system to compose with (12 tone, quartal, secundal, etc.) All systems make for a fairly straightforward process.
I.e., if the harmonic scaffolding is there, any conjunct melodies or counterpoint are fairly easy to construct.
But the same is true for any system. Take quartal harmony. The harmonic scaffolding is there. And given that 5 contiguous perfect fourths yield the pitches of a pentatonic "scale", any conjunct melodies or counterpoint are fairly easy to construct.
 
#26 ·
Again, CP is not alone this regard. It is a system just as any other system to compose with (12 tone, quartal, secundal, etc.) All systems make for a fairly straightforward process.

But the same is true for any system. Take quartal harmony. The harmonic scaffolding is there. And given that 5 contiguous perfect fourths yield the pitches of a pentatonic "scale", any conjunct melodies or counterpoint are fairly easy to construct.
Yes, you're right. The book, and this idea, is exploring the features of tonality. It's not my purpose to argue with you.
 
#27 ·
It is, if you want to know what tonality is from an 'outside the box' perspective. But isn't that true of anything? In order to know what it is, you must compare it with what it is not.
 
#32 · (Edited)
Anyway, this is just the first of four claims about tonality that Tymoczko makes. At this rate, we'll never get to number two. This seems typical of the argumentative and confrontational aspect of the internet. It seems that everybody is stuck in their own little world, and many are revealed to be 'not at peace' with themselves; thus, it often ends in bickering over inconsequential details, misunderstandings, and more bickering. My advice to everyone is: do your thinking (homework, practice) before you post.
 
#33 · (Edited)
I downloaded the pdf of the book (for free here: https://www.academia.edu/18164732/Geometry_of_Music?auto=download)

The example he used to make the claim in the thread title was in pages 12 and 13, not the example you quoted.

Here is a real life example which proves him wrong (the fugue starting at 1:30). The fugue subject uses only major triadic notes and at certain times with more than one voice). First of all, it has unchanging harmony at times, which Dmitri (the writer, not Shostakovich :D) said could only be accomplished with chromatic clusters. (ie. Conjunct melodies which step by 2 semitones maximum are not required for counterpoint which proves him premise wrong) Secondly, it's a great piece with beautiful harmony and nothing wrong with the counterpoint.

 
#34 · (Edited)
The example he used to make the claim in the thread title was in pages 12 and 13, not the example you quoted.
That's totally misleading, verging on lying.

This site will not allow me to post PDF images from my computer on this site (I seem to have run out of privileges in that regard), so the 'corrected' quotation only refers to the figures without showing them. In the OP, I thought that would be confusing. That's why I left those references out of the first quote, and EdwardBast jumped on it.

Additionally, those figures on p. 12 and 13 refer to a different part of the text, which I did not quote at all.

You two guys are really something else!

Here is a real life example which proves him wrong (the fugue starting at 1:30). The fugue subject uses only major triadic notes and at certain times with more than one voice). First of all, it has unchanging harmony at times, which Dmitri (the writer, not Shostakovich ) said could only be accomplished with chromatic clusters. (ie. Conjunct melodies which step by 2 semitones maximum are not required for counterpoint which proves him premise wrong) Secondly, it's a great piece with beautiful harmony and nothing wrong with the counterpoint.
Then you're misreading Tymoczko. He didn't claim that diatonic harmony could not be static. (Shostakovich sounds static to me quite often :D ), or that static harmony could only be accomplished with chromatic clusters.That's just the way the examples turned out.

You don't understand the example, or the principle behind it, obviously.

It sounds to me like you're arguing just to be arguing.
 
#36 ·
If you actually read the chapter, it’s not a strong claim to say that harmony and counterpoint constrain one another. The example on 12-13 compares counterpoint derived from a single major triad vs a three note semitone cluster and then observes that you can’t get conjunct melodies (defined as moving in whole or half steps) from the major triad without introducing passing tones that will then define a scale. The cluster has the opposite problem, you can obviously make conjunct melodies from the chord tones, but there is no way to link the ends of the cluster with a small number of passing tones - the cluster does not imply any sort of a scale. He certainly does not say that unchanging harmony can only be accomplished with clusters, so the Shostakovich example is irrelevant
 
#38 ·
If you actually read the chapter, it's not a strong claim to say that harmony and counterpoint constrain one another. The example on 12-13 compares counterpoint derived from a single major triad vs a three note semitone cluster and then observes that you can't get conjunct melodies (defined as moving in whole or half steps) from the major triad without introducing passing tones that will then define a scale. The cluster has the opposite problem, you can obviously make conjunct melodies from the chord tones, but there is no way to link the ends of the cluster with a small number of passing tones - the cluster does not imply any sort of a scale. He certainly does not say that unchanging harmony can only be accomplished with clusters, so the Shostakovich example is irrelevant
Thank you, Bwv 1080; you have proven that you understand what Tymoczko is getting at, and that you have a superior musical intelligence. Kudos! :tiphat:
 
#41 ·
It seems as though Tymoczko is laying out some theorems the way a mathematician would, and will go on to develop a deeper argument in the book. His first claim about harmony and counterpoint is straightforwardly true, by definition. He isn't saying anything about tonal music "writing itself" or anything like that.
 
#42 · (Edited)
You'd argue with a fencepost, Iso. What I said was in context: "CP classical music is a self-fulfilling system which is virtually 'automatic' in nature; a no-brainer for composers like Mozart, Handel, Vivaldi, and Haydn. The diatonic scale, which divides the octave up so evenly, and fits together in such a closely-related cookie cutter fashion, is an easy environment in which to do counterpoint and compose conjunct melodies for; they practically compose themselves. You can look in any direction and find a closely related chord or voice which is a member of a chord."

I think that statement is true, and stand behind it. And I think Tymoczko implies the same thing.

When you take it out of context, it sounds more radical. I'm not sure what you're disagreeing with, or if it's just that you don't like what I post, which is fine with me. I like to take some license, and be provocative. If you want to discuss the ideas, do so, but stop your "internet complaining" which is so characteristic of internet 'discussions.' I'm bored with your negativity.
 
#48 ·
Why do you suppose that is?
 
#52 ·
Yes, especially all those parallel major triads; those really sound contrapuntal.
 
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