A New Theory of Harmony? – Isocords & Palindromes

Hell, starting with different tuning systems/frequency arrays and playing with different interval spaces, overtones, etc, that sounds all very interesting if maybe not 100% new (but who cares?).

Yeah, I wouldn't even know where to begin "ditching" the 12-tone system since most all music is understood as operating within the frequency parameters of a thoroughly established system of pitches. But, I do like the idea of doing away with Octaves and attempting to broaden the intervals out to larger spaces just to make room for using more overtones.

Both quotes suggest that both of you would surely profit from reading Sethares' book on Tuning, Timbre, Spectrum, Scale. You'd find a lot of your ideas already realized in his investigations!

I'm going to play around with that idea a bit and post it when I get time, but that's generally my initial thought about how to implement it. Simple? Yeah, but we've gotta start somewhere.

I'd support very much these ideas. Go ahead and do so!

In any case, I'll go to post soon some concrete ideas on how to develop composition means (methods) out of the isocord theory.

@ Hansen: All I'm saying is you need to be more thorough in explaining the framework of your theory, not just stating a bunch of analysis and telling people how it can unify tonal and atonal music. Your theory has to stand APART from everything else before you can start comparing/applying it TO everything else.

I'll surely try to follow this advice in my subsequent posts.

Before going into further detail of the theory, I'd like to present some examples of precursors of using equal intervals in music.

The first one is as early as the beginnings of western music in the Middle Ages. Singing simultaneously in octaves, fifths, or fourths, parallel chant, is a first step in developing poylphonic music. A straightforward example is the following example of "Nos qui vivimus", a parallel organum from the 9th century.

As can be seen, the upper and lower voices sing in perfect fifths in parallel. However, this is not yet a true symmetric sound since between lower and upper voices there's a fourth instead of a fifth

Your idea excites me...It also reminds me of something Sch

Interesting interpretations of symmetry which you presented. However, you seem to apply a concept of symmetry somewhat different from mine.

Let's take your example C-D#-E-G-Ab-B. In my interval notation this would read as 3

Are you working in Equal temperament?

It sounds like this would work better with unequal intervals.

Are you working in Equal temperament?

It sounds like this would work better with unequal intervals.

Basically, isocord theory presumes 12-tone equal temperament for correct re-spelling of transpositions and inversions. For example, C-E-G# transposed to Ab-C-E would sound slightly different (i.e. "out of tune", so to speak) if played in just intonation.

Nevertheless, it would be an interesting exercise to study isocords in just intonation. Which kind of symmetric chords would really sound "pure" and which not, especially from the same interval class? I'd suppose that all 12-tone isocords (m2nds, M7ths, p4ths, p5ths) would become rather impossible to be represented with small integer ratios throughout. Possibly, low-numbered isocords like a4ths/d5ths ("tritone" isocords), M3rds/m6ths, and m3rds/M6ths would be easier to realize in just intonation. In any case, it will heavily depend on the proper choice of a suitable axis of symmetry for a particular set of isocords. It's worthwile to use Leonhard Euler's "Tonnetz" representation (1739) for studying isocords in just intonation.

I'll follow this line of thought in future explorations of the isocord theory

In subsequent posts I'll present some explorations in isocord composing. So let's start with the first one, a short piano piece on minor seconds and some variants, i.e. transpositions and inversions of m2nds.

Here you may listen to the piece, Isocord Exploration 1.mp3, and this is the associated score, Isocord Exploration 1.pdf.

Have fun with this kind of exploring symmetric sounds!

An interesting feature of isocords is that the maximum number of distinct tones in a full isocord depends on the interval used (see the table in the Isocord Theory article). In Isocord Exploration 1 the minor seconds isocord and its inversion, the major seventh isocord, are used. As a full chord, both have a maximum number of 12 distinct tones, but I've used them differently regarding the number of tones included, varying from 3 through 12 tones for the m-2nds and only 4 tones for the M-7ths.

The reason is simply compositional variety: Successive build-up of 6 sustained tones of a m-2nds in mm. 1–3 and 13–15 as opposed to chromatic scales of 12 tones which sound well as a 12-tones m-2nds when played with pedal down on the piano. Of a special flavor is the successive build-up of sustained m-2nds of 3 through 11 tones in mm. 30–32, resolved in a diminished seventh chord, which happens to be just another isocord of m-3rds. This leads to a little coda of contrapuntally distributed m-2nds of 4 x 3 tones, concluded with resolutions in minor triads.

The compositional problem with major sevenths isocords is that they cannot be played as full chords on the piano. So you may realize only some kind of arpeggio of M-7ths in suitable sustained figurations as done with 4 tones each in mm. 9–11 and even more in mm. 22–24 and 26–28. Note that the latter examples show that it's equivalent whether you build isocords upward or downward.

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YC Music Articles: Isocord Theory

IMO, that's just a shiny name. I personally don't need one more new, special theory for something which is around for decades. With knowledge in acoustical principles, it follows automatically. Hindemith shew this very well (even better) in his theoretical book, where the reader can also get some understanding of why this is so.

IMO, that's just a shiny name. I personally don't need one more new, special theory for something which is around for decades. With knowledge in acoustical principles, it follows automatically.

Could you elaborate on that "which is around for decades"?

My second example of exploring isocords uses major seconds with transpositions and inversions.

Furthermore, it introduces a new kind of symmetric chords which can be derived from isocords by means of isocord transformation, as I would like to call it. Look at the chord of bar 42, which is B3-D#4-A4-G5-Db6-F6. It shows an interval structure of 4-6-10-6-4 semitones which obviously is symmetric in both directions, upward and downward. However, the intervals are not of equal size as defined in Isocord Theory. Nevertheless, this "mixed m-7ths chord" is uniquely based on a plain minor sevenths isocord, namely D#2-C#3-B3-A4-G5-F6, with interval structure 10-10-10-10-10. You only have to transpose the lowest two tones in such a way that the mixed interval structure 4-6-10-6-4 results. This process is an isocord transformation. You can build any mixed-symmetric chord by means of this process as long as it contains the necessary tones of a plain-symmetric isocord and results in a bi-directional mixed-symmetry.

As you can see – and hear when listening to it – the second part of Isocord Exploration 2 is composed solely of mixed-symmetric chords by isocord transformation.

Here you may listen to the piece, Isocord Exploration 2.mp3,

and this is the associated score, Isocord Exploration 2.pdf.

Have fun in listening - and studying - this piece!

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YC Music Articles: Isocord Theory

Well now that there's some actual music to talk about (I guess), I have to say that nothing has changed and it's exactly as I predicted way back. You got clusters used as a percussive element (ala Cowell), mixed in with traditional 3rd based chords.

What you're calling "resolution" here is nothing but shoving in a traditional chord in between all the other stuff. I don't see why any of the new names you came up with are necessary to explain or analyze this at all. In fact, it's very, very, easy to analyze it without your theory since everything that happens here already HAS a name or is very easy to sum up using regular terminology.

Nothing new is going on either, so do explain why all this complicated renaming system of yours is necessary or even useful? I mean, besides mental masturbation on your behalf?

Well now that there's some actual music to talk about (I guess), I have to say that nothing has changed and it's exactly as I predicted way back. You got clusters used as a percussive element (ala Cowell), mixed in with traditional 3rd based chords.

What you're calling "resolution" here is nothing but shoving in a traditional chord in between all the other stuff. I don't see why any of the new names you came up with are necessary to explain or analyze this at all. In fact, it's very, very, easy to analyze it without your theory since everything that happens here already HAS a name or is very easy to sum up using regular terminology.

Nothing new is going on either, so do explain why all this complicated renaming system of yours is necessary or even useful? I mean, besides mental masturbation on your behalf?

SSC is right.

I question your education, people skills, and background when you try to do something like this. I really do. Then you unsrupulously, and stubbornly, argue this to death and question skepticism. You don't question skepticism bud; you prove skeptics wrong. You don't do that.

It is getting really old for this to be treated as fact, and is sort of insulting to see an article about it on here.

Come on YC mods... how seriously do you want this site to be taken?

Well, SSC, that's an easy game: Since there's some actual music to talk about, exactly show me, measure by measure, where

Well, SSC, that's an easy game: Since there's some actual music to talk about, exactly show me, measure by measure, where

It is cool that you have a system for yourself... but it isn't cutting-edge or whatever you're trying to do with it. There has been similar harmonic systems that fall under umbrella systems from the early 20th century. The fact that you try to pass it off as anything other than that is the arrogance in this thread.

Well, at least you concede that I have a system for myself. That's just a natural way to start with when trying to move on to pastures new. But now, tell me what you have a card up on your sleeve about "similar harmonic systems that fall under umbrella systems from the early 20th century". I didn't find one (despite all the care taken so far).

What I try to pass off is the systematics of a theory which has its roots even in the earliest of Western music (see Precursors of equal interval usage) and some scattered practice in 19th and 20th century's composition. But still lacks the thoroughness of general applicability (instead of falling under "umbrella systems").

Well, at least you concede that I have a system for myself. That's just a natural way to start with when trying to move on to pastures new. But now, tell me what you have a card up on your sleeve about "similar harmonic systems that fall under umbrella systems from the early 20th century". I didn't find one (despite all the care taken so far).

What I try to pass off is the systematics of a theory which has its roots even in the earliest of Western music (see Precursors of equal interval usage) and some scattered practice in 19th and 20th century's composition. But still lacks the thoroughness of general applicability (instead of falling under "umbrella systems").

That's because you're not tying anything to particular aesthetics. Of course things will sound different when you compose in a different way with the same technique. SSC already has said what your latest piece consist of harmonically... go back and read.

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That's because you're not tying anything to particular aesthetics. Of course things will sound different when you compose in a different way with the same technique. SSC already has said what your latest piece consist of harmonically... go back and read.

Particular aesthetics? Again, I have my own, "an aethetics for myself" (in all of my compositions, including those completely different from the pieces in this thread). However, my things sound different not for using some "same technique" from odd "umbrella systems" but for adhering to my own theory (and expanding on it where necessary).

What SSC said about my use of harmony is really beside the point. I got my "clusters" because it's a piece with major seconds and its variants. And the "mixed in with traditional 3rd based chords" is nothing but a consequent application of the minimal tone-step rule as defined for isocord resolution in Isocord Theory. Where and when to apply the features of this theory in an actual composition is solely the decision of the composer.

Well, to me this sounds like a normal piece based on the whole tone scale with an occasional diatonic chord as a resolution.

As for the parallel isocords: This is a common technique, Debussy and Messiaen used it for example, it's called "planing". You take a certain chord structure and then play transpositions of it in succession. The only difference is that you decided to stick to "(transformed) isocords" but you don't really use them in a different way. So I fail to see the innovation. Also I don't get what this has to do with dodecaphony.

So I fail to see the innovation.

You really didn't fail! Your comment is primarily a fine validation of the applicability of the theory to historical material.

On the other hand, my "explorations" are not necessarily unique compositions of a "new flavor" but more of a "showcase" to the systematic usage of the theory's features and the ex post factum usability of its terminology.

Also I don't get what this has to do with dodecaphony.

In one of my final explorations I'll deal exactly with this

I should probably wait to hear what your final culmination will amount to, but I just have to step in here and say... this is becoming rhetorical for me. It's not that your theory doesn't have merit, you're just limiting your scope and the potential for "Isochords". And this is what people are commenting on, that your theory is really just cummulative.

Now, I realize that you have to start 'somewhere', but any interest others seem to have in this theory is diminishing... regretably, I must admit I haven't really been paying much attention. Why, you may ask? Because I'm not learning anything about sound that I don't already understand in a more comprehensive form from existing theoretical principles. There's nothing 'new' here, nothing 'refreshing' or 'inspiring' in what you're doing with this idea. Do I think the concept has potential? Sure. But I've already mentioned other ways you could approach this and you've had no interest in entertaining those ideas.

That's cool. Just don't be surprised by the resistance you're seeing right now. This isn't insightful or revealing in any real way where it concerns sound and composition, and arbitrary designations of new terminology is only more of a nuisance given that there's utterly no context in which to consider that terminology more useful than that which we already use.

And finally, in what way do we guage what is or isn't an "Isochord" work as opposed to any number of other styles? Your theory is so general and the sound pallette so ordinary, there's really nothing that distinguishes music created in this manner from music created from traditional tonal methods.

I said this before. In fact, I said it various times. It's totally fine that you have your own system for writing music. Yeah, you can call chords cheeseburgers and your melodies airplanes; do whatever you want, for yourself.

The thing is, all that you wrote in that example can be done quite as well by not following your supposed system. I'll briefly explain:

In the start, measures 1 to 7 you have clusters in a rhythm, then a triad in the same range as the cluster. The bass moves the first two times chromatically a semitone to the triad, while the third has a major second (d to e.) Clusters are based on stacked major seconds as far as I can see. Nothing particular there.

Then measure 7 has an octave+5th in bass.

Then you repeat the same model pretty much until measure 15. Then you have a motive going on the right hand and the clusters are an accompaniment figure, setting a simple rhythmic pulse. Then you repeat the same model until measure 31, where you do the C#/Db enharmonic for no discernible reason.

32 and 33 are arpeggios built on 7ths, and you repeat the same thing. Then in 38 there's the good'ol mixture technique with the parallel intervals (7ths in this case.) I'd have written them differently too, specially the starting one. B#? Is that really necessary? Might as well have written C - Bb, Bb - Ab, etc etc.. which you actually do on the RIGHT hand in the same measure. Is there any particular reason why this inconsistency is there? Why not keep the same model?

42 has has some bitonal harmony. Even then, most of it is still based on the typical triad anatomy with added/subtracted notes. The low chord is a D7 minus the 5th, right hand one can be seen as a G based chord with an altered 5th, no third and a minor seventh.

Then you repeat that some, and... that's it.

Yeah well, I hardly think anyone needs any new theory to write any of the above. Nor to even make sense out of it, or compose anything like it. Like I said, very, very, simple stuff to take apart.

I mean, you can still use it for yourself if you want, sure. But really if I didn't know this was written using your system there beforehand, I would have no way to have guessed it had ANY system at all.

Likewise, the whole stacking things can give some particular sounds and so on, however there needs to be much more to it than just that. Here, as I've said before, it's just a huge layer of theory where really it's not necessary.

I should probably wait to hear what your final culmination will amount to

I said this before. In fact, I said it various times. It's totally fine that you have your own system for writing music. Yeah, you can call chords cheeseburgers and your melodies airplanes; do whatever you want, for yourself.

The thing is, all that you wrote in that example can be done quite as well by not following your supposed system. I'll briefly explain:

In the start, measures 1 to 7 you have clusters in a rhythm, then a triad in the same range as the cluster. The bass moves the first two times chromatically a semitone to the triad, while the third has a major second (d to e.) Clusters are based on stacked major seconds as far as I can see. Nothing particular there.

Then measure 7 has an octave+5th in bass.

Then you repeat the same model pretty much until measure 15. Then you have a motive going on the right hand and the clusters are an accompaniment figure, setting a simple rhythmic pulse. Then you repeat the same model until measure 31, where you do the C#/Db enharmonic for no discernible reason.

32 and 33 are arpeggios built on 7ths, and you repeat the same thing. Then in 38 there's the good'ol mixture technique with the parallel intervals (7ths in this case.) I'd have written them differently too, specially the starting one. B#? Is that really necessary? Might as well have written C - Bb, Bb - Ab, etc etc.. which you actually do on the RIGHT hand in the same measure. Is there any particular reason why this inconsistency is there? Why not keep the same model?

42 has has some bitonal harmony. Even then, most of it is still based on the typical triad anatomy with added/subtracted notes. The low chord is a D7 minus the 5th, right hand one can be seen as a G based chord with an altered 5th, no third and a minor seventh.

Then you repeat that some, and... that's it.

Yeah well, I hardly think anyone needs any new theory to write any of the above. Nor to even make sense out of it, or compose anything like it. Like I said, very, very, simple stuff to take apart.

I mean, you can still use it for yourself if you want, sure. But really if I didn't know this was written using your system there beforehand, I would have no way to have guessed it had ANY system at all.

Likewise, the whole stacking things can give some particular sounds and so on, however there needs to be much more to it than just that. Here, as I've said before, it's just a huge layer of theory where really it's not necessary.

OK, you said this before and you said this again.

You may be right that what I wrote in that example "can be done quite as well by not following your supposed system". And then you explain carefully.

Well, I appreciate your analysis very much. But do you see how long-winded your description is? In contrast to this, my explanations given in the score are descriptively short and strictly substantial regarding the content of the music.

Let me give just some short remarks for clarification:

The "clusters" are well-defined M-2nds as plain-symmetric chords.

"the C#/Db enharmonic for no discernible reason" in bar 31 is well understandable as leading tones to the subsequent D (right hand) and C (left hand).

The runs of parallel m-7ths in mm. 38-41 are enharmonically written in such a way that they're uniformly readable as stepwise running 7ths within each measure (thus avoiding double # or b).

All "mixed-symmetric chords" in mm. 42-52 are indeed "bitonal", if you like. But what's more important (and well audible), from the point of Isocord Theory, is the bi-directional symmetry of the resulting sounds.