A New Theory of Harmony? – Isocords & Palindromes

There's an interesting empirical fact of general music theory: Superimposing equal intervals, one above another, yield vertically symmetric sounds which are more or less dissonant, depending on the particular interval used. Let's call those equal-interval, or iso-interval, sounds isocords (comparing well to the usage of discords for dissonant chords and concords for consonant chords – cp. Dolmetsch Online - Music Theory Online - Triads & Chords).

Let's look at some examples.

Minor thirds chords:

B-D-F (= diminished triad, a 3-tone chord)

B-D-F-Ab (= diminished seventh chord, a 4-tone chord)

There are no more higher-toned minor thirds chords from B since the next minor third above Ab would be Cb which is enharmonically equivalent to B.

Minor sevenths chords:

B-A-G (a 3-tone m-7ths isocord)

B-A-G-F (a 4-tone m-7ths isocord)

B-A-G-F-Eb (a 5-tone m-7ths isocord)

B-A-G-F-Eb-Db (a 6-tone m-7ths isocord) – which is the highest-toned minor sevenths chord from B.

Major seconds chords:

B-C#-D# (the 3-tone M-2nds isocord from B)

or

Bb-C-D-E-F#-G# (the highest-toned M-2nds isocord from Bb)

etc. –– Explore some other isocords, e.g. those which really build up to symmetric 12-tone sounds.

An interesting feature of isocords is that they're all (apart from primes/octaves isocords) "atonal", so to speak: They have no tonal center to them nor do they imply one when played in progressions. Insofar they are a new kind of harmony which might be useful for contemporary composing. In particular, for 12-tone composition as well: Instead of solely building a 12-tone row you can have, for each tone of the row, a distinct isocord built on it, thus yielding a 12-isocord row. Just experiment with this idea. And if you're ingenious enough you may even build an all-interval 12-tone row of isocords!

The most exciting aspect of isocords however, at least for me, is the possibility to combine their "atonality" with traditional tonality in a very strict way by introducing just one voice-leading rule, the rule of minimal tone-steps resolution: In resolving a dissonant isocord, conduct semitone or wholetone steps, or leave a tone unchanged, to succeed in a consonant sound (major or minor harmonies, or "void" sounds by avoiding the critical third of a triad, or even plain unisons of primes/octaves). For instance, you all know the resolution of the diminished seventh chord, the 4-tone m-3rds isocord, into possibly four different major/minor chords (under suitable enharmonic respelling of the original m-thirds isocord) – and with the minimal tone-steps resolution there are even more concords resulting from a particular m-thirds isocord.

With these features of isocords it is possible to write music which is both modern and traditional: Use isocords as much as you want, with or without minimal tone-steps resolution, and switch to common practice harmonies by deliberately applying the single voice-leading rule (and vice versa).

This kind of new harmony theory of isocords is indeed a unique way of reconciliation between dodecaphony and functional harmony, if you wish to do so as a contemporary composer.

I don't understand how this can make "music both modern and traditional" if these ideas are already [over] 100 years old. You're Mr. Darmstadt and I'm SURE you've studied Berg and Schoenberg and Webern.......right?

Just like old times, eh?

An interesting feature of isocords is that they're all (apart from primes/octaves isocords) "atonal", so to speak: They have no tonal center to them nor do they imply one when played in progressions.

So they're just like any major and minor triad? Context determines harmonic functions except in cases such as characteristic dissonances (arguably), and even then the associations would be very hard to see (or impossible, even) if the context worked against it. No single chord by itself can have a "harmonic function," as you of course realize, without other chords that give it a context.

C-E-G, otherwise known as a C major triad, can be how many things exactly? Dominant? Subdominant? Tonic? With it alone, can we tell which function it may be? Clearly not. As for implying a tonal center when played in a progression, isn't this a little vague? If it's ANY progression, there are millions of ways of making that chord completely vague (Debussy, Bart

Just like old times, eh?

So they're just like any major and minor triad? Context determines harmonic functions except in cases such as characteristic dissonances (arguably), and even then the associations would be very hard to see (or impossible, even) if the context worked against it. No single chord by itself can have a "harmonic function," as you of course realize, without other chords that give it a context.

C-E-G, otherwise known as a C major triad, can be how many things exactly? Dominant? Subdominant? Tonic? With it alone, can we tell which function it may be? Clearly not. As for implying a tonal center when played in a progression, isn't this a little vague? If it's ANY progression, there are millions of ways of making that chord completely vague (Debussy, Bart

Schoenberg used 4ths stacked on top of each other in the chamber symphony and one of the six little piano pieces. Smith Brindle uses them as examples in his muscial composition book.

Really the main thing is to see what results you can get out of the technique...

I don't understand how this can make "music both modern and traditional" if these ideas are already [over] 100 years.

Yes, the ideas of functional tonality (not the term but the case) are 300 years old and the ideas of dodecaphony about 100 years old. But the systematic idea of building iso-interval chords

The prefix iso is used everywhere in nomenclatures. The term iso-interval chord (or "Isointervallakkord" in German) is well-established in music theory

I invite you to write a few little pieces ("isocordal miniatures"). Then we'll see how well you understood theory and method. Feel free to do whatever you like.

I invite you to write a few little pieces ("isocordal miniatures"). Then we'll see how well you understood theory and method. Feel free to do whatever you like.

Maybe I should've been more blunt about it, so here you go:

...and most importantly nobody is going to pay attention until you have a good portion of examples of actual music (WRITTEN BY THE PERSON PROPOSING THE METHOD) using this.

Seriously now, don't be so lazy.

Germany's days are over

Maybe I should've been more blunt about it, so here you go:

...and most importantly nobody is going to pay attention until you have a good portion of examples of actual music (WRITTEN BY THE PERSON PROPOSING THE METHOD) using this.

Seriously now, don't be so lazy.

No problem. Visit #11 of this YC page http://www.youngcomposers.com/forum/atonal-competition-blm22-subs-20189-2.html and tell me what you have to say.

No problem. Visit #11 of this YC page http://www.youngcomposers.com/forum/atonal-competition-blm22-subs-20189-2.html and tell me what you have to say.

Alright, so let's get to the main issue I have with your chord anatomy theory:

Stack 5ths together within an octave and you get major second formations (clusters.)

Stack major 6ths within an octave and you get minor 3rds (diminished chords.)

Stack minor 6ths within an octave and you get major 3rds (augmented chords.)

Stack major 2nds/9ths and you get clusters.

Stack minor 2nds/9ths and you get clusters.

Stack major 7ths within an octave and you get minor 2nds (clusters.)

Stack minor 7ths within an octave and you get major 2nds (clusters.)

Can't stack 8ths within the same octave, and intervals above the octave are redundant.

Can't stack Tritones within the same octave, and across octaves yields the exact same notes over and over.

Notice 4ths are missing from that list?

Now, I'm sure you're familiar with Hindemith's quartal harmony ideas and pieces. There's a good reason why single out 4ths out of the intervals from which to build chords. 4ths in both configurations (within the octave and across octaves) have very characteristic properties which are not redundant.

This is one of the reasons why quartal harmony actually developed as a harmonic language, there's an actual difference generated by the fact that chords built in 4ths are neither clusters or interchangeable with traditional chord formations. For example:

Stack 4ths together in the same octave and you get a sound which is neither the complementary intervals nor a cluster form (there's enough room to hear each note even if your chord is 4-5 notes big.)

However, it all depends also in how you're using it. For example 4th suspensions are chords formed in 4ths however that's not how they are understood in traditional context (4 must go to 3, forming a triad.) It wouldn't be hard to see 5ths being used the same way. However, this is all ignoring "equal intervals" as in reality the chord within the same octave is composed of complementary + different intervals. A chord supposedly built on 5ths (say, C-G-D) within the same octave is actually composed of a 2nd + 5th from C (or 2nd - 4th from D). It would be hard to claim that the only way to this chord is through stacking 5ths and likewise with the other examples.

So, considering the options, there is actually very little room to move in if you are strictly only considering chords built on the same intervals (most MUST be across octaves), as either some intervals yield naturally typical 3rd formations or boil down to cluster forms. Instead of trying to come up with a method of chord anatomy, you might as well concentrate more on how these chords actually relate to eachother.

And that's where this example piece you posted I think is the weakest. Sure you show some possible chord setups (many of which are either cluster forms, octaves or simple typical augmented/diminished chords.) Since you're talking about strict voice movement, I really don't see it in the example. If it's there, you should point it out.

Better yet, why not make a new piece for, say, a solo keyboard instrument or SATB where it's simpler to follow what you're doing and actually demonstrate what you mean with voice leading changes.

Thank you for your detailed analysis. It's a worthwhile contribution since it helps to clarify some important issues of the isocord theory. Let me do this in order:

Alright, so let's get to the main issue I have with your chord anatomy theory:

Stack 5ths together within an octave and you get major second formations (clusters.)

That's not quite true

Thank you for your detailed analysis. It's a worthwhile contribution since it helps to clarify some important issues of the isocord theory. Let me do this in order:

That's not quite true – you get mixtures of major and minor seconds (and other intervals if you don't take enough 5ths). In the end, with all 12 tones of 11 5ths stacked above each other (i.e. the isocord of the circle of 5ths) you get the chromatic 12-tone isocord of m-2nds if stacking all 5ths within an octave.

However, this kind of stacking 5ths isocords (and 4ths isocords as well) within an octave is not allowed by the rule of building isocords as symmetric sounds. They would lose their symmetry property.

IE; you get clusters and obviously in a non-equal order of interval progression. I already said that, and that's fundamentally a problem. Likewise, 5ths across octaves I can see as distinct independent chord anatomy, but everything else simply doesn't work.

These examples are basically correct – apart from stacking 9ths because the building intervals for tone-distinct isocords are only the 11 intervals within the octave, from m-2nds through M-7ths (or, preferably, 13 intervals, including the prime and the octave to produce arbitrary big 1-tone primes [spread over multiple instruments] or octaves [what I've done in my Isocordal Piece]).

However, there's an interesting point to these examples: They show a kind of relationship between isocord classes which is either close or more distant, depending on the (mirror-related) distance of the intervals within the octave.

See this table of isocord formations (the stacked numbers indicate semitone distances of intervals, the dash says "stack next interval"; the other symbols should be self-evident):

p-1mes: 0

m-2nds: 1–1–1–1–1–1–1–1–1–1–1

M-2nds: 2–2–2–2–2

m-3rds: 3–3–3

M-3ths: 4–4

p-4ths: 5–5–5–5–5–5–5–5–5–5–5

a-4ths: 6

d-5ths: 6

p-5ths: 7–7–7–7–7–7–7–7–7–7–7

m-6ths: 8–8

M-6ths: 9–9–9

m-7ths: 10–10–10–10–10

M-7ths: 11–11–11–11–11–11–11–11–11–11–11

p-8ves: 12

This symmetric table reveals the relationship between isocord classes: String length shows similar isocords (where, mirror-inverted, closer strings show closer relationship and distant strings more distant relationship). String length also shows the possible number of different tones of an isocord (minimal 1 for primes and octaves, 2 for the tritone, maximally 12 for perfect fourths / fifths and minor seconds / major sevenths). Of course, an isocord need not include all different tones; it only has to conform to the rule of strict symmetry. These features are important for the operations of inversion and transposition of isocords. However, this is a different aspect which I'll deal with at another time.

A final remark to this table: Contrary to the discussion of intervals in dodecaphony, the number of intervals is not reduced to the set of complementary intervals within an octave (namely 6) because the "layer-sensitivity" of the 13 intervals of the octave, i.e. the actual distance between the constituent tones of an interval (its "ambit"), is of concern. Therefore a major seventh is not equivalent to a minor second, but complementarily distance-related (with similar sound quality, however).

I didn't mention this before, but I think by now it should be obvious. Unisons and octaves are not and never will be considered "chords" for any practical purpose, unless words stop meaning anything. They are called just that, unisons and octaves. Likewise, all of this STILL sounds extremely redundant since you're taking this enormous work to build simple sounds out of... repeating intervals. MOST of the results are completely within traditional standards, the rest are pointless to relabel (tritones, clusters, etc.)

In essence, this seems by and large like a waste of time.

If you think of fourths, no, not missing. Quartal chords are used in mm. 5, 21, 25, 29, 35, 37, 38, and 42/43.

I mean in my example.

Quartal harmony is dealt with in isocord theory as well as fifths sounds (up to 12 tones big – only just feasible on the keyboard of the 88-key piano!). But only, as you can see from the table above, in their strict symmetric form. Ordinary inversions of quartal chords (as is the case in 20th century harmony theory) are non-issue of isocord harmony (only doubly directed inversions – up and down simultaneously of the ambit tones – are feasible; think about this as a hint to the matter of inversion and transposition). –– Might be enlightening what YC member Michel Edward had to say about quartal harmony since I know that he's very keen – and knowledgeable – on this subject matter.

You're sort of missing the point. Clusters also envelop intervals across octaves, that's the "spread" of the cluster. Once you start reaching a point where individual notes are lost to the mass (as it would start happen with 6+ note chords, specially with the same intervals.)

Just as well, if supposed "isochords" are only considered so if they're stacked across octaves (not within the same octave) then, like I said previously, MOST of the options are gone. Furthermore, combination of multiple chords together generate yet again different non-equal formations. You really have to define what you're technically "allowing" and "forbidding" because as it is it seems you can't do practically anything and several intervals yield clusters or traditional sounds (along with several other "chords" which are not really chords at all.)

That's actually one of my next posts – to show how isocords relate to each other through inversion and transposition.

You're quite right with this observation. I deliberately left off the rule of minimal tone-step resolution in Isocordal Piece because I had to write an atonal composition. As soon as you use this rule you depart from atonality and touch the tonal domain of functional harmony. However, how neat atonality meets with tonality by the use of isocords and minimal tone-step resolution – this will be my next task to accomplish.

Inversion alters the makeup of the chord and unless I'm understanding you wrong here it would break up the "equalness" of it! On the other hand if you allow inversions, essentially your idea has no coherency.

I'm personally not a fan of re-labeling just for the sake of re-labeling. If you putting traditional augmented chords, all sorts of clusters and even quartal chords in the same bag, something's wrong. Unless there's a VERY good reason to call augmented/diminished chords, clusters, octaves and unisons anything other than what we're calling them, I don't see the point.

Another issue with the whole thing is when you start addressing horizontal formation (arpeggios, line building.) In which case adhering to the larger intervals (5ths, 6ths, etc) will quickly put any instrument well out of its range and furthermore limit what it's doing to a simple jump sequence. IF NOT, then there's absolutely no point in, again, labeling anything differently if intervals can be combined in different ways as... that's what it's always been about.

By what I'm seeing in your explanations, you do little to address the absolute redundancy that comes with both the "strict" and my possible interpretation of what you're saying. It leads to a gross generalization of a lot of things (mixing clusters with traditional chords, octaves and unisons) and I REALLY don't see how in any way it can be actually useful.

I'll stick by what I said, you should show the relationship bit next as the chord anatomy bit remains un-interesting and irrelevant thus far.

Your lengthy reply indicates that you still not yet have understood the concept of isocord (note: not isochord!). I repeat its original definition from my post #1 of this thread: Superimposing equal intervals, one above another, yield vertically symmetric sounds which are more or less dissonant, depending on the particular interval used. Let's call those equal-interval, or iso-interval, sounds isocords.

Note, that I talk about sounds and not about chords

Well obviously I noticed you wrote "cords" however considering that as far as I saw it dealt with grouping notes together to form...chords. So, really, what's the deal? You really should explain if this is a chord anatomy idea or a horizontal/lineal thing. I'm by now confused enough that I'm not seeing much of a point continuing.

I can also call a theory "fnarf" but unless I can somehow explain it in a way that it makes sense considering existing musical terminology, it means absolutely nothing. You can't just conjure up new terminology without explicitly explaining how it relates to existing terminology and/if it will replace it, why.

Furthermore since you're talking about HARMONY (as in, vertical aspect) and as far as I've seen treating the entire thing as chords for all intents and purposes, what do you expect? Your example piece as full of... what should I call them if they're not "chords" yet look exactly like what I would normally call chords...? Are arpeggios not arpeggios anymore either? How much stuff are you renaming and why? Single notes are now "isocord" unisons? ALL of them?

Honestly you really need to be more clear. Much, much, much more clear.

I can also call a theory "fnarf" but unless I can somehow explain it in a way that it makes sense considering existing musical terminology, it means absolutely nothing. You can't just conjure up new terminology without explicitly explaining how it relates to existing terminology and/if it will replace it, why.

actually you can. well, i don't know if you can in this particular example, but (since i think you try to universalize) you can in very many different language-games. new theory doesn't (in a very strict sense) have to deal with old theory and even more with its termonology. for example, myths are replaced by scientific procedures without any explanation of mythic terminology. and if you say that inside science (knowledge procedures)should be different, well, maybe, but they aren't.

all it takes is a princple and operator that can 'translate' old terms (of 'facts') into a new system, and loosely so. and even more often, new terms are tried to be translated into an old language only because of psychological problem of changing human thinking fast enough. that is, mutations don't necessarily depend on environment. or - new is possible, regardless of its relation to old.

actually you can. well, i don't know if you can in this particular example, but (since i think you try to universalize) you can in very many different language-games. new theory doesn't (in a very strict sense) have to deal with old theory and even more with its termonology. for example, myths are replaced by scientific procedures without any explanation of mythic terminology. and if you say that inside science (knowledge procedures)should be different, well, maybe, but they aren't.

all it takes is a princple and operator that can 'translate' old terms (of 'facts') into a new system, and loosely so. and even more often, new terms are tried to be translated into an old language only because of psychological problem of changing human thinking fast enough. that is, mutations don't necessarily depend on environment. or - new is possible, regardless of its relation to old.

OK so from now on I expect people to write using my fnarf theory. What is fnarf? It's a thing! You put notes together, etc. No need to deal with previous theory (even "notes" is previous theory terminology,) right? Then no need (no WAY) to explain it either!

Have fun writing using fnarf!

Protip: You can't explain the new without using the old. It's impossible. That's like having a dictionary explain what words mean WITHOUT using other words!

words are not one. there's no meta theory of words.

while certain theory is using certain concepts to explain and create new things. it's inseperable. it's not that one proceeds simply from falsifications of old or new theories.

the new conceptual sytem brings new pespective on the facts, and, as such, creates these facts by forcing theory.

plus, one could continue with your 'fnarf' theory if it had enough content (and-or, in time, create this content).

i don't think there's rigid and one empitrical-material world so we could simply have one certain method of going on from there. if it is so, then theories proceed to get more practical space-space in history due to some heterogenous factors that depend more on human factor than on certain objective logic. if you happen to push your fnarf thing as far as to obtain some significant practical space, your theory is pretty much valid, regardless of its relation to an old one(s). the relation is a 'trivial' thing and rather institutional-social than objective.

that is, it took a death for galileo to force his theory onto an old field of preceding cosmology theories even before it had any real and tested content.

more abstractly speaking, form precedes content and so, it cannot be judged by old theories.

it took socrates to die to force an idea of philosophy as thing which is completely independent of myth.

that is, it takes a rupture,a schism, a break, precisely, to force new theory, and its relation to old ones is an artefact and in more open and free societies it would come easier and more obviously 'good'. actualy, to some extent, non-replaceale theories could exist simultaniously without any tension, except for social and practical - that is productive and economic.

the need to explain new using old, or building a relation to it, stems not from the fact that new is dependent on old(which it is not), but from simple psychological rule of psychic and social economy. in time new replaces or lives together with old not by explaining it, but by winning a right (practical) to produce in limited space-place of appearent world.

That's nice and dandy. However the fact remains that unless you use terminology people understand to describe your crap, it means nothing.

Again, fnarf.

I don't get the point of this. Would you care to explain? And yes, I've read every post in this thread and still don't get it.

P.S. SSC, would you like to see my first fnarf piece? I'm looking for advice on how to improve it, and I want to make sure it abides by all the fnarfian rules.

P.S. SSC, would you like to see my first fnarf piece? I'm looking for advice on how to improve it, and I want to make sure it abides by all the fnarfian rules.

SURE!

It will start playing in 5...4...3...2...1...

That's nice and dandy. However the fact remains that unless you use terminology people understand to describe your crap, it means nothing.

Again, fnarf.

it's not nice and dandy, it takes years and decades (even centuries) of forcing new theories.

my point was simply directed at your pseudo-falsificationist linear notion of theoretical (conceptual) change. theory is being constructed much earlier than it has already been incorporated into institutions.

at some points it may need creating new institutions.

at extreme points it needs to simply get done with old crap: theories, institutions and such(because, simply, the richness of new theory is way over the top and there's no need relating it to old, except for academical games).

there's no unforced transition from old to new in which you would see the light of old (tradition) shining through. at best you may coexist, at worst you simply throw the old away.

so, basically you're asking the guy hansen to do a thing which is of no use for creating new theory.

especially since it is art field and ruptures are much easier there than in wider fields such as cosmology or physics.

and to repeat, you can come with new scraggy not explaining it in terms of the old scraggy. you change facts together with creating theory. that is - the same concept can have a new content (to avoid using all the new symbols). plus, not all logical constructions are science based, so they are used transitively, which means - you can understand new things because of the wide variety of games where logical constructions hold. it doesn't mena that the content of games is the same. anecdotically speaking, basketball is not explained in terms of football. nor is relativity in terms of newtonian mechanics.