A New Theory of Harmony? – Isocords & Palindromes

There's a nice thread on Your favorite bichords where you'll find some interesting bi- and tri-tonal chords. Some of them can be described as palindrome chords (see post #12) which have an attractive sound character of their own.

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.

I'm not too sure about this. Why should'nt isocords played in progression imply a tonal center? Just curious.

I'm not too sure about this. Why should'nt isocords played in progression imply a tonal center? Just curious.

Let me specify this: Plain-symmetric isocords, which consist of equidistant intervals only, have no tonal center, neither per se nor in progression (except for primes/octaves). Only if you apply the minimal tone-steps rule to an isocord and resolve it to a consonant minor or major chord, then you may understand the resolved isocord as having functioned as a kind of dominant, resolved to a tonic. However, the tonality lies in the resulting tonic chord only.

Maybe it's more appropriate to speak of non-functionality of isocords instead of "atonality". "Atonal" isocords can be built more easily by bi-symmetric chords consisting of successive dissonant intervals only. But then you don't need the "atonality" concept, you may just say "dissonance" or "discord".

Okay, to me a progression of 3-Note-5th-Isocords can be Tonal.

for example the chord progression AEB, FCG, GDA, AEB

has to me the tonality of A. such chords could be interpreted as classical NinthChords with the 3rd left away, which are pretty common in tonal music.

But it really depends on context and you obviously want to use them in an atonal way , which is fine. Just wanted to say that I don't think that Isocords per se have to be atonal.

Besides that, now that your theory also includes all kinds of palindrome chords, this is getting a bit more interesting and I'm looking forward to your next posts.

However I have the problem that, as for now , you didn't really give us any ideas on how to actually use palindrome chords in practice. Do you just imagine using them as an accompaniment and having a single-voice melody above? How do you structure the melodic material when using isocords, is it in any way dependend on the used chords? Is it possible to write contrapuntal music based on your theory?

In another thread you said

In a way, I've proposed a theory in which quartal harmony can be dealt with, at least with regard to its interval setup, transposition, inversion, and its relation to other harmonies.

So how does one actually analyze the relation of palindrome chords to other harmonies?

Let me specify this: Plain-symmetric isocords, which consist of equidistant intervals only, have no tonal center, neither per se nor in progression (except for primes/octaves). Only if you apply the minimal tone-steps rule to an isocord and resolve it to a consonant minor or major chord, then you may understand the resolved isocord as having functioned as a kind of dominant, resolved to a tonic. However, the tonality lies in the resulting tonic chord only.

Maybe it's more appropriate to speak of non-functionality of isocords instead of "atonality". "Atonal" isocords can be built more easily by bi-symmetric chords consisting of successive dissonant intervals only. But then you don't need the "atonality" concept, you may just say "dissonance" or "discord".

I think you're missing the primary ingredient needed to create tonicization. The all-important tritone relationship is what creates a 'tonal' center. A G7 chord 'resolves' to a CMaj or Cmin BECAUSE the B-F tritone collapses (or expands) to the C-E (or E-C depending on inversion of the chord). Can you have a tonality form through the Isochord theory? Yes, if you use some kind of Palindrome harmony that creates this collapsing tritone relationship.

That is the essence of what 'resolution' is in tonality, the almighty tri-tone resolution.

Let me deal with some particular points of your recent posts with the help of some quotes.

Now that's at least something to consider... Palindrome harmonicization. I'm sure it's probably been used before, but in the way you're describing it, it is much more interesting. I'm somewhat encouraged.

There's an interesting Wikipedia article, Palindrome, including detailed indications of its use in music (even more interesting might be the German Wikipedia article, Palindrom, since it presents the mathematics of palindromes also).

Besides that, now that your theory also includes all kinds of palindrome chords, this is getting a bit more interesting and I'm looking forward to your next posts.

However I have the problem that, as for now , you didn't really give us any ideas on how to actually use palindrome chords in practice.

Basically, the theory is descriptive as well as prescriptive with respect to the musical material you can achieve with it. How you use this material in your composition, how you develop your own rules to handle it, how you go on to combine it in new contexts

Hopefully, in its latest version isocord theory/palindrone harmony will not limit chord building in any way. Plain-symmetric chords ("isocords" in their proper sense) are possible as well as bi-symmetric chords ("palindrome harmonies" in any form as long as they comply with the symmetry property). Furthermore, with the help of the minimal tone-steps rule you have the possibility to achieve any non-symmetric chord of any flavor (major, minor, altered, cluster, mixture, bi-/polytonal, modes, etc.).

Let me explain what I mean by 'restrictive'.

Isochord Theory does not account for Augmented 6th Chords that are Asymmetric. There are hundreds of ways that chords can be Asymmetric, just like A,C,E,G# is Asymmetric (intervals are m3,M3,M3) or how about an atonal example like C3,B3,C#4,E4,G4,A4,D4 (intervals are M7,M2,m3,m3,m2,P4). These are reasons why focus on symmetry is so restrictive. You have to account for asymmetric interval content for Isochords to be useful or practical to the styles you're dealing with...

And with that restrictiveness, you're talking about a theory that attempts to 'bridge worlds' being too strict to apply to either of these languages. This is why I have constantly tried to stress that you stop trying to create a bridge and start making this a truly individualized style of music with a unique language and try to differentiate it from other styles instead.

You're right, your atonal example C3:11-2-3-3-2-5 (= C3,B3,C#4,E4,G4,A4,D5) isn't feasible with isocord/palindrome harmonies, at least not in one step with the minimal tone-steps rule. (But all your other examples are doable.)

However, how did you build up your C3:11-2-3-3-2-5 chord? By a rule? Just by intuition? Or merely at random, just for fun?

At least a ray of hope is the middle part of your atonal chord, B3:2-3-3-2.

How did you actually come up with using palindrome chords? What makes them so interesting to you compared to other chords? What special properties do they have concerning their sound? (Not that I don't believe that there are good reasons for using them - i would just like to know your point of view)

I've been thinking a bit about palindrome chords, their relation and the way one could construct/analyze such chords.

The terminology I use is spontaneously made up, so don't take it too serious!

As Hansen pointed out in the thread on bi-chords:

major triads superimposed with minor triads (or vice versa) result in palindrome chords.

Some examples from the same thread:

3-4-1-4-3

The minor chord has the interval structure 3-4 , the major chord 4-3. Note that the minor chord is the retrograde/mirror image of the major chord(btw. I find it fascinating that in classical harmony the minor triad was indeed treated as a counterpart to the major triad, being its intervallic mirror image).

The interval 1 serves as the symmetry axis.

4-3-7-3-4

Here we have the major chord 4-3 in the lower register and the minor chord 3-4 in the upper octave. This time 7 is the symmetry axis, so the triads are farther apart.

Generally palindrome harmonies can be analyzed as a chord and its retrograde being superimposed, with the distance between the two chords being the symmetry axis.

(However, not in all cases they should be interpreted this way, later I'll tell you why!)

I'll refer to the interval serving as the symmetry axis simply as axis of the palindrome chord .

I'll mark the axis with a [ ]. The part LEFT from the axis will be called "lower structure" of the chord, the part RIGHT from the axis will be called "upper structure". Upper and lower structure are related by retrograde.

The axis is the interval between the highest note of the lower structure and the lowest note of the upper structure.

Let's take a look at our chords this way:

3-4-[1]-4-3

Lower Structure: 3-4

Upper structure: 4-3

Axis: [1]

Generally a palindrome harmony can be created from any

chord/interval structure by using the following formula:

chord(prime form) - [Axis] - chord(retrograde)

Let's derive a palindrome from the harmonic cell 2-5!

2-5-[Axis]-5-2

The axis may be any interval you like!

So you can create from any palindrome chord new palindromes , simply by changing the axis interval!

For example: 2-5-[4]-5-2

Lower structure: 2-5 Sus2 chord

Upper structure: 5-2 Sus4 chord

Axis: [4]

We derive new palindrome harmonies from this chord by changing the axis.

2-5-[8]-5-2

2-5-[3]-5-2

The axis can also be 0 (prime)!

2-5-[0]-5-2

In this case the highest note of the lower structure is the same as the lowest note of the upper structure(they overlap).

Usually you would describe such a chord as 2-5-5-2.

However, in this context I prefer to add the 0 as the axis, to make it easier to analyze the palindrome property and the relationship to the chords above.

Note that the four chords above are all palindromes related to each other.

The upper and lower structure is in all cases the same, the only difference is the axis! Changing the axis is a simple way to create related palindromes from a single chord (Palindrome transformation).

There is another type of transformation. This only works when the lower/upper structure itself is NOT symmetric.

The only thing you have to do , is to SWAP the lower and upper structures,which yields a new palindrome chord related to the first one.

So: 2-5-[4]-5-2 becomes 5-2-[4]-2-5

Or: 3-4-[1]-4-3 becomes 4-3-[1]-3-4

Again,you can also change the axis of this new chord!

So as you see, with the formula and these two types of transformation you can create a whole universe of related palindrome harmonies from a small harmonic cell (in my example: 2-5).

Let's take a look at another way of creating palindrome chords:

Symmetric chords with periodic interval structure

periodic-interval structures are what I like experimenting with in my own compositions. Here is an example:

2-5-3-2-5-3

2-5-3 Is a period in this case!

However, this example is not palindromic.

You can construct palindromic periodic interval structures by using symmetric periods!

Like 3-2-3

With this pattern you can construct palindromic periodic chords like

3-2-3-3-2-3-3-2-3

With my axis approach this could be analyzed as:

3-2-3-3-[2]-3-3-2-3 based on the structure 3-2-3-3.

However, in a musical environment dominated by 3-2-3 cells, it might make more sense,to analyze this chord as a periodic palindrome derived from 3-2-3.

It's really a matter of context.

A special case of periodic palindrome chords are

chords with alternating interval structure.

The interval structure of such chords alternates between two intervals.

Examples: 3-2-3-2-3

Or: 4-3-4-3-4-3-4

Or based on the 2-5 cell: 2-5-2-5-2

I'm very fond of working with such chords. I think that alternating between two intervals can give harmonies a very unifying/coherent sound.

Also note: 4-3-4-3-4-3-4 (could be interpreted as Cmaj7+Dmaj7) would be traditionally described as a pretty dissonant chord (it even contains a C-C#-D cell.). However,when alternately stacking major and minor thirds, the overall sound can be pretty pleasing and it gives a nice floating impression(at least to me). Even when such a chord contains all 12 chromatic notes it sounds completely different from a cluster. The presence of so many consonant intervals(major and minor thirds) between consecutive notes kinda dilutes the

dissonance within the chord.

btw. Also Isochords are not just the most primitive type of palindrome chords but also the simplest periodic chord structures.

That's all I have to say as for now, I really have some more ideas but I'm getting a bit tired of writing.

I'm looking forward to find out about how to create melodies and counterpoint and how to analyze chord progressions, when working with your theory, Hansen!

An excellent essay on palindrome harmony, DAI!

As an aside to your axis idea for palindromes with an even number of intervals

An excellent essay on palindrome harmony, DAI!

As an aside to your axis idea for palindromes with an even number of intervals

Nice interlude, AA!

WHAT? Are? You? Talking? About?

Nothing but palindrome patterns, inspired by DAI!

Have you even looked into the overtone series material yet to see 'why' such sonorities might sound so 'pure' and 'clean' within your aesthetic?

No, not yet, of course. You may know

Meanwhile I've uploaded a completely re-written YC Wiki article, Isocord Theory.

I also uploaded a new YC Wiki article, Palindrome Harmony, which complements the Isocord Theory essay.

In my opinion, both articles blend in well with each other. So I would be very interested in your opinion about this, by now more comprehensive, theory of sound construction with isocord and palindrome harmony.

And this takes all of about 5 minutes to explain... funny how there are 140+ posts on this thread topic.

So, you want to make Palindrome Harmony a sub-theory (or subset) of Isochord Theory. M'kay...

Can't say there's much to talk about with Isochords. There's more to say about Palindrome Harmony. Seems a little lopsided to me. You're still talking about a sonority of pitches ordered through structures that pretty much create really only one type of sound. This is a good start, but I think you're going to need to account for more sonorities by creating them and discussing them.

Most theory is generally created AFTER the music, where in this case, all your theory happens BEFORE you've written or discovered the music.

Music theory is pointless.

Music theory without MUSIC is pointless, rather!

Music theory without MUSIC is pointless, rather!

My thoughts exactly!

There is no music in music theory.

I agree.

Music theory is pointless.

Music theory without MUSIC is pointless, rather!

There is no music in music theory.

My thoughts exactly!

I agree.

Why so impatient? We've time enough! (Life is short, art is long. But – just wait a bit longer.)

Nevertheless, I like this list of mutual comm!

And this takes all of about 5 minutes to explain... funny how there are 140+ pages on this thread topic.

Well, it took 140+ posts in this thread until I actually accepted the idea that symmetric chords built on uneven intervals (now "palindromes") are a legitimate candidate of the theory – an idea, which I had at the very beginning of my considerations, but rejected then.

So, you want to make Palindrome Harmony a sub-theory (or subset) of Isochord Theory. M'kay...

More precisely, isocord harmony is a subset of palindrome harmony, but Isocord Theory is more easily to develop and to explain (in less than 5 minutes, if you like).

BTW, what do you think about the fact that isocord harmony is a special case of palindrome harmony, but the deducibility of palindrome harmony from isocords (i.e. by "isocord transformation" or "palindrome re-transformation") is another fact (of operability, to be precise)? That's the (philosophical) question of subjection of a (part of a) general theory under a special case of it. Isn't this a nice example of a logical lemma?

Can't say there's much to talk about with Isochords. There's more to say about Palindrome Harmony. Seems a little lopsided to me. You're still talking about a sonority of pitches ordered through structures that pretty much create really only one type of sound.

IMO, there are two common types of sounds, non-symmetrical (of traditional and most of 20th c. harmony) and symmetrical (isocord & palindrome harmony). But the cardinality of palindrome sounds is larger than that of traditional/modern sounds: Every non-symmetric sound can be turned into a symmetric sound by complementing it with its exact retrograde (similar ideas are discussed by Persichetti 1961, 173-175). So we have a wider range of sonorities in palindrome harmony than in traditional approaches.

Most theory is generally created AFTER the music, where in this case, all your theory happens BEFORE you've written or discovered the music.

OK, my music will come after the theory – and, hopefully, possibly from many others in the future? Who knows?

At least, I'll continue with my "Explorations of Isocord and Palindrome Harmony" in subsequent posts.

IMO, there are two common types of sounds, non-symmetrical (of traditional and most of 20th c. harmony) and symmetrical (isocord & palindrome harmony). But the cardinality of palindrome sounds is larger than that of traditional/modern sounds: Every non-symmetric sound can be turned into a symmetric sound by complementing it with its exact retrograde (similar ideas are discussed by Persichetti 1961, 173-175). So we have a wider range of sonorities in palindrome harmony than in traditional approaches.

What is it that we can or should infer from these 'two common types of sounds' as they relate to one another. Also, you're really only dealing with one dimension of sound... sonority. If you're really looking for a prescriptive way of dealing with sound (you mentioned creating palindromes in both melody and harmony), have you considered taking that further to elements of rhythm, form, and timbre? How about dynamics?

If, in your theory, symmetry is tantamount to resolution or consonance, then also consider what asymmetry might mean to this theory and how you can use it more freely. For example, what would it involve to create a large scale work of symmetry with its climactic moment operating in an asymmetric way.. and how would that impact the theory?

Could symmetry and asymmetry be an additional dimension of music, and if so, how could you take that dimension to further extremes? These and many more questions should await you, and you should be prepared to deal with them. Not only that, but you should be able to represent these in your theory both in concept and in the pedagogy you create and use to explain your theory.

That's a profound list of further inquiries into the theory, indeed! Its impact is

Here's a little thing of a minor thirds exploration, including a nice intersection in classical style.

Here's the piece, Isocord Exploration 3.mp3,

and its score, Isocord Exploration 3.pdf.

Have fun in listening!

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YC Music Articles: Isocord Theory & Palindrome Harmony

Now come on. You want to demonstrate the potential of your theory with a piece that is 95% based on common practice theory? As AA said, you should (as for now) not bridge traditional and "modern" music but try to create an own distinct sound world with palindrome chords.

Now come on. You want to demonstrate the potential of your theory with a piece that is 95% based on common practice theory? As AA said, you should (as for now) not bridge traditional and "modern" music but try to create an own distinct sound world with palindrome chords.

Any way, my intent is, among other things, to bridge traditional and modern music. My little thing of a minor thirds exploration is just a tiny example. You could do as well another thing of minor thirds exploration which would sound very modern

You've done so much to try and sound 'traditional' or 'modern' in an attempt to 'bridge' these sonorities. Unfortunately, nothing you've done up to now sounds unique or even a genuine attempt at creating 'music'. All you're really doing is subverting the pedagogical structure of theory with these 'experiments' without providing what I or anyone else might call a 'genuine work of music'.

Create MUSIC, your music, Isocord Music... whatever... but make it 'definitively' Isocord Music that sounds completely different than traditional or 'modern' music. Do something with THIS THEORY that no one can do with traditional or modern techniques. Create sonorities with THIS THEORY that sound nothing like traditional or modern music. Then you'll have something that is worth understanding... until then, it's just theory.