Recently I've been learning about alternate analysis techniques used for modern compositions and one that I've recently learned about the Hanson system partiality described below:

Explanation of Hanson system symbols

Now as shown in the example above a Dmadd9 gets reduced to a p2mnsd because of the intervals between the notes.

What I don't understand is how this analysis is used to give meaning to groups of notes since it seems that different groups can result in the same analysis. For example, wouldn't a C, E, F and C, G, A♭ yield the same analysis? So how can one efficiently use this system? Are there times when this type of analysis is more appropriate then others?


3 Answers 3


The fact that this method will yield the same results for some sonorities is a feature, not a bug! This is an analytical notation rather than a performance notation, and as such, part of its purpose is to find similarities and identities between sonorities that are different on the surface. This particular p2mnsd harmony is, as you say, a Dmadd2, but there are many other harmonies, which, though different, have fundamental similarities. Most obviously, for example, Fmadd2 is an extremely similar harmony that would also be called p2mnsd. This is something everyone does with more familiar harmonies—we acknowledge that C Major chords and F# Major chords are, in some sense, fundamentally the same kind of harmony when we say they are both Major chords despite the fact that they share no actual pitch classes.

However, Hanson's approach will also point out similarities beyond these more obvious comparisons. For example, if you turn that Dmadd2 chord upside-down—that is to say, if you, for example, were to build a M2, m3 and P5 below D instead of above it—you would have an Gadd4 harmony. This too is a p2mnsd, despite the fact that is has a somewhat different sound. A method such as Hanson's points out that there are still some underlying similarities between these seemingly different harmonies.

Hanson's method will also eliminate functional distinctions based on enharmonic spellings. Both an Ab7 chord and a German Augmented Sixth chord in C minor will have the same pitch classes, but in common practice tonality they will function very differently. But we already have a theoretical apparatus to deal with those distinctions; Hanson is interested in being able to find new potential relationships irrespective of pre-existing grammars or even whether they are familiar triadic harmonies at all.

It's important to note that Hanson's descriptive apparatus is essentially identical to interval class vectors in set theory, and has been effectively superseded by it. p2mnsd is the same thing as the interval vector <111120>; they impart the same information but the interval vector will be recognizable to far more people. I also think it's a little clearer about the fact that there are no tritones in the harmony since there's a zero for any unrepresented interval classes. In Hanson's notation you have to notice there aren't any t's.

Finally, there are a few harmonies that Hanson's method will show as being identical that aren't transpositions or inversions of each other. For example, the notes C, Db, E, and F# would be pmnsdt in his notation, and so would C, Db, Eb, and G. However, try as you might, you will never be able to transpose or invert the first set of notes and generate the second set. In traditional set theory, the interval class vectors for these two sets are (of course) also identical: <111111>. However, the first set is (0146) while the second is (0137). Forte called these sets that are different but share the same interval content "z-related sets" (I believe "z" stands for "zygotic").

With either notation though, it's clear in this example why such analytical methods can be useful. pmnsdt and <111111> both make it clear that these are particularly interesting sets because, despite the fact that they only have four notes, they have every possible interval contained within in them. They are sometimes called "all-interval tetrachords," and both analytical notations give you a sense of their general soundworld as well as their compositional possibilities. The diatonic set (C,D,E,F,G,A,B) would be labelled p6m3n4s5d2t or <254361>, both of which beautifully illustrate the diversity and asymmetry of the set as well as the abundance or dearth of common tones at various different transpositional levels. It's especially instructive in contrast to the whole-tone scale collection which is highly symmetric and starkly either/or: m6s6t3 or <060603>


The Hanson system appears to be a means of efficiently expressing an interval-class vector chart commonly used in Set Theory analysis.

An interval-class vector chart shows all of the possible interval classes for a given set. As is the case with set theory, analysis has little to do which how the notes are spelled or which notes are used, but rather the space between the notes.

Normally, this chart is confined to a couple of rows with boxes. The notation system you describe, seems to be a way of taking the analysis out of the boxes and putting them in the composition.

That being said, I have never heard of this system before, so it has likely not caught-on in the theory world because it is by-and-large impractical for functional analysis. No-one needs to have mobile interval-class vector charts because we already use sets. Besides, using using such charts is only helpful if the composer originally used those charts to compose the piece.

Hope that helps.

  • Given the origins, we might assume this system was taught as Eastman during Hanson's tenure. I found a PDF of his book here.
    – NReilingh
    Feb 19, 2015 at 13:38
  • I think a composer's language is generally dynamic and in a sense cannot be measured exclusively by theoretical analysis. You can certainly design algorithms to read a composer's catalog, build up the structures necessary to use the same 'language', and then churn out more songs that sound the same. But is that really what the composer is doing? Composers are constantly listening to new things and trying to form new perspectives so that the linguistic structures they have used in the past aren't as prevalent in future compositions. To varying degrees of success. Feb 19, 2015 at 22:32
  • That is to say nothing about subconscious intent the composer may codify, only to realize they have done so at a later time. How many of us have written a catchy hook only to realize it's almost total plagiarism? Even famous artists do it all the time, unintentionally as often as not. Although, algorithmic approaches make no distinction between what was conscious or unconscious intent, as they all stem from more-or-less the same brain structures in some fashion. Feb 19, 2015 at 22:34
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    @DarrenRinger I think people may be interpreting my comments a little too extremely. My point is simply: functional analysis isn't helpful if the music isn't harmonically functional; it doesn't tell you anything about the piece. This is similar to Einstein's paraphrased thought about a fish being judged by its ability to climb a tree. Darren, I definitely agree with your thoughts, and this is the point of theory - to try and explain how the composer created the music, and more importantly, why it works. Feb 20, 2015 at 2:28

I must reiterate my passionate plea that this text not be considered a "method" nor a "system" [of composition]. It is, rather, a compendium of harmonic-melodic material. (Hanson, Harmonic Materials of Modern Music, xi)

Hanson intended his notation as a concise, descriptive language for sets of intervals and their generative possibilities. In the book laying out his ideas, Harmonic Materials of Modern Music (at achive.org), he explains that his intention is primarily to create a catalogue for composers, allowing them to easily explore the harmonic implications of a particular interval set.

I hope that this volume may serve the composer in much the same way that a dictionary or thesaurus serves the author. It is not possible to bring to the definition of musical sound the same exactness which one may expect in the definition of a word. It is possible to explain the derivation of a sonority, to analyze its component parts, and describe its position in the tonal cosmos. (viii, emphasis original)

It would not seem necessary to explain that this is not a "method" of composition, and yet in these days of systems it may be wise to emphasize it. (Ibid.)

The following describes Hanson's intentions for his system as a taxonomy, expanding on the categories admitted by common-practice theory.

[The common-practice era's] fundamental scales are ... the seven-tone major and minor scales; and the only chords which it admits are those consisting of superimposed thirds within these scales together with their "chromatic" alterations. The many other combinations of tones that occur in traditional music are accounted for as modifications of these chords by means of "non-harmonic" tones, and no further attempt is made to analyze or classify these combinations.... I have, therefore, attempted to analyze all of the possibilities of the twelve-tone scale as comprehensively and as thoroughly as traditional harmony has analyzed the much smaller number of chords it covers. This vast and bewildering mass of material is classified and thus reduced to comprehensible and logical order. (ix)

Hanson does acknowledge the usefulness of his ideas as analytical tools, but only insofar as they provide a descriptive language for identifying and classifying the intervalic content and interval-set relationships of a composition.

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