# In what ways is a set's inversion different than its prime form?

It is clear to me that an inversion shares the vast majority of its features with the prime form.

They have the same interval vector, same spread-evenness, the same number of generic intervals, the same number of imperfections, the same sameness and coherence quotients (as discussed by Carey 2007), the same number of trichords (or any other n_chord for that matter), more or less the same roughness measure (Vassilakis 2010), and so on.

With that being said, the very distinction between the prime form of a set and its inversion implies that there is some "difference" between them.

What are some (or the most prominent) features that set apart an inversion from its prime form?

(looking for anything, including structural features, acoustical features, mathematical features, social features, whatever distinction you can think of, I am interested in).

Many thanks!

Carey, N. (2007). "Coherence and sameness in well-formed and pairwise well-formed scales." Journal of Mathematics and Music 1(2): 79-98.

Vassilakis, P. N. and R. A. Kendall (2010). Psychoacoustic and cognitive aspects of auditory roughness: definitions, models, and applications. Human Vision and Electronic Imaging XV, International Society for Optics and Photonics.

The difference between the minor triad and the major triad is a good case study. People are often surprised when they hear that they have the same interval vector, since they "sound so different".

The commonplace that minor and major triads, which have the same prime form, sound very different, can be put into question. I'm pretty sure that it takes quite a while for music students to reliably tell apart minor and major triads by ear. However, if we compare, say, the minor triad and a three-note cluster (comprised of C, C#, D in some inversion or transposition, Forte number 3-1), I am pretty certain that people with no musical background at all can learn to tell them apart pretty much immediately. So yes, the major and minor triad are different, but by far not as different as many of the other possible three-note chords/chordioids.

The reason that minor and major triads do not sound all that different is that they have the same interval vector, etc. etc.. Now to get to your question: what feature makes that they nonetheless sound different? It is a good question, and actually a bit mysterious to me. The best answer I can offer is in the way the overtone series of the three tones interact. On most instruments, one cannot hear very much of the overtone series. So it's important not to overemphasize the contribution of overtones to the overall sound of chords.

In trying to work this out I added the first five harmonic overtones to each of the three chord notes of a major and a minor triad. The resulting pitch collections are not inversions of each other and have a different interval vector, so this could be a good answer to your question.

The first five overtones are three different octaves, a fifth, and a major third. So on the level of the pitch class, we only need to add the fifth and major third to all three of the chord notes.

For C major (C, E, G), we get (C, D, E, G, Ab, B).

For C minor (C, Eb, G), we get (C, D, Eb, E, G, Bb, B)

As mentioned, these are not inversions of each other. (They do not even have the same cardinality.)

In other words, although the fundamentals of the major and minor triad have the same interval vector, as soon as you start to consider overtones, some divergences start to occur, which could explain the difference in sound.

As a sidenote: This train of thought would predict that major and minor chords made with pure sine waves would be harder to tell apart than the same chords made with more harmonic content. (Perhaps interesting to test this empirically?)

Finally, perhaps a simpler answer is that minor and major chords are typically embedded differently in harmonic progressions. And they also have different melodic implications. Perhaps that is the simpler explanation.

• Your answer goes exactly along the lines of what I was looking for. I like your experimental proposal as well, it would serve as a good measure to test your hypothesis. Thank you for your thoughtful response! Jan 3, 2021 at 23:11
• I just re-read your answer two years later and I am still in awe of its insight. Thanks again for taking the time to write! Mar 25, 2023 at 18:37

It depends on the set. For diatonic subsets the differences are essential, and underpin much of the Western tradition. e.g. (037) is both the minor triad and it’s inversion the major triad. Similarly (0258) as prime can be voiced as a half diminished seventh or a minor triad with added sixth, or as the ‘Tristan-chord’; all of which sound and function completely differently to each other, and differently again from the inversion, which we know as the dominant seventh. With more complex sets it’s the opposite problem. By ear you’d be hard pushed to tell if you were listening to 6-z4, 6-5 or 6-z6, however they were spaced, let alone whether they were in prime or inversion.

• I understand that the inversion "sounds different" than the prime form. But I am after tangible, even quantifiable features. Can you think of any? Oct 1, 2020 at 15:08
• In other words, "It sounds different" is absolutely right, but it's not really tangible. If it acts differently, can you think of a way to formalize the unique behavior of a given set and its inversion? Oct 1, 2020 at 15:15
• For instance, by definition(-ish), the prime form is more compact than its inversion. That's one quantifiable feature I can think of. Oct 1, 2020 at 15:16
• i read the articles you cited and can’t think of anything useful re application to sets - sorry
– user71850
Oct 1, 2020 at 20:09

Prime form is just a canonical way of specifying all the pc-sets that are T- and I-related. It's the most "compact" way to express a set: in normal form and transposed to start at 0.

For example, see "Set Class and Prime Form" on OpenMusicTheory.com.

In order for a pitch-class set to be transpositionally or inversionally related to some other pitch class set, they must share the same collection of intervals....All pitch-class sets that are transpositionally and inversionally related belong to the same set class, and they are represented by the same prime form. (italics original)

• I absolutely understand how the prime form and its inverse are defined. But that's not really what I'm asking. I'm looking for tangible, quantifiable features that set them apart. For instance, you can say that what sets a set apart from its transposition is frequency. or what sets a mode of a set from another is its relational context. What, then, sets the inverse apart from its prime form? What features? How can they be measured? Oct 1, 2020 at 15:07
• The kind of differences you're looking for don't apply to arbitrary sets. For example, the sets {0, 6}, {0, 4, 8}, {0, 3, 6, 9}, {0, 2, 4, 6, 8}, and {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, T, E} all share the quality that their prime forms and inversions are identical. Oct 2, 2020 at 7:02
• Right, and in that regard they are non-invertable because their inversion maps back to themselves. That's the case with the diatonic set as well. So of course my question does not apply to non-invertable sets. Oct 2, 2020 at 15:41
• @MichaelSeltenreich you must analyze this as sounding by intervals, not by pitches. So you may predict some inversion differences. Oct 10, 2020 at 23:12
• A sets inversion has exactly the same interval vector by definition. Oct 11, 2020 at 20:15