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I have been studying Paul Hindemith's "The Craft of Musical Composition, Book 1: Theory", trying to apply Hindemith's principles to my own harmonies and melodies.

As you may know, Hindemith's two fundamental concepts, Series 1 and 2, are derived from natural acoustic phenomena--the overtone series and combination tones. Series 1 shows the mutual relationship between any given tone and the fundamental tone; Series 2 shows the relative "strength" of intervals in decreasing order.

The appeal of Hindemith's Series 1 and Series 2 for me is their apparent universality--the implications of these relationships, set forth by nature, apply for melody as well as for harmony. And from a physical/acoustics/theoretical perspective, it makes sense to me.

However, I am having trouble using Hindemith's framework to improve my own compositions. The following are some aspects of Hindemith's method I am having trouble with:

  • knowing the proper way to determine the root of a melodic phrase.
    Should we be trying to identify best intervals? Triads? (I want to differentiate this from determining the tonal center, which he does provide guidance on)
  • knowing a "good" degree progression (the succession of chord-roots of a progression) from a "bad" degree progression. From my prior training, I tend to consider the basics of functional harmony--expanding the I chord through inversions, building progressions based on the circle of fifths, moving from tonic to subdominant to dominant and so on. Despite reading Hindemith's work, I am not exactly clear on what his prescription for crafting compelling harmonies is.
  • Identifying the "best" interval in a chord, for example: in the chord spelled, C E F A C# E G#, the root of the fifth A to E is apparently the root of the chord (according to Hindemith), although, is C# to G# not also a fifth?
  • Hindemith believes that roots are supports of larger harmonic structures, and as such they must exhibit tonal coherence if the chord-successions which take place above them are to be understandable; what would be a "coherent" succession of roots?

I will leave my specific questions there for the moment, though I could conjure up more.

Can you please provide an example of a melody being altered/improved based on Hindemith's theory? How about a chord progression being altered/improved based on Hindemith's theory? I am desperately trying to apply his logic to my compositions, but I just can't bridge the gap between theory and practice here. Thank you very much.

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    In my answer, I joined your second and fourth bullet points because they seemed to be asking a similar question: how to "know a good degree progression" and "what would be a coherent succession of roots." If I've misunderstood your intention with those bullet points, please let me know. – Richard Apr 12 at 2:04
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    Fascinating question, I‘ve read it several times and I will try to give an own answer like Richard. But first I am reading the biography of Hindemith - I‘ve never listened to his music, the only title I know by hearing is Mathis, der Mahler .... Then I‘ll read the 3 books about the craft of musical composition, analyze the fugues of the ludus and see whether I’ll find some application in my own writing. May be I‘ll be through in a year or two. :) Thank you for the input and inspiration! – Albrecht Hügli Apr 17 at 16:01
  • Thank you @AlbrechtHügli. I'm so happy to be able to ask questions in this community where people like yourself don't hesitate to engage with (what I find to be) challenging concepts. Best of luck! – 286642 Apr 17 at 20:52
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This is a little hard to answer, because Hindemith was (somewhat famously) idiosyncratic, inconsistent, and unclear with his own theories and analyses. But I’ll answer the best I can.

Throughout this answer, I’ll reference three texts in addition to Hindemith’s own Craft of Musical Composition (Unterweising im Tonsatz):

  • The Music of Paul Hindemith by David Neumeyer (1986)
  • The Music and Music Theory of Paul Hindemith by Simon Desbruslais (2018). This book is especially helpful for his Chapter 1, "An Unterweisung Critical Commentary,” where he outlines some of Hindemith’s errors in deriving his intervals, typically due to flubs in the overtone series. Early versions of Hindemith’s work had all of these errors, but some were corrected in future editions. At the very least, cross-referencing your copy with this chapter is a helpful exercise for someone really serious about Hindemith study. This book is from a notoriously expensive publisher, though, so hopefully you can access it through a university or another institution, either online or as a hard copy.
  • And Pieces of Tradition: An Analysis of Contemporary Tonal Music by Daniel Harrison (2016). The inclusion of this last one may be unclear, but Harrison’s approach to analyzing this repertoire is heavily influenced by Hindemith’s own approach (along with the approaches of a few others). In fact, a large part of his notational system is taken from Hindemith:

enter image description here

Now, on to the questions!

Determining the Root of a Melodic Phrase

Of prime importance for Hindemith in terms of melody was the step progression. In other words, the best melodic lines could all be reduced to a succession of different stepwise motions, as shown in this example from Harrison:

enter image description here

According to Neumeyer:

[I]t is the obvious long-range step-progressions that provide the framework for the melody and the basis for analytical decisions about it. The consistency of Hindemith’s practice, especially in the forties, greatly facilitates analysis of melodic activity. (71)

So I think the focus should be less on finding the root of a melodic phrase and more on embellishing clear step progressions in your melodies.

Also important to note is how closely Hindemith integrated his understanding of melody and harmony. So although Hindemith never made this explicit, it seems to follow from his theories that the tonal center of a given progression would also be the tonal center of the melody played along with it. This seems to be confirmed by Hindemith’s own analyses at the end of Craft I, because the only musical example where a tonality is not definitively given is that of the Dies irae chant: the only single-line (that is, unaccompanied by harmony) example in the book!

Determining a Coherent Succession of Roots

I’m afraid my answer here will be disappointing, because Hindemith quite simply never made this clear. The overarching logic is relatively straightforward, as Harrison states:

Hindemith in particular then reimagined the succession of dissonances and consonances as “fluctuation” among chord groups. Resolution, in this scheme, is motion from a more to a less dissonant chord group. (4)

But anything more specific was never explicitly stated, as evidenced by both Harrison:

[U]sing harmonic fluctuation as a pedagogical rather than an analytic tool...is all the more apparent in Hindemith’s well-known graphic analyses, found at the end of volume 1 of the Craft; his prosing of analytic results is meager, and no judgment calls are discussed, nor any find points of practice. (His analysis of Schoenberg’s op. 33a produced mostly group IV chords, for example, but this uniformity and lack of fluxion prompted no comment from him.) (57–58)

As well as Desbuslais:

[H]ow might Hindemith’s Unterweisung relate the student to free composition? Regrettably, the practical application of Series 1 and 2 is not uniformly clear in Hindemith’s writing. (42)

Determining the Best Interval of a Chord

This one, thankfully, is relatively easy to answer!

Neumeyer conveniently summarizes the rules regarding chord roots and how they were changed and strengthened between Craft I and Craft IV. But your conundrum is pretty succinctly cleared up:

if two intervals of the same type appear, the one whose lower note is nearer the bass has the chord root (56)

The logic should be clear, as well: since Hindemith’s theory is completely based on the harmonic series, the lower-sounding note will therefore have more command as the chord root.

In your example, then, you have two perfect fifths: A/E and C♯/G♯. The chordal root will be either A or C♯, depending on which note is closer to the bass.

Hindemith’s Revisions

There’s quite a lot to say here, but thankfully Debruslais devoted an entire chapter to this very issue: see his “Theory-based Revisions” that begins on p. 177, including a list of revised works on p. 178.

He divides his chapter into three case studies: examples with “minor revisions,” like that given in Example 5.11 below; examples with “similar basic outline, yet significant background revisions”; and a final example of “a fully reworked song, bearing no similarities to the original,” like that given in Example 5.17 below. Since this material takes up an entire chapter, I hope these two examples will be enough to whet your appetite.

enter image description here

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Hindemith was a brilliant mind, and he was the first theorist I ever really read back as an undergraduate; his discussion of the overtone series really spoke to me as a brass player, because that was my bread and butter! He's not only important as a composer and performer, but he has also rightfully gone down in history as the first individual to really make a systematized attempt at understanding highly chromatic tonal music. But despite this brilliance, his writings did unfortunately lack the clarity to transmit all of his ideas to future generations.

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  • I found Desbruslais as well ... you've saved me a lot of studying! – Aaron Apr 12 at 3:00
  • Thank you @richard for the thorough and thoughtful answer. I really appreciate it. – 286642 Apr 13 at 0:51
  • 250+50+10! ... and interesting titles by Neumeyer and Debruslais. I‘ve also found Debruslais thesis: The Identity, Application and Legacy of Paul Hindemith’s Theory of Music. This was one of the most interesting questions and best answers here in. – Albrecht Hügli Apr 17 at 16:06

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