Harmony, as it evolved with the times, isn’t, the main common aspect of Bach and Hindemith’s music. However as Hindemith wrote a theory book about his harmony and I found a thesis by Simon Desbruslais1 talking about this theory book ‘Unterweisung im Tonsatz’ applied to several of his pieces. I thought it would be interesting to illustrate some of the ideas. Especially, because these structures and incorporations strengthen my claim of his tonal hierarchy, working with tonal centres and secondary tonal centres, ‘clusters’ based on scales, as well as the mathematical mind behind all the proportions of the tonal relations.

In 1937, Hindemith published his first edition of ‘Unterweisung im Tonsatz’ discussing his approach to harmony. This edition included an appendix, in which Hindemith listed compositions that best represented his discussed harmonic theory in practice. This appendix also included works written before the theory book was published, including his Sonate für Bratsche allein, Op.25 No.1, written in 1922.

Hindemith’s main goal with writing ‘Unterweisung im Tonsatz’ was attempting to find a way to order chromatic music, putting extended tonality in a theory. This main goal also connects us to his objective world view, as it shows, as William Thomson wrote: “… his belief that all music is the manifestation of rational and pre-ordained relationships which inhere in musical tone.”2.

The main directions Hindemith touched on in his Unterweisung were the order of melodic and of harmonic relationships, all based on the acoustic and scientifically observable phenomena of overtones. In the book, we can see how Hindemith transformed the traditional system of harmony and melody. His whole system is based on mathematics as he calculated frequencies of notes from formulas. This connects very well with Hindemith’s personal aspiration that music theory and science would eventually converge into some universal form.

When considering where Hindemith got his inspiration for his own personal harmonic theory, we have to take into account the wide range of repertoire he performed as a member of the Amar Quartet. This repertoire regularly included octatonic and whole tone collections as well as the modernist aesthetics of his contemporaries Stravinsky and Schoenberg. Next to this, although there’s no clear evidence of this, many sources mention that a clear influence from others such as Heinrich Schenker (1868-1935), Hermann von Helmholtz (1821-1894) and Boethius (middle ages) can be found, as they believe he built his ideas with the methodology and synthesis of their theories in mind.3 It is also believed that his reading of late 19^th-century psychology, by writers such as Carl Stumpf (1848-1936) and Theodor Lipps (1851-1914), lead Hindemith to some more general conclusions about musical perception.4

Hindemith’s major development of theories that came before, was his shift from using intervals as the units from which the whole tonal system was built up, rather than chords. This caused him to view chords as the product of the intervals which constitute them. In his theory, he is adamant about every chord having a clear root. This was established by his overtone series and order of tension of intervals. His overtone series in a scale row starting from C was: C – G – F – A – E – E flat – D – B – D flat – B flat – F sharp. His ranking of intervals from least to most tension was the following: octave – fifth – fourth – major third – minor sixth – minor third – major sixth – major second – minor seventh.– minor second – major seventh – tritone. This again shows his affinity with the perfect intervals, especially the fourth and the fifth, often laying at the basis of his music as he viewed them as neutral. These relationships between notes concerning intervals and overtones together with Hindemith’s strong desire for a root, connects us back to pitch centricity, discussed more in detail later in this chapter, as one note still has importance over all others in a chord, as the others according to the overtone row and interval order have a diminishing relationship to this main note.

This is especially valuable when discussing some prevalent harmonic ideas from Hindemith’s Unterweisung present in his Solo Sonata Op.25 No.1. As the sonata was written in a very short period of time, with the 4^th movement on March 7^th 1922, 2^nd and 3^rd movement on March 9^th, followed by the 1^st and 5^th movement on March 18^th, the day it premiered, it suggests that the writing might have been more intuitive, rather than including pre-planned structures and harmonies.

Now, I will discuss some specific harmonic ideas found inside each movement as read in Desbruslais’ thesis on Hindemith’s Unterweisung.5[5]

In the first movement, many octatonic pitch collections can be found, meaning the tones of an octatonic scale clustered in a few chords combined in a short excerpt, in this case often one to two bars. In this movement, which is the shortest of the sonata Hindemith included 9 different octatonic pitch collections out of the 29 existing. Which shows he liked to use many of the tones of the collections in a short amount of time. The first collection can already be found already in the first bar of the piece, showcasing all 8 notes in just 3 chords. This concise use showcases Hindemith’s intuitive composing style as this was one of the movements he composed during a relatively short train journey. Another important harmonic aspect that Hindemith used throughout his life that we can find in this movement are pitch hierarchies. In this first movement the F#  is the obvious tonal centre, due to its many reappearances but especially as almost every time it returns it’s preceded by a leading tone F natural. (see Fig. 1)

The F# is also supported by the C#, a fifth above the F#, which we can see as the secondary pitch centre or even dominant. This conclusion we can draw from the voice leading of the main motive, as seen above.

This concept of pitch hierarchies helps to evoke the sense of ‘traditional’ harmony.

In the second movement, as seen in the first two bars, there’s motivic treatment of the perfect fifth and the minor seventh. This seventh returns in combination with an octatonic scale. Throughout the rest of the movement we see mostly hexatonic tone collections in the outer sections and mainly octatonic tone collections in the middle section. (see Fig. 2)

The third movement is the one from which we can derive the least connections as it’s very unstable in its use of pitch collections. It can most generally be described as chromatic. There is also a structural significance of the pauses or fermatas in this movement connecting back to the pitch hierarchies. All pauses land either on our pitch centre E or our secondary pitch centre B, again located a fifth above the primary pitch centre.

Hindemith’s fourth movement was the first one written of this sonata. It also shows the most structurally and harmonically thought out result. We can see that the main areas in the structure are based on the four open strings of the viola. Many chromatic tendencies can be found in the interval patterns. Linear patterns, including large scale pitch structures, are present over the C pedal.

In the fifth and final movement, there are many references to all material from the previous movements. Contrary to Hindemith’s approach in some of the previous movements, the pitch collections here are formed over several bars within slightly longer phrases. Also again many octatonic pitch collections are included.

How do we view this in relation to Hindemith’s structure?

When looking at the general picture, the pitch hierarchy serves for our most interesting idea. Reflecting back on my structural idea that the final movement would be an epilogue, the pitch centre of the fourth movement is a C, as seen in our first and final notes of this movement. In the final movement, this C, which is also this movement’s pitch centre, reappears as its final note, making an even stronger case for this movement serving as an epilogue. The octatonic pitch collections could also be an optional connection between all the movement as our four different pitch centres could be part of the same octatonic scale, but as we only have four out of eight tones this can only be assumed.

The Solo Sonata connects to Hindemith’s Unterweisung, due to the early traces of the main ideas that can be discovered in there, such as clear voice leading and quartal pitch collections. Quartal and quintal harmony is, very simply put, the art of building chords by stacking fourths or fifths, not necessarily meaning it only consists of those intervals, just that those intervals predominate. The only prevalent element that didn’t stay with Hindemith to reach his theory book were his octatonic scales and octatonic pitch collections. Surprisingly, as Hindemith had a strong interest in music by Stravinsky and Bartok which often included this scale.

All this proving, that even though the system is much removed and advanced from traditional harmony in Bach’s time, the all-important hierarchy of harmony, the relation that some harmonies and notes serve to resolve and others create tension, remained. The mathematical mind of Hindemith is shown once again in the building of this system also connecting us back to Bach. So even in the not directly similar elements, the connection between our two masters still remains.

Fig. 1: F natural going to F# when reaching the main motive – F# and C# as bass voice in the main motive

Fig. 2: Minor sevenths in combination with the octatonic scale

Page out of Hindemith's Unterweisung dividing chords into different categories (depending on their inclusion of a tritone)

Analysis Hindemith's Solo Sonata Op.25 No.1 (as support to see our pitch centres)

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