I ended my lecture at the Jerusalem Academy of Music and Dance with the following statement: “Musical analysis is like a detective story: there is always a dead body and there is always a murderer. The interesting is, how he did it and why”.
I (the detective) started my investigation of the Hindemith sonata for double bass and piano, as one of my students played it and experienced intonation problems. This student was outstanding in all theory and ear training classes and he always played well in tune. I was determined to find out what the problem was and how I could help him. The solution to my students' problem was rather simple and it is already revealed in the title of that lecture: Tonality and A-tonality in the Sonata for Double Bass and Piano by Paul Hindemith. Once I realized the conflict between the tonal elements of the piece and the a-tonal elements, as I will demonstrate soon in the analysis of the second movement, I have asked my student to play equally tempered half steps and resist all temptations of tonal inflections. The problem was gone right away and I have used this recommendation for many other pieces as well.
Here, I have solved another murder case, and I know how he (Hindemith) did it, and you will know as well in just a minute but is this what really matters?
Last June one of my favorite actors passed away. I can still remember myself as a teenager watching Peter Falk as Lt. Columbo. The interesting thing about a good detective story is actually the detective's theory. Will he be able to prove his thesis? Isn't interpretation about using our intuition, finding the clues, putting them all together and creating a reality that works for us?
So here is my theory: It is all about Mahler's First Symphony.
I was thinking about Hindemith sitting down to write a Sonata for Double Bass. What would be the most famous bass piece he knows? What would he consider to be the “greatest hit” of double bass?
I would say that like many others at that time, he would think about the opening solo of the third movement in Mahler's first Symphony.
Let us go straight to my analysis of the second movement and see if my theory works.
This short movement is not only in the character of a Scherzo but a paraphrase of the traditional Scherzo and Trio form.
Measures 1-28 can be considered the first part of the Scherzo.
Throughout the "Scherzo" sections, the register of the Double Bass is about two octaves lower than the piano. This type of texture might raise questions about the various functions of the Double Bass in the music: is it just melody? Does it have a harmonic function?
Ex. 1- mm. 1-5 Bass Melody (sounds an octave lower)
mm. 1-5 Bass Melody (sounds an octave lower)
The Double Bass part in the example above is clearly in G and mostly diatonic.
The Piano part in those measures consists of a G pedal in its lowest voice, which gives a strong feeling of tonic.
The two upper voices move chromatically in parallel fourths, and the third voice, also moves chromatically, but in contrary motion to the upper two.
Example 2- mm. 1-5 (in the analysis staff the Double Bass is presented in concert pitch).
The chord that is circled in m. 1 is a G major/minor chord. On one hand, it represents the tonic, but on the other hand it is also a typical sonority of the octatonic scale.
Another element that has a strong presence in the first five measures is the Perfect Fourth: it has a motivic function as seen in the first two measures of the Double Bass melody, and it is also present at least once in each one of the accompanying chords.
In the first five measures, Hindemith hints at the compositional tools to be used in the entire movement:
Perfect fourth as melodic and harmonic interval
The second part of this section is characterized by the sound of octatonic scales.
In the analysis we see five octatonic scales simultaneously:
The Bass Part and the lower part of the Piano are in descending motion, while the three upper parts are ascending in parallel fourths, thus forming quartal chords.
In the Double Bass melody, the motive of the fourth from mm.1-2 continues. The first three measures return this time leading to an A major/minor chord which can be interpreted as a secondary dominant (v/v):
Example 4- mm.10-12
A Major/minor Chord is, of course, a characteristic octatonic sonority.
This leads us to the second half of the phrase: Here the Double Bass part moves chromatically. D is the only note missing in the chromatic sequence of twelve tones. The high D in m.15 (the highest note so far) completes the entire chromatic scale.
The B flat in m. 18 can be seen as part of an octatonic motion starting with the C in m.13, going to the B flat in m. 18 and then to the G sharp and G in m.21
The long pedal on F sharp (lower voice in the piano), and the pedals on E, give a strong feeling of a dominant 9th prolongation. The dominant 9th harmonic quality is supported by other notes from the chord that are always present: (D and A in m.13,
C and D in m.15,).
Example 5-mm. 13-21
The section ends with another statement of the opening phrase, this time staying in the tonic:
Example 6- mm. 22-26
The three measure transition contains the same elements that we have encountered so far: As can be seen in the diagram, the overall sonority is diatonic.
Example 7- mm. 26-28
The notes that have stems are also members of the octatonic scale.Each one of the three note sets is composed of a perfect fourth and a major second.The circled sets belong to both the diatonic and the octatonic scales.In the middle of the passage we see three chromatic notes in a row: B, B flat, A.The B and the B flat, positioned in the center of the phrase are the major/minor thirds of the tonic G.
Measures 29- 51 can be seen as the second part of the Scherzo:
Example 8- mm. 29-32
The Double Bass part moves in leaps of major sevenths which give a chromatic sound. As the example above shows, when we separate the part into voices we see that each voice moves in fourths.
The three note sets, each contains a minor second and a perfect fourth and can be seen as a variation of the ones we have seen in mm. 26-28 (Example 7)
The chromatic sound changes in the second half of the phrase into a very diatonic one with almost functional harmony:
Example 9- mm. 33-35
Next, as Hindemith has done in the previous section in mm. 22-26, he brings back the opening motive of the major seventh.
Example 10- mm.36-37
Example 11- mm.38-42
In this short transition we have a strong statement of the G major/minor quality:
The long pedal in the bass supplies the B natural. The structural notes of G minor are marked with stems up in the upper staff.
The fourth interval is elegantly camouflaged in the right hand of the piano, and shown in the top analysis line with solid slurs.
The piano left hand plays whole/half step ascending figurations of which the first notes form descending whole/half step figurations. (marked with stems down)
With the passage ending with a high F sharp and a G sharp in the second voice (A flat a measure earlier) we expect a resolution on G.
But instead we get the "repeat "of the second half of the Scherzo:
Example 12- mm. 43-46
This is very similar to what we have seen before and the second part of the phrase is identical to the first time:
Example 13- mm. 47-49
Example 14- mm. 50-51
In contrast to mm. 27-28, this passage here, has a very chromatic sonority created by the fact that this passage does not contain any fourths (interval class 5) which can also be a part of an octatonic set but they also give a more diatonic or tonal sonority.
Incidentally, the pitch D is the only note missing in this whole chromatic passage.
We might speculate that since G is a note common to both octatonic sets, the note D might ruin the balance of the two interwoven octatonic scales by emphasizing the dominant tonic relationship.
Measures 52-77 are the Trio:
Here is the first time that the piano goes lower than the Double Bass. The whole texture is dark and low and the voices are very close together.
The music implies a center of F minor:
We have a long F pedal, another quasi pedal on C in the piano parts, and the Double bass melody that always starts on A flat and goes to F.
Example 15-mm. 52-55
It is well worth mentioning that the set F,A flat, B flat, [0,3,5] is also a subset of the octatonic set. The notes D flat and D natural are also members of the same octatonic set: C sharp (=D flat), D, E, F, G, A flat, B flat, B natural.
Measures 56-59 again juxtapose diatonic and octatonic materials:
The dashed slurs represent the octatonic elements and in the lower staff of the analysis we see the structural notes from the Double Bass part forming a G major chord together with the A sharp we get the major/minor quality.
Example 16- mm. 56-59
The theme of the trio repeats now, in measure 60, and again we find the chromatic, diatonic and octatonic elements: it is important to mention that although it seems like we have moved to a new center of F (minor), which is implied in the Double Bass part, everything is somehow connected to our main center G: The chromatic motion goes from F sharp to G and back, and then from B flat to B (our major/minor third).The middle part moves around the D (dominant) and the Double Bass part moves from the low A flat (m. 60) through the octatonic set an entire octave to the higher A flat at the end of m.67.
We should now pay attention to the A flat, being the upper chromatic neighbor to G.
Example 17- mm.60-67
Measures 68-71 feature the interval of the fourth, along with chromatic and octatonic elements. Although the melody in the Double Bass is still in F minor,Musically it sounds like it is time to reach D, as the dominant of G:The E flat at the beginning of m. 68 in the bass part as well as the high D flat (the highest note in the whole section (the highest note in the movement is the D next to it in mm. 33 and 47) both hint at D. The upper voices also end in the same way E flat and D flat at the end of m.71.
Example 18- mm. 68-71
Example 19- Measures 72-75 close the F minor section and are supposed to lead us back to G:
The lower voice in the analysis shows a plagal cadence. The B flat also functions as part of the major/minor sonority in conjunction with the diatonic upper part . the middle part moves chromatically from D flat to B.
(The fourths are marked with dashed slurs ).
Again we have two bars of transition similar to mm.50-51:
Example 20- mm.76-77
The passage is a combination of two octatonic sets.Here as well, Hindemith avoids the D and arrives at the G from the upper chromatic neighbor, A flat. A short Da Capo ends the movement with a long ostinato on the G major/minor harmony.
Example 21- mm.77-end
As we have seen this movement is characterized by a variety of colors and sonorities which are created by the following means:
Registration: Hindemith juxtaposes the low register of the double bass with a very high register of the piano. This helps the bass to be heard as well as adds to the humoristic character of the Scherzo. In the Trio section (mm. 52-76) the double bass is placed within the range of the piano which creates a dense texture and a darker sonority.
Articulation: The short percussive sound of the piano chords in the beginning (mostly on the off beat), is contrasted by the legato runs that function as connecting passages between sections. (mm.26-29, 50-51, 76-77)
Pitch Organization: the thematic and motivic materials derive mainly from four sources:
Tonality- There is a strong feeling of a tonal center (G major and minor)
Chromaticism- Chromaticism may be regarded as a separate element as shown in the analysis above but may also be seen as a result of the following:
Octatonicism- This is the element that holds the entire piece together.It manifests in all the compositional aspects: melody, harmony, motivic structure (extensive use of sets [0,1, 3] and [0,2,3]) and counterpoint.
Perfect fourth interval- Historically the status of the perfect forth has changed from consonant to dissonant and back. In tonal music, the fourth also has a pivotal role between tonic and dominant. Interval class 5 may imply tonality through the tonic – dominant or tonic- subdominant relationships. As we have seen in Ex. 14, which is highly chromatic, Hindemith doesn't use any fourths there. On the other hand, it can also be used in ways that imply a-tonality such as forth chords. In this discussion of tonality and octatonicism it might be worth mentioning that forth chords do not belong to the octatonic set. And although they give us the feeling of a-tonality, they can be easily manipulated by inversion into [0, 2, 6], (suspended 2 chord), which has a highly tonal color. We might say than, that in this movement the forth assumes its historical role of being a pivotal interval.
Going back now to my Mahler theory: Isn't the interval of the forth strongly present in our Mahler example? It is there from the beginning timpani solo and throughout the movement. The forth is also the melody of the last two measures of our solo, D,A,D.
What about the strong octatonic element in the Hindemith sonata? Well, as we know the octatonic set is made of a sequence of alternating half steps and whole steps just like the beginning of our famous Mahler solo: D,E,F [0,2,3]
and later in the melody G.A.Bb.
Quod Erat Demonstrandum.
I think I made my point but just like Peter Falk as detective Columbo used to say while scratching his head, “I have one more little thing to show you…”