There is an essay on Bach's Inventions and their history, etc., by Dr. Yo Tomita. Each Invention uses a different key that was important in the Well Temperaments favored during Bach's time and therefore exhibits key color; unfortunately, key color disappears when the piano is tuned to today's Equal Temperament. The Inventions were initially written for his oldest son Wilhelm Friedemann Bach when he was nine years old, around 1720. They were subsequently updated and taught to other students. Today, students should be able to start learning the Inventions before age seven because we have figured out how to practice them.
explore how and why he composed them. Bach used advanced musical concepts in harmony, counterpoint, etc., that music theoreticians are still debating to this day, while Hanon, Czerny, etc., wrote "lesson music" just for their finger training value. Below, we examine the Inventions at the simplest structural level, which leads to an astounding
Discovery: each Invention is based on a small number of parallel sets (PSs)! Now,
you might say, "Any composition can be decomposed into PSs, so what's new?" The new element is that each Invention is based only on one or two chosen PSs, starting with the simplest and introduced in order of increasing complexity with increasing Invention number. Before I made this discovery, I had catalogued the PSs according to complexity [(10) Parallel Sets Catalogue], and Bach basically used the same sequence of complexity! To demonstrate this, we list these PSs below for each Invention. In order to concentrate on simple PSs, Bach avoids the use of thirds and more complex intervals (in one hand); thus he wanted his students to master simple PSs before the more complex ones.
I use the term "linear" to denote PSs in which the fingers play sequentially (e.g., 12345), and "alternating" when alternate fingers play (132435). PSs are joined to form "motifs" in these Inventions. Because the motifs were created using specific PSs they were not chosen because of their musical content, but were chosen for their pedagogical value and the music was then added by the genius of Bach. Thus only Bach could have achieved such a feat; which explains why Hanon failed. Another reason why Hanon failed was that he did not know efficient practice methods while Bach did.
Only one representative member of each PS is listed below for each Invention; Bach used them in many variations, such as reversed, inverted, etc., a composition tool called symmetry operations [(67) Mozart's Formula, Beethoven and Group Theory].
List of the PSs in each Invention (listed for the RH; LH is similar):
#1: 1234 and 4231 (linear followed by alternating); this was a mistake because the first Invention should deal only with the simplest (linear) sets. Accordingly, in a later modification of this Invention, Bach replaced the 4231 with two linear sets, 432,321. This modification provides the strongest evidence for my thesis that Bach used PSs as the structural units in the Inventions. This modification teaches how to cope with different rhythms without introducing new PSs. We now have an explanation of why there are two Invention #1s!
#2: Linear sets as in #1, but with a wider variety of conjunctions. An added complexity is that the same motif, appearing at different times, requires different fingerings.
#3: 324 and 321 (alternating followed by linear). A short alternating set is introduced.
#4: 12345 and 54321 with an unusual conjunction. These longer linear sets and the unusual conjunctions increase the difficulty.
#5: 4534231; full blown alternating sets.
#6: 545, 434, 323, etc., the simplest example of the most basic 2-note PSs joined by one conjunction; these are difficult when the weak fingers are involved. Although they are simple, they are an extremely important basic technical element, and alternating them between the two hands is a great way to learn how to control them (using one hand to teach the other). Arpegic PSs (531) are introduced.
#7: 543231; this is a combination of #3 and #4 and is therefore more complex than either one.
#8: 14321 and the first introduction of "Alberti" type combination 2434. Here, the progression in difficulty is created by the fact that the initial 14 is only one or two semitones which makes it difficult for combinations involving the weaker fingers. It is amazing how Bach not only knew all the weak finger combinations, but was able to weave them into real music. Moreover, he created situations in which we had to use the difficult fingerings he wanted us to practice.
#9: The lessons here are similar to those in #2 (linear sets), but are more difficult. The PSs are strung together into longer motifs and played with more difficult ornaments.
#10: This piece consists almost entirely of arpegic sets. Because arpegic sets involve larger distances between notes, they represent another progression in difficulty, especially for youngsters. Bach was obviously aware that arps are harder than scales.
#11: Similar to #2 and #9; difficulty is increased by making the motifs longer than for the preceding pieces. In all the preceding pieces, there is only a short motif followed by a counterpoint section which makes it easier to concentrate on the PSs.
#12: This one combines linear and arpegic sets, and is played faster than previous pieces.
#13: Arpegic sets, played faster than #10.
#14: 12321, 43234; a more difficult version of #3 (5 notes instead of 3, and faster). #15: 3431, 4541, difficult combinations involving finger 4. These finger
combinations become especially difficult to play when many of them are strung together. The above list shows that:
(i) There is a systematic introduction of increasingly complex PSs.
(ii) There is a progressive increase in difficulty, with emphasis on developing the weaker fingers.
(iii) The "motifs" are carefully chosen PSs and conjunctions, selected for their technical value.
The fact that motifs, chosen for their technical usefulness, can be used to create some of the greatest music ever composed is intriguing. This is nothing new to composers: that famous melody in Beethoven's 9th symphony is just a play on the
simplest structure, the major scale. To the average music aficionado who has fallen in love with Bach's music, these motifs take on special significance with seemingly deep
musical value because of the familiarity created by repeated listening. It is not the motifs themselves, but how they are used in the composition that produces the magic. If you look only at the motifs, there is hardly any difference between Hanon and Bach, yet no one would consider the Hanon exercises as music. The music consists of the motifs and the counterpoint section, so named because it acts as the counterpoint to what is being played by the other hand. The counterpoint serves many purposes, such as creating music and teaching a myriad of technical lessons.
Thus music is created by some "logical" sequence of notes that is recognized by the brain; we shall explore this idea more fully in the [(68) Theory, Solfege] section.
Bach's lessons: Bach was not a trained analyst, teacher, or writer, in the sense that
he left no instruction manuals on how to teach. His most proficient language was music. Therefore the best he could do was to embed his lessons into his compositions: the human language was woefully inadequate (how would one describe key color??). The structures of his Inventions suggest that he was aware of most of the major principles of this book (PSs, segmental and HS practice, quiet hands, HS memorizing, etc.) because, without them, the Inventions can be devilishly difficult to memorize and to play. Of course, you can say that about practically any piece of music, but in Bach's case, it is
extreme. Thus it is important to "read" those embedded lessons: they indicate that he may
have been the greatest piano teacher that ever lived. He also tried to exhaustively cover all scales, key colors, etc., and may even have left us instructions on how to tune the chromatic scale (Larips.com, , Bach's temperament).
(47) Mozart's Rondo, in Sonata No. 11 in A major, K331 (300i)
The term sonata has been applied to many types of music and does not have a unique definition because its definition evolved with time. In the earliest times, it meant music or song. Until Mozart's time, "sonata" meant instrumental music with one to four parts: Sonata, Minuet, Trio, and Rondo. A sonatina is a small sonata. There is also a sonata allegro, initially developed as the first part of a sonata, symphony, concerto, etc.; it generally contained an exposition, a development, and a recapitulation. The sonata
allegro is important historically because this basic structure was gradually incorporated into most compositions. Curiously, no part of this sonata (K331) is in sonata allegro format (Hinson, P. 552); instead, the Sonata part is replaced by a theme and 6 variations. Variation V is Adagio and should not be rushed. Then comes a break in the form of a minuet-trio, a form of dance. The minuet originated as a French court dance with 3 beats and was the predecessor of the waltz. The waltz format also includes mazurkas that originated as Polish dances, which is why Chopin composed so many mazurkas. The (Viennese) waltzes have the accent on the first beat; the Mazurkas have the accent on the second or third beat. Waltzes started independently in Germany as a slower dance with three strong beats; it then evolved into the popular dances that we now refer to as
three voices in this trio, a violin, viola, and cello. Trios gradually went extinct as quartets gained popularity. Both the minuet and trio in this sonata have the time signature 3/4. Thus every first beat carries the accent; knowing that it is in a dance (waltz) format makes it easier to play the minuet-trio correctly. The trio should have a totally different air from the minuet (a convention in Mozart's time); this change in air gives the transition a refreshing feel. Don't forget the "Menuetto D. C." (Da Capo, which means return to the beginning) at the end of the Trio; thus you play minuet-trio-minuet. The final section is the Rondo, which has the general structure ABACADA. . . , that makes good use of a catchy melody, A. This entire sonata is sometimes referred to as a variation on a single theme, which is probably wrong, although the Rondo resembles Variation III, and the Trio resembles Variation IV.
We now discuss how to practice the Rondo. It has the structure
(BB')A(CC')A(BB')A'-Coda. The time signature is a lively cut time; can you figure out the key of BB'? The rest of this Rondo is in A, as is the formal key of this sonata. This Rondo starts with the "B" structure, constructed from a short motif of only 5 notes, repeated twice with a rest between them in bars 1-3; it is repeated without the rest in bar 4. He uses the same motif as a conjunction between these repetitions at the end of bar 3. It is then repeated at half speed in bars 7 and 8 and the last 2 bars provide the ending. Bar 9 is the same as bar 8 except that the last note is lowered instead of raised; this abrupt change in the repeating pattern is one way to signal an ending. The half speed units are disguised by adding two grace notes in the beginning, so that, when the entire B is played at speed, we only hear the melody without realizing that the whole section is created using only one 5-note motif! He repeated the same motif 8 times in 8 bars to compose one of his famous melodies. He multiplied a one second motif by 16 times, and because the BB' uses B twice, the total multiplication factor is 32 times! The entire sonata consists of these repeated sections that are 8 to 10 bars long. There are several sections that are 16 or 32 bars long, but these are multiples of the basic 8 bar sections. This explains how he composed such a long sonata using a minimum of basic materials — he took advantage of the most basic property of music - repetitions. More details are presented in the section on structural micro-analysis of repetitions in (67) Mozart's Formula, Beethoven and Group Theory. This type of analysis is helpful for memorization and mental play – after all, mental play is how he composed them!
The technically challenging parts are (1) the fast RH trill of bar 25, (2) the fast RH runs of bar 36-60, (3) the fast broken RH octaves of bars 97-104, and (4) the fast LH Alberti bass of bars 119-125. Examine them and start practicing the most difficult one first. The broken chords in the LH (bar 28, etc., and in the Coda) should be played very fast, like grace notes. Match the first note of the LH broken chords with the RH octaves.
(1) For fast trills, go to (35) Trills and Tremolos. Don't try to learn bar 25, HT by slowing it down. Make sure that the HS work is completely done using bars 25 and 26 as a practice segment, then combine the 2 hands at speed. Always learn to combine things
HT at speed (or close to it) first, and use slower speeds only as a last resort because if you succeed, you will save lots and lots of time and avoid forming bad habits. Advanced pianists never have to combine hands by slowing down. In fact, slowing down HT
accurately is more difficult than combining the hands at final speed; it makes no sense for beginners to use the more difficult method.
(2) Use PSs to get up to speed quickly for bars 36-60, being extra careful to get the fingering right. To prevent missing notes, use staccato practice.
(3) The broken octave sequence of bars 97-104 is not just a series of broken octaves, but two melodies, an octave and a half-step apart, chasing each other.
(4) Use PSs for the fast Alberti [(9) Parallel Sets (PSs), Conjunctions, Cycling] of bars 119-125.
How do you make music that sounds like Mozart? Simply follow the expression markings on the music. For Mozart, each marking has a precise meaning, and if you follow every one of them, including the time signature, etc., the music becomes an intimate, intricate conversation.
Let's examine the first 8 bars in detail.
RH: The first four notes (bar 1) are played legato followed by an eighth note played
light staccato. The staccato creates an anticipation that something is coming and the rest heightens the anticipation. This construct is repeated; then the 4-note motif is repeated at double speed (2 motifs per bar) in bar 4, and climaxes at the C6, bar 5, played legato
firmly. The two following staccato notes form the conjunction to the ending bars 5-8,
played staccato, sustaining the level of excitement. The series of falling notes in bars 8-9 brings this section to a close, like a vehicle slowing down to a stop. It is Allegretto, and therefore should not be played too fast. Meanwhile:
LH: The LH accompaniment provides a rigid skeleton; without it, the whole 9 bars
would flop around aimlessly. The clever placement of the ties (between the 1st and 2nd notes of bar 2, etc.) not only emphasizes the cut time nature of each bar, but brings out the rhythmic idea within this exposition; it is like a fox trot dance – slow, slow, quick- quick-slow in bars 2-5, repeated in bars 6-9. Because every note must be staccato in bars 6-8, the only way to bring out the rhythm is to accent the first note of each bar. After all the preceding staccatos, both notes of bar 9 (both hands) are legato and slightly softer in order to provide an ending, and both hands lift at the same instant.
The strategic placing of legato, staccato, ties, accents, etc., is the key to playing Mozart, while accurately maintaining the rhythm. A typical example is the last 3 chords at the end — it is an unbelievably simple device (a hallmark of Mozart): the first chord is a staccato and the remaining two are legato. Play it any other way, and the ending becomes a flop. Therefore, these last 3 chords should not be pedaled although some scores
(Schirmer) have pedal markings on them — no wonder students have trouble bringing out the Mozart in the music. Better pianists tend to play this Rondo without pedal.
produce music that is uniquely Mozart.
After you are comfortable HT without the pedal, add the pedal, which is optional. In the section starting at bar 27, the combination of broken LH chords, RH octaves, and pedal (as indicated in most editions) creates a sense of grandeur that is representative of how Mozart created grandeur from the simplest constructs. The pedal is in principle inappropriate here because most octaves must be played staccato with strategic legato octaves needed for the expression. Think of the oxymoron of a staccato octave that is pedaled! Mozart correctly indicated the staccato but wrote no pedal markings (pedals weren't invented yet). Less pedal is always looked upon by the pianist community as indicating superior technique; if you have the technique, the music can come out better without it because you have more control.
The rest of this sonata (preceding the Rondo) is beautiful Mozart and a lot of fun to play. Because it is so long, I did not memorize it, so that I could use it to practice (sight) reading, because it is relatively easy.