Rhythmicana 1
(jump to Rhythmicana I - rhythm, Rhythmicana I - clusters, Rhythmicana II - part 1, Rhythmicana II - part 2, Rhythmicana III)
Rhythmicana and New Musical Resources
Rhythmicana is a three-movement solo piano work written in 1938. Composed while Cowell was in prison, the work was dedicated to and first performed by Johanna Beyer (1888-1944), a German composer and pianist living in New York City.
Other than a few dynamic and tempi markings, Cowell gives very little interpretive instruction in the score. Perhaps the best resource for performing this work is Cowell’s book, New Musical Resources. New Musical Resources came about at the suggestion of Charles Seeger, one of Cowell's only formal instructors in music. Seeger urged Cowell to write a theoretical discussion of his "tone clusters." First published in 1930, Cowell's book explains the use of the overtone series throughout musical history and suggests ways it can apply to modern composition. In part II, Rhythm, Cowell suggests that composers can apply the overtone series not only to melodic and harmonic ideas, but to rhythmic ideas as well.
In Ex. 1 below, we see a section of the partial series for a pitch C with the relative period of vibration for each partial. For those not familiar with the phrase “relative period of vibration,” let me explain. Every key on the piano produces a different pitch. Each pitch sounds higher or lower depending on the frequency of the pitch, or the number of times the pitch’s wavelength goes by in a second. As the wavelength becomes shorter, the pitch becomes higher. The picture below, in Ex. 2 shows two separate waveforms. The “period” of a waveform is the length from any one peak (top of a wave) to the next peak. With a higher pitch, the period is smaller than in a lower pitch.
Frequency is the number of times the wavelength or period is perceived in one second, measured in hertz (abbreviated Hz). The pitch A, above middle C, is measured at 440 Hz. This means that in every second, the wave oscillates 440 times. Further, the frequency of the pitch A, an octave higher than A440, is 880 Hz and the frequency of the pitch A at an octave lower than A440 is 220 Hz.
So far we have discussed the difference in frequency between different pitches produced on different keys. When talking about harmonics, we concern ourselves with pitches produced or perceived on only one key, that of the fundamental pitch. Again, let's say that we have the pitch of A, whose frequency is measured at 440 Hz. The easiest perceivable harmonic is the next highest C, one octave above, which has a frequency of 880 Hz, or double the fundamental. The next most easily perceived pitch is the G an octave and a fifth above middle C. The frequency of this pitch is three times the frequency of the fundamental, or 1320 Hz. 2
Now we can look back at Cowell’s picture in Ex. 1. Cowell states, "...whereas at no instant within the second of time do the vibrations coincide, at the end of that period all the vibrations coincide." In other words, in a given length of time, such as one second, the fundamental pitch oscillates 16 times. The octave above that, the first harmonic, oscillates double that or 32 times. The fifth above that, the second harmonic, oscillates triple the fundamental, or 48 times, etc… At the end of the period, the waveforms again coincide.
Ex. 1) New Musical Resources, p. 47
Ex. 2) Waveform examples: The top wave represents a higher sounding pitch and the bottom wave a lower sounding pitch. The brackets represent one period within the waveform.
As suggested in his graph on p. 48, Cowell proposes using the same idea with complex rhythms. Assuming one whole note per measure or period, Cowell divides that measure into three, four and five beats all meeting up once a measure just as all the vibrations of the partials of C coincide at the end of the period. 3
Ex. 3) New Musical Resources, p. 48
Cowell also suggests applying the overtone series to meter. Just as the vibrations of a partial series coincide periodically, the downbeats of varying meters can arrive intermittently. We will see this again in our discussion of Rhythmicana III.
Specifically, he finds the usage of tuplets 5 tiresome saying, "...instead of calling a note occupying a third the time of a whole note a "half-note triplet," why not refer to it as a third-note?" 6 The chart below shows Cowell's new suggestions:
Ex. 4) New Musical Resources, p. 58 7
Cowell is not blind to the challenge of learning “fifth notes” or “thirteenth notes.” He states, "An argument against the development of more diversified rhythms might be their difficulty of performance. It is true that the average performer finds cross-rhythms hard to play accurately; but how much time does the average performer spend in practising them?...Surely they are as well worth learning as the scales, which students sometimes practise hours a day for years. By experiment we have observed that such rhythms as five against six...can be quite accurately performed by the devotion of about fifteen minutes a day for about six months." 8
Particularly with the smaller divisions with which we are already familiar, the learning of new note durations would be possible for performers with concentrated practice. If we look at Cowell's “EXAMPLE 10” (Ex. 5 below), one could first translate this into our standard tuplet usage (Ex. 6 below). With time, why couldn't a performer internalize this idea of a "third" note?
Ex. 5) New Musical Resources, p. 59, with alternating “third” and quarter notes
Ex. 6) Cowell’s example rewritten as triplets (with a duple left hand part as reference)
Despite his suggestions, few of Cowell's pieces made use of this new notation. The solo piano piece Fabric is one of the few examples. Whether by Cowell's hand or that of the publishers, there is some concession on the part of the rhythm. The music uses common tuplet groups of 3, 5, 7 or 9. In the score, each group is first given the expected "7", "5" etc... marking above or below. There is no discussion of the meaning of the noteheads. Unless a person were to seek out New Musical Resources, it would seem that the new noteheads signify very little.
Ex. 7) Fabric, mm. 1-2
Although we also see Cowell's rhythmic ideas in Rhythmicana, we see none of the new noteheads. Perhaps when he wrote Rhythmicana in 1938, Cowell felt that performers were not ready or willing to tackle this new notation.
Rhythmicana I
Rhythmicana I - rhythm
The first movement is labeled Impetuously with a metronome marking of 
. The opening dynamic is forte and the LH has two legato markings per measure. Beyond that, the performer is given little other instruction. The LH rhythm is simple enough, a collection of quintuplet eighths followed by four standard eighths.
Ex. 8) Rhythmicana I, m. 1
The RH however, often contains contrasting divisions of the beat. Similar to the discussion in New Musical Resources, the asymmetrical subdivisions are often complete within each measure. In other words, both hands come together at the downbeat of every measure, or in a few circumstances, every other measure, assuring that the performer will have the correct rhythm at least once a measure!
Although the mathematical element of the rhythms may seem daunting at first, many of the difficult subdivisions are repeated. The traditional approach to solving subdivisions, that of finding a common denominator (Ex. 9), will not work well for many of the rhythms in this piece. Certainly one can use the handwritten method for the three against four in m. 2, but in mm. 11-12, where we see a quarter note nontuplet (9) against sets of quintuplet and four standard eighths, a calculator makes figuring out the subdivisions easier. This will be explained below.
Ex. 9) 3 against 4 handwritten method
Let us try applying the calculator approach to the rhythm in m. 2. Performers must decide if they want to fit the RH in between the LH rhythms or vice versa. Why? We have to fit one rhythm into the other, dividing either three into four or four into three, and each version will give us different arithmetical results.
I find it easiest to fit the RH into the LH as the LH has the most consistent rhythm throughout the movement. In beat 2, we have a quarter note triplet in the RH and four eighths in the LH. When we divide four into three, we get 1.33 (rounded to the nearest hundredth). Therefore, a triplet eighth note in the RH will occur after every 1.33 notes of the LH quintuplet. Keep in mind that we would not count the first note, as the hands line up on the downbeats and our “1” is actually the second LH note. If it is easier, one could put “2.33” and “3.66” underneath the RH chords instead of “1.33” and “2.66.”
Ex. 10) Rhythmicana I – m. 2, beat 2
Let us try a more difficult rhythm; one that would benefit more from the calculator. In m. 3, we have a half note triplet rhythm in the RH over a quintuplet and a set of four standard eighth notes in the LH.
Ex. 11) Rhythmicana I, m. 3
In order to determine the subdivisions, I find it easiest to think of this as two separate measures. Keeping the RH rhythm constant, I create two new measures: one consisting of only quintuplets in the LH and another consisting of only standard eighths in the LH. Keep in mind that after we calculate the subdivisions, we will only use half the calculations, the first half of the LH quintuplet calculations for the first half of m. 3 and the second half of the LH standard eighth note calculations for the second half of m. 3.
Ex. 12)
As the smallest subdivision in the RH is an eighth note, I think of the half-note triplet in terms of its eighth note equivalent: twelve triplet eighth notes. In Ex. 13 below, I replace the half-note triplet rhythms from Ex. 12, with sets of twelve triplet eighth notes.
Ex. 13)
Now we can calculate the subdivisions. In the first measure, two sets of eighth note quintuplets equals ten quintuplet eighth notes. Ten divided by twelve (triplet eighth notes) is 0.83 (rounding to the nearest hundredth). Therefore, one RH eighth note will occur after every 0.83 LH notes and we can now fit the RH in between the LH for beats one and two (Ex. 15).
Next we try the second half of the measure. We again have twelve eighths in the RH, but eight standard eighths in the LH. Eight divided by twelve is 0.67, giving us a RH note every 0.67 LH notes (Ex. 16). Finally, we have the subdivisions needed to perform this measure. As mentioned above, one need not use “2.5” and “3.3.” You may use whatever number you want in the “TEN” placement (i.e., 1.5 and 2.3, 0.5 and 1.3, etc…). It is the decimal that is most vital.
Of course, this may seem like a tedious task, but it provides the performer with a precise view of the rhythm. I find it much easier than "guessing" the correct rhythm or using the handwritten method. Nevertheless, let’s try the handwritten method to compare.
Ex. 14) Arithmetical divisions for eighth notes in m. 3 (rounded to the nearest tenth or hundredth). The numbers in bold represent where the RH notes fit in between the LH.
10/12 0.83 1.67 2.5 3.33 4.17 5 5.83 6.6 7.5 8.33 9.17 10
8/12 0.67 1.33 2 2.67 3.33 4 4.67 5.33 6 6.67 7.33 8
Ex. 15) Rhythmicana I, m. 3 – LH quintuplet subdivision
Ex. 16) Rhythmicana I, m. 3 – LH eighth note subdivision
The handwritten method requires that one find the lowest common denominator between two rhythms, typically a rhythm in the RH and a rhythm in the LH. Since we have three different rhythms in m. 3, an eighth-note triplet (in the RH) and a set of quintuplet eighth notes and a set of four standard eighth notes (in the LH), we should again divide the measure in half, creating two separate measures. However, as will be explained below, we will not have to double the LH rhythms.
Therefore we have:
Ex. 17) (lightened notes will not be needed)
Now we break down the RH to its lowest equivalent, that of twelve eighth notes.
Ex. 18)
For the first measure, we have a group of twelve over a group of ten. The lowest common denominator (more precisely, the “least common multiple”) between ten and twelve is sixty. Since the highest common factor is two, and we only need half of the measure, we can split this in half. Therefore, we have six over five, giving us a lowest common denominator of thirty. 9 This means that we first need to write out thirty noteheads evenly divided.
Ex. 19)
Then, we place the rhythms within the thirty noteheads. For the RH, a note will be placed after every five noteheads and the LH will come after every six noteheads.
Ex. 20) Rhythmicana I, m. 3, 1st half: score and handwritten method
The second half of the measure is easier. In fact, it is much easier than using the calculator. We have twelve over eight, giving us a lowest common denominator of twenty-four. The highest common factor is four, but we can only reduce this in half, to six over four, because we need at least four notes in the LH.
Ex. 21) Rhythmicana I, m. 3, 2nd half: score and handwritten method
(The beats in this edition do not line up visually. The RH quarter note chords should occur later in the measure.)
Someone could argue, why bother with the calculator method? When it comes to rhythm, performers do not typically think in decimal points and if half of the measure is simply a rhythm of six over four, why bother? I can see two major benefits to using the calculator method.
- Although some of the rhythms in this movement are easier with the handwritten method, there are not ALL easier with the handwritten method. For example, the rhythms in mm. 11-12, a two-measure quarter note nontuplet in the RH, two sets of eighth note quintuplets and four standard eights in the LH would be excruciating with the handwritten method.
Ex. 22) Rhythmicana I, mm. 11-12
- The handwritten method is difficult to transfer to the score, particularly with the more complex subdivisions. Do you really want to write thirty noteheads into the music? Decimals, however, are easily transferable onto a score (as seen again below).
Ex. 23) Rhythmicana I, m. 3
Rhythmicana I - clusters
The clusters are not easy. Unfortunately, there is not a fingering that allows for a pure legato of the LH passages. I use a combination of techniques.
First, I try to connect the clusters in groups of two making sure to give a slight accent to the downbeat of the groups of five and four so that the divisions are clear.
Ex. 24) Rhythmicana I, m. 1 – fingering for groups of two
Second, I use the 'chicken-pecking' sort of method, hopping around to hit each cluster. In other words, I do not try to so much to connect any cluster to another, but rather, keeping my hand in the same sort of position (i.e. like the beak of a chicken!), I peck up and down the keys. With both methods, the pedal is a necessary ally.
Ex. 25) Rhythmicana I, m. 1 – fingering for “chicken-pecking”
Rhythmicana II
This movement, labeled Andante, also leaves many decisions up to the performer. In the beginning, it seems that the melody, denoted in the alto part, is the primary feature; it is the only line in the opening with a clear instruction, Alto legato espressivo. However, the complex rhythms will occupy so much of the performer's energies, that perhaps the asymmetrical subdivisions should be the primary melodic lines. I chose the following melodic hierarchy: alto, bass, soprano and in m. 17, when Cowell marks the soprano line Soprano ben marcato, the top line is brought out.
What about the dynamics? Again we get very little instruction. There is a mpp with the Alto legato espressivo, a marking meant for the alto part. Translated as mezzo pianissimo, this is not the most common of dynamic markings,. I think we can assume that mpp should be louder than pp just as mp is louder than p. If nothing else, the mpp must be louder than the ppp at the top of the staff, whether or not it is louder than a pp.
Ex. 26) Rhythmicana II, mm. 1-4
Ex. 27) Rhythmicana II, mm. 25-28
The music in the last three measures, mm. 40-42, does not appear in the original A section, and there seem to be missing stems in m. 41. By looking at the stemming in every other measure, the rhythms of the soprano and alto lines each add up to a whole note. In m. 41, the soprano line equals only 7/11 of a whole note and we still need 4/11 of a whole note. By looking at similar passages at the end of the B section, (m. 23), one can assume that the F-sharp and the A-natural should also be a part of the soprano line. By changing this, the soprano rhythm in m. 41 becomes the reverse of the soprano line rhythm in m. 23.
Ex. 28) Rhythmicana II, m. 23 – The soprano and alto rhythms each add up to a whole note.
Ex. 29) Rhythmicana II, m. 41 – The soprano line is missing 4/11 of a whole note. I suggest making the F-sharp (third eighth note in the RH) and the A-flat (fifth note in the RH) quarter notes in the soprano line.
(Rhythmicana II - part 2)
The second movement also demonstrates some of Cowell’s rhythmic ideas from New Musical Resources. Again we have rhythms that meet up on the downbeats of each measure although we have larger subdivisions between the hands. In the right hand, we have groups of thirteen or eleven and in the left hand, groups between four to nine notes in length. There are a variety of ways to tackle these rhythms and I suggest trying a combination of several.
1) Begin practicing with a mathematical breakdown of the divisions. The handwritten method, using a lowest common denominator, is pointless with these matchings. The smallest lowest common denominator would be 44 (11 over 4) and the highest would be 117 (13 over 9). (For more information on the handwritten method, see Rhythmicana I discussion above.)
Instead, you will need a calculator and a little patience. Below is a chart that lists the breakdown of each grouping (13 against 5, 13 against 7, etc...). You can use this as recommended in the rhythmic discussion of the 1st movement. Also see Ex. 31 below the chart.
Ex. 30) Chart of RH/LH groupings, RH divided into LH
RH: 13 13 13 13
LH: 5 7 8 9
______ ______ ______ ______
0.384615 0.5384615 0.615384 0.692307
0.38 0.54 0.62 0.69
0.77 1.08 1.23 1.38
1.15 1.62 1.85 2.08
1.54 2.15 2.46 2.77
1.92 2.69 3.07 3.46
2.31 3.23 3.69 4.15
2.69 3.77 4.31 4.85
3.08 4.30 4.92 5.54
3.46 4.85 5.53 6.23
3.85 5.38 6.15 6.92
4.23 5.92 6.77 7.62
4.62 6.46 7.38 8.31
RH: 11 11 11 11 11
LH: 4 5 6 8 9
__ __ __ __ __
0.36 0.45 0.54 0.72 0.81
0.36 0.45 0.54 0.72 0.81
0.72 0.90 1.09 1.45 1.63
1.09 1.36 1.63 2.18 2.45
1.45 1.81 2.18 2.91 3.27
1.81 2.27 2.72 3.63 4.09
2.18 2.72 3.27 4.36 4.90
2.54 3.18 3.81 5.09 5.72
2.90 3.63 4.36 5.81 6.54
3.27 4.09 4.90 6.55 7.36
3.63 4.54 5.45 7.27 8.18
Ex. 31) 13 against 7
In this example, I have added “1” to each calculation to more easily visualize where the notes lie in the measure.
Original calculation of 7/13:
0.54 1.08 1.62 2.15 2.69 etc…
New “+1” version:
1.54 2.08 2.62 3.15 3.69 etc…
Practice excerpts of similar combinations. In other words, practice all of the 13 against 7 measures, then the 11 against 8 and so on…
2) Use a metronome. This proves tricky as the lowest setting on a basic metronome, 
, is still too fast for any one measure. Further, division of the beat is difficult. Nevertheless use the metronome in two ways.
a) Setting the metronome to a very low setting, (I use 
), 10 I practice the LH rhythms, thinking two big beats a measure. Therefore, depending on the subdivision, with every metronome beat, you have either 3.5 notes every half measure, (with septuplets), 4 notes every half measure (with eight eighth notes) or 4.5 notes every half measure (with nontuplets). Once you master the LH, try putting the two hands together with and without the metronome. 11
Ex. 32) Rhythmicana II part 1, mm. 1-9, LH (AUDIO)
I find this method, that of thinking two big beats a measure, the most helpful. In my score, I have drawn a line down the middle of each measure so I can see the halfway marker. This way, I can tell that a group of thirteen in the RH must have six but not seven beats before the second metronome drop.
Ex. 33) Rhythmicana II, sample septuplet
b) As septuplets are the most frequent LH figure, use a single septuplet note (or a Cowell 7th note) as the beat. Pick a faster pulse, such as one note of a septuplet = 120. In other words, there will be seven metronome clicks per measure, lining up with the septuplets in the LH. Try it with the other LH divisions (i.e., one "9th" note to a beat, one "6th" note to a beat, etc...) or if you are feeling brave, try to play the entire piece using just one of the LH divisions as the constant. For example, the metronome will equal a "7th" note, and you will play the "9th" notes against that. The challenge is really no different than having an "8th" note as the constant, just less familiar.
Below are two audio examples of the first two measures of the piece. In both examples, one note of a septuplet = 120. In the first “wrong” example, the rhythm of the right hand is double the LH. In other words, it is a set of 14 pitches (with one missing at the end). The second example demonstrates the accurate rhythm.
Ex. 34) Rhythmicana II, mm. 1-2 AUDIO 1 AUDIO 2
3) Using the chart in Ex. 28, practice a “looser” version of the subdivisions. For example, there is one note of the set of thirteen in between the first two setptuplet notes, but there are two notes from the set of thirteen in between the second and third septuplet notes. Aim to have the right amount of notes from the set of thirteen in between the septuplets notes (either one or two for each septuplet note).
Ex. 35) Rhythmicana II, m. 1
4) Lastly, "feel” your way through the measures. Aim for a smooth line in each hand. See if you can play the piece at different tempi. Only attempt this after you have mastered a rhythmically accurate version.
Overall, this movement requires a fair amount of attention. It is easy enough to work up any single pair of rhythms (i.e. 13 over 7 or 11 over 8) but to keep the rhythms accurate throughout the piece is challenging. Over time, it really does get easier.
Rhythmicana III
In this movement, Cowell uses separate time signatures for the right and left hands, making the matter of measure numbering more difficult. For the purpose of this discussion, I will cite both sets of measure numbers.
Again, Cowell gives little instruction in the score. There is a tempo marking, Allegro Vivace and an opening dynamic, f, given separately for each hand’s entrance. There are also a few articulation clues. We have downbeat accents in the LH that continue for only two measures and the performer is left to decide if there should be a simile throughout. There are also legato markings for the RH in the A sections. Other than these simple indications we must again look to New Musical Resources for clues.
This movement is in an A-B-A form. At the beginning, the LH has a time signature of 5/4 and the RH is given 3/4. Starting in m. 24, in the B section, in the RH and m. 15 in the LH (also the beginning of page two in the published version), the meters are reversed. The 3/4 is now in the LH and the 5/4 in the RH. At the return of A, starting in m. 36 in the RH and m. 35 in the LH, the time signatures are the same as in the beginning.
In New Musical Resources, Cowell relates the use of different meters to the vibration length of partials in the overtone series. The meters of 3/4 and 5/4 meet up periodically just as the partials do. If we look again at Cowell’s chart (Ex. 36), we can see how the vibrations of different partials separate and meet again after a certain amount of time. Cowell suggests that meters could meet up in much the same way. In ex. 37, we see Cowell’s written explanation of differing meters within a piece.
Ex. 36) New Musical Resources, p. 47
Ex. 37) New Musical Resources, p. 72
From the beginning of Rhythmicana III we see Cowell’s ideas in action as we have measures of 3/4 in the RH and measures of 5/4 in the LH. The downbeats of these measures only meet together every fifteen beats. The example below, from the beginning of the movement, shows how the measures line up after fifteen beats, designated by full bar lines (rather than those barlines that simply cover one staff). The first two LH measures are not included in the first count of fifteen beats. They are more of an “upbeat.”
Ex. 38) Rhythmicana III, opening
After the two measure introduction in the LH, the time signatures meet up every five measures in the right hand and every three measures in the left. This is misleading however, as the downbeat of the right hand melody is shifted. It is felt more on beat two, demonstrated by the quarter rest in m. 4, and by Cowell’s phrasing throughout.
This shift of the RH downbeats gives us the sense that we never really arrive, even when the meters do. It is not until the start of the B section (m. 24 in the RH and m. 15 in the LH) that we get a simultaneous downbeat between the hands that feels like an arrival.
In addition to the meters, Cowell's phrase groups differ between hands. In the A section, the LH clearly has two measure phrases (Ex. 39) while the RH is more in a three measure group (Ex. 40).
Ex. 39) Rhythmicana III, LH, mm. 1-4
Ex. 40) Rhythmicana III, RH, mm. 3-10
In the B section, the meters are reversed, and the phrases are different. Here, the LH uses four bar phrases that restart with the D octave every fourth bar.
Ex. 41) Rhythmicana III, LH, mm. 15-22
The RH appears to be in four bar phrases as well, although this is less clear and arguments could be made for other phrasing.
Ex. 42) Rhythmicana III, RH, mm. 24-31
Again, the lack of any articulation, dynamics or other performance suggestions leaves a lot to question. However, notice that the basic composite rhythm in the RH, page two, is very similar to that of the LH on the first page.
Ex. 43) Rhythmicana III, RH, mm. 24-25 and corresponding composite rhythm
Ex. 44) Rhythmicana III, LH, mm. 1-2 and corresponding composite rhythm
Similarly, the LH rhythm in the B section is in a simple 3/4 pattern, just as the RH rhythm was in the A section. Perhaps Cowell wanted these rhythms to be brought out in a similar way so that we could recognize this continuity.
In any case, more positively, the lack of musical instruction in the score leaves the performer with numerous options. Here are a few ideas:
The A sections, with the f markings, octaves and accents in the LH clearly demand a full sound.
As there are no pedal markings or specific phrasings in the LH, perhaps Cowell wanted the bass sixteenth notes to be clear and crisp. Although Cowell indicates an allegro vivace tempo, the sixteenth notes are not easy and I play this piece perhaps a little slower than Cowell intended, around quarter note = 112. Further, I could envision a satisfying performance at a tempo anywhere between quarter note = 100 to quarter note = 140.
1. Rhythmicana was also an early name of a previous work by Cowell, Concerto for Rhythmicon and Orchestra, involving the Rhythmicon, an instrument created by Cowell and Leon Theremin. See William Lichtenwanger, The Music of Henry Cowell: A Descriptive Catalog (Brooklyn, NY: Institute of Studies in American Music Monographs: Number 23), 132-3.
5. A “tuplet” is a generic term that includes any rhythm that uses “tuplet” in its name. For example, a “quintuplet,” (group of five notes) or a “nontuplet” (a group of nine notes) are both types of tuplets. We also consider “triplets” (a group of three) a type of “tuplet,” although if one wanted to be consistent, we could call them “trituplets.”
7. These “new” notes of Cowell’s are similar to those in shape-note hymnals and were in existence long before Cowell. Cowell makes no mention of this tradition in New Musical Resources but it would be difficult to argue that Cowell is stealing these notes from the tradition of shape-note hymnals. Cowell is using his notes for duration or rhythmic purposes, and his usage is quite extensive. The shape-notes are for pitch only, and there are only seven different notes, one for each of the notes in the solfege scale (do, re, mi, fa, etc...).
9. If you are like me, and a little lazy when it comes to arithmetical calculations, I found an excellent calculator online at http://www.easycalculation.com/hcf.php.
10. If you have a more advanced metronome, you can probably find more advantageous settings. My metronome, a standard electronic model, has a tempo range between 40 and 208 beats per minute. A Boss DB-88 or “Dr. Beat” metronome ranges between 35 and 250 and I am sure there can be found metronomes with wider ranges of tempi. I also have downloaded a metronome for my laptop from the following website, although it is currently only available for Mac OS X: http://home.earthlink.net/~mistercow/AquaGnome.html. It ranges from 40-400 beats per minute.
11. For a more detailed approach, you could also divide the LH in half again. For example, instead of thinking seven quarter notes in two metronome beats, you could think fourteen eighth notes in two metronome beats. This gives you seven beats per metronome click and you play a written note every other "14th" note.