The Chord Catalog of John Cage and Paul Zukofsky

[PZ dictated the following six days before his death:]

“This Cage cards text was intended to be a collaboration between me and John. I always wanted to get the cards published and tried to write a long disquisition, but from a combination of circumstances, including loss of interest, just couldn’t manage it. When John died, I decided I really should get off my ass, but I found I couldn’t deal with his foundation. So I waited, and then Merce – whom I liked very much – also died. After further delay, I finally wrote my text, which I didn’t really like much and still don't, and I got all the charts ready. They are not really voluptuous, and frankly I don’t know how to make them interesting. I wanted to write about the relationships between chance and the I Ching, but finally I threw up my hands because that’s not really what I do, and there are others better qualified… and now this illness.

“In a way, it’s interesting because you’re alive, and then one part goes, and then another, making it harder and harder to be intelligible… [hums a melody in exact rhythm and intonation] What is that? Some child’s tune… ‘around this wandering path?’ ”

[It was “Ashes, ashes, we all fall down.”]

Introduction & Basics

How John Cage came to compose various pieces for me has been described elsewhere.¹ This text surveys the “Chord Catalog” (hereinafter CC), which served as the basis for the aggregates (“agg”(s)) of Cage’s Freeman Etudes (FE).

Construction of those “agg”s involved my being asked what pitches I could play given certain circumstances; and my responses. Most of these exchanges were copied onto index cards (i-cs), whose accretion was misnomered the CC², for "Ceci n'est pas un CC"; i.e. the CC is potentiality. It does not contain finished “chords”; and while all FE agg”s derive from the CC, the CC is not a collection of “agg”s that appear in the FE! Blame for the existence of the CC may be laid at the feet of the physics of strings.

Basics

String players produce different frequencies (aka “pitches”) by using their left-hand fingers to depress (“stop”) a string at various points along the length of the string. Dividing the full length of the string into four equal parts, the first quarter-length encompasses the range of a perfect fourth; the next quarter-length covers a perfect fifth; the third quarter-length an octave; and the last quarter-length range is theoretically infinite. Put another way, the first octave of pitches uses 1/2 the total string length; the second octave 1/4 of the total length; the next octave 1/8, etc. 

Because of this geometry, the interval size that can be spanned by two fingers increases as one leaves "first position”; and the precise amount of change is hardly ever evident, even to “experienced” violinists.³ Some intervals, difficult, even impossible, in low positions, become, in high positions, quite easy; and the transmutations are not uniform, as the angles and distances (of the arm - hand - fingers relative to the body of the violin) vary, both because of where one is on the string; as well as the specific strings involved.

Hence John’s need, and desire, for the CC.

Procedure

In creating “agg”s, John would first decide which events (of an already determined sequence) were to be “agg”s; and how many strings would be used for each “agg”.

For each event-that-was-to-become-an-“agg”, John would decide a starting string (always referred to as string “X”); a specific pitch on that string; and a subsequent string choice (always referred to as string “Y”). He would then ask what was “available” (i.e. playable) on string “Y”?⁴

I would play the note specified for string “X”; experiment; and reply with a collection of pitches playable on string “Y”. An "open" string (i.e. no fingers used, resulting in an E, A, D, or G, depending on “Y”) was always available; but I was usually able to supply other pitches, from which collection John could select. John would then use I-Ching operations to determine which pitch (of my collection) would be used on string “Y”.

For greater-than-two-string-”aggs”, the process would start as above; and after a two-note-”agg” had been formed, a new question would be phrased: pitch “X” is on some string; pitch “Y” is on some other string; what is available on yet a third string (“Z”); and I would respond. For four-string “agg”s, the process would be continued by John saying what was on “X”, “Y” and “Z”, and I would provide any (usually highly restricted) possibilities; and at some (fairly timely) point, after enough responses had been accumulated on to large sheets of paper, (most) answers would be copied (by John) onto i-cs.⁵ 

General Description

The physical CC consists of 65 manila tabbed i-cs, categorizing 546 three-inch by five-inch non-tabbed i-cs. All i-cs are housed in a cardboard box in which such i-cs were purchased. Tabbed i-cs are color-coded to indicate various string combinations (red, yellow, blue, and green represent the E, A, D, and G strings respectively).

This web version of the Chord Catalog uses folders to separate the 65 categories. Folder numbers are identical to PZ category numbers.⁶

Clicking on a folder unfurls the contents. Manila tabbed category i-cs appear first; followed by the specific question and response i-cs within each category. Within a category, or folder, i-cs are arranged as I found them, generally (but not always) ascending by first note “X” within the category, in the order John seemed to have preferred.

Table 1 arranges the 65 tabbed i-cs in four panels i.e.

“agg”s where the first pitch choice (“X”) is always on the first (aka E or I) string (PZ #s 1 - 15)(JC # 2 - 16);

the first pitch choice (“X”) is always on the second (aka A or II — see the ochre band at the top of the col.) string (PZ #s 16 - 31);

“X” is always on the third (aka D or III) string (PZ #s 32 - 48);

“X” is always on the fourth (G - IV) string (PZ #s 49 - 65).

Each panel shows three sub-groups: black type is used for two-string “agg”s; red type for three-string “agg”s; and blue for four-string “agg”s.

The 546 three-inch by five-inch i-cs are, with three exceptions, all written on one side.⁷ I-cs are of two colors i.e. white or blue. White cards are of two types — ruled on one side; vs both sides unruled. Blue cards are all ruled on one side. Blues were used after John "ran out" of whites. I numbered (most often in the lower part of the i-c, and in original CC order) the i-cs, using a format of numbers to the left and right of a period (.).⁸

String information on i-cs was at first, color-coded. Later, Roman numerals were used. For some i-cs, responses were notated as “agg”s; for other i-cs, pitches were written separately. I believe those differences to be arbitrary.

Each i-c contains (towards the left) either a single pitch, or pair, or triune, of specified pitches. Every i-c implies a question, answered at the right, where one finds the possibilities that can be combined with the specifics at the left. Responses can be single pitches; and/or bounded ranges, i.e. for token 1.01, the pitches on the second string that could be combined with the F-natural harmonic on the E-string were either the open “A”-string; or else any pitch starting from the B-flat up thru the sharpened D-natural. That any pitch within the boundaries is acceptable is indicated by the slanted line between the limit pitches.⁹ Fingering information is not provided, i.e. that the pitches of this range must be fingered by the 2nd and 3rd (violin) fingers was, for John’s purposes, irrelephant.

As regards my pitch ranges: for initial string “X” on the second, third or fourth string I used 2.5 octaves. “X” on the first string used a somewhat greater range.

Ranges for string “Y” (to form a 2-string “agg” after “X” was specified) conform to (minimally reduced) standard violinistic usage.

Possibilities for string “Z” (i.e. the third string to be added to already specified strings “X” & “Y”) vary as a function of the fingers involved. For cases where “X” & “Y” could be played by adjacent finger-pairs¹⁰ possibilities could remain somewhat large. Where “X” & “Y” had to be played using finger-pairs 1 & 3, or 2 & 4, possibilities were more restricted; and for “X” & “Y” requiring finger-pairs 1 & 4, possibilities were determined solely by the movability of the second and third finger, something almost always close to non-existent, due to neurology and the musculature of the left hand. 

Ranges for four-string “aggs” (“X”, “Y”, and “Z” being specified) were most often extraordinarily limited.¹¹

CAVEAT: Note that the i-c ranges provided are only “snapshots” of my capabilities at the time, i.e. when my hand was both fairly flexible and “prepared” for the task. On some other day, let alone decade; at some other temperature; under some myriad of other conditions; there could easily have been different results. There must also be inconsistencies between i-cs, as the samples were collected in two large tranches, i.e. one for the first two FE “Books”; and then, after a substantial hiatus, the additional samples needed for John to complete “Books” 3 & 4; all samples being co-mingled without indication of when they were created. While the entirety of samples probably most likely sheds light on the geometry outlined above, any single sample should be treated with “sly circumspection”, i.e. the CC was not a controlled experiment.

Original Order of the 65 Manila-tabbed I-cs

There are 60 possible combinations, with permutations, of four things (in this case, violin strings) taken two, three, or four at a time i.e. twelve two-stringed combinations (on Table 1, in black type); 24 three-stringed combos (in red); and 24 four-stringed (blue).¹² While each panel of Table 1 has 3 two-stringed; 6 three-stringed; and 6 four-stringed combinations (not counting erroneous duplicates), the three colors show that the internal order within panels is not consistent, i.e. while the agreement between the far left and right panels (strings I and IV playing “X”) is quite good; and the agreement between the two middle panels (strings II and III playing “X”) is nearly perfect; the agreement between the outer and inner panels is poor. I do not know if this discrepancy is intentional, or happenstance. The outer panels both have “X” on outer (E & G) strings; while the inner panels both have “X” on inner (A & D) strings. That seems a poor reason for different internal “cat.” orders; but the consistencies within outer and inner panels argues against happenstance.¹³

It is imperative to understand the “flow” within panels, i.e.for each panel of Table 1, a three-stringed combination (henceforth category, or “cat.”, synonymous with “folder”) descends from only one of the two-stringed “cat.”s within the same panel. Four-stringed “cat.”s descend from only one three-stringed “cat.”. “Cat.”s NEVER “cross” panels.

Table 2, details the flow of PZ “cat.”s 1 – 15. 

The three possible two-stringed “cat.”s of “I playing X” (# 1, 3 and 11) are at the top.

“Cat.”s # 2 & 6 are directly beneath. Both are “I playing X, II playing Y”; but with mutually exclusive prospects, i.e. “what is available on III” (# 2) vs. “IV” (# 6).

Once # 2 or 6 is determined, # 2 must flow to # 5; # 6 must flow to # 7. There is no choice!

The “flow” also applies to “cat.”s #3 and 11 (at the top), where #3 can flow to either #4 or 9, (and #11 to either #12 or 14); but each of #s 4, 9, 12 & 14 can only flow to a single blue “cat.”. Similar branching applies to the remaining three panels of Table 1 (i.e. “II PLAYING X”; “III PLAYING X”; “IV PLAYING X”). Specific examples may clarify.

Consider i-c 1.01, i.e. the first i-c of folder #1, “I PLAYING X, WHAT IS AVAILABLE ON II”. “Starter note” “X” is a fourth harmonic (based upon F-natural on the E-string), with the ask “what is playable on the A-string”. Once the response was supplied, and the additional pitch (on the A-string) selected, only string(s) III or IV remained, i.e. the process could ONLY branch from “cat.” #1 to “cat.” #2, or # 6.  

In the case of cat. #2 (“I PLAYING X, II PLAYING Y, WHAT IS AVAILABLE ON III”), i-c 2.02 shows an expansion of i-c 1.01, keeping the starting F-natural harmonic, and adding a B-flat selected from the range provided on i-c 1.01. A response range for “WHAT IS AVAILABLE ON III” is provided (the green pitches at the lower right). I-c 2.03 shows a variant (i.e. the harmonic is gone, but the sharpened F natural is only millimeters up the fingerboard). I-c 2.04 is also somewhat similar, essentially a “transposition” up the fingerboard. 

As “cat.” #2 can ONLY branch to #5, i.e. “I PLAYING X, II PLAYING Y, III PLAYING Z, WHAT IS AVAILABLE ON IV”, we now inspect cat. #5, and find no exact continuation of i-c 2.02. On the other hand, i-c 5.01 is a clear continuation of i-c 2.03 (which we previously considered a variant of i-c 1.01 and 2.02).

If we now return to the branch that leads from “cat.” #1 to “cat.” #6 (“I PLAYING X, II PLAYING Y, WHAT IS AVAILABLE ON IV”), we find no i-cs with the F-natural harmonic of i-c 1.01. There will therefore be no such F-natural harmonic found in cat. # 7.¹⁴

Returning to Table 1, and the distributions thereof: the “# of i-cs” col. shows that tabbed i-cs have as few as 0, vs as many as 44, i-cs. More specifically: the rows of the matrix below (table 3) represent the four panels of Table 1 (i.e. initial string choice (“X”) on the E, A, D or G strings). Matrix col.s show two-stringed, one pitch specified i-cs (i.e. “X/?”); three-stringed, two pitches specified (“XY/?”); and “XYZ/?” “cat.”s. Numbers within matrix “cell”s are the number of i-cs, i.e. there are 74 i-cs with “X” on the E-string, with only “X” specified; there are 10 i-cs with “X” on the D-string, with pitches specified for “X”, “Y” & “Z”.¹⁵

Table 3:

That the col. totals line shows a preponderance of samples with only “X” specified (243/546 = 44.5%) is to be expected, as there must be an “X/?” before an “XY/?” can be constructed. The almost equally robust 212 (38.8%) samples of the “XY/?” col. can be similarly explained, given the need for progenitors of “XYZ/?”.

Within the two-, three-, and four-stringed groups, a very few “cat.”s account for a large percentage of their group, i.e.:

of the 12 two-stringed “cat.”s (243 i-cs), the 127 i-cs of “cat.”s #1, 50, 18, and 17 represent approximately 52% of all two-stringed samples.

The four (out of 24) largest three-stringed “cat.”s (#2, 4, 23, 35) account for approximately 49.5% of the 212 samples. Adding “cat.” #51 (17 samples) takes the percentage to 58%.

As for the 24 four-stringed “cat.”s, #s 15, 7, 8, & 28 account for 49/91 i-cs, or almost 54%, of the samples. The remaining 20 “cat.”s never contain more than 4 samples.

I-cs where “X” is an E or G string may be favored i.e. there are 325 i-cs with “X” on the E & G strings vs 221 i-cs with “X” on the A & D. The emphasis is even more extreme for the top line itself, with 220 i-cs of “X” on the first string.

All of the above (and other observations) raise questions (not addressed) regarding expected I-Ching probabilities.

For Violinists
The Importance of String Choice Order

String-choice-order in “agg” construction is normally not of primary importance for composers. For violinists, string-choice-order in the CC is yet another albatross attached to our neck. Tables 4, 5, 6 & 7 explore this.  

Table 4 displays six sets of two-stringed string “reversals” (such as the pair of “cat.”s #1 & 17). Each pair involves the same strings, but the order of string choice is reversed; and as initial string-choice affects, and usually restricts, succeeding finger placement flexibility and functionality, results can differ for the same pairs of strings.

Not only can one compare within pairs (such as #1 & 17), but one can compare between pairs.

Comparing between the [I + II vs II + I] pairs, with the [I + III vs III + I pairs], and the [I + IV vs IV + I] pairs, allows one to ask how one’s fingers may be discombobulated by small lateral distances.

Comparison between pairs [I + II vs II + I] vs [II + III vs III + II] vs [III + IV vs IV + III] explores left-elbow placement, and how that affects left-hand plasticity.

To the pairings of Table 4 we add the questions of Table 5 i.e. to “cat.”s #1 & 17 we add “what is available on III, or IV” (“cat.”s #2, 6, 19 & 25), etc. This allows us to ask how string accretion affects our ability to manipulate our digits.

Table 6 rearranges the same six groups of Table 5 in to four groups of I/II/III; I/II/IV; I/III/IV; and II/III/IV. We do this because, for the group of I/II/III (“cat.”s 2/19/4/39/21/36) (as example), the only internal differences are due to string-order-choice; but ask how and if the results differ, and generalize the concept to the other string triunes.

In addition, Table 6 foreshadows Table 7, where the data is rearranged by the final string choice, but with three strings already determined (Tables 5 & 6 only have two strings determined). All these differing perspectives provide information (sometimes subtle, often crucial) that violinists MUST appreciate and savor.

Some Other Considerations

The CC demonstrates that aspects of classical technique can fail in music such as the FE. (The closed-minded should avoid reading what follows.) Problem areas include:

A Movable Left Elbow

Tradition tends to fix left arm segments in space. Left hand re-configurations are made almost exclusively with fingers, or, in extremis, a bit of wrist, although that is in the wrong plane. Weakest muscles bear the greatest load. Constant deformations strain the hand complex.

The so-called “reasoning” behind this approach is that minimal physical motion equates with physical efficiency, which equates with some imagined concept of mechanical elegance. That minimum motion may imply efficiency or elegance can, in some cases, be so; but minimum motion as a goal unto itself, without a thought to the components of the machine, while utilizing parts of the machine well beyond capacity (thereby creating the potential for strain, if not strain itself, and breakdown), is a perfect example of “backsidebeforeness” — unless one wishes to break the machine as efficiently as possible. 

Rather, envisage a system where, for each and every “agg”, left-arm segments have an almost unique place in three-dimensional space. The upper arm moves the left elbow in an arc, allowing wrist, hand and fingers to approximately maintain the same shape for all strings (which is useful for sonic, mechanical, and organizational reasons). While the elbow now traverses a distance far greater than that previously traversed by the fingers, any additional “work” or “energy” required (i.e. calorie, or oxygen, consumption), is more than compensated for by the use of large, less easily tired, muscles, which reduces the risk of finger/hand fatigue and/or strain.

Additionally: a “movable elbow" is crucial for the tuning and intonation of certain intervals, especially the perfect fifth, which can be varied from almost a large tritone to a minor sixth, depending upon how far to the right or left of some central point the left elbow is moved. I also believe it imperative to consistently play on the pads of the fingers, and not on the tips (only partially because of intonation concerns), and the only way to do so is to move the left "elbow". A movable elbow can be useful, and advantageous, in classical music as well; but its use is less crucial. For music such as the FE, there is little choice.

To clarify left elbow motion, consider: 

1— One String - Four Fingers: on the E-string, with ONLY one finger in contact with the string, successively place the fingers 1, 2, 3, and 4. Be certain to lift each finger before placing the next finger. The left elbow should swing in an arc from “left” to “right” as each successive finger is engaged. For finger pattern 4, 3, 2, 1 the swing is “right” to “left”. The elbow movement compensates for the third and fourth fingers’ extra distance from the violin neck.

2— Four Strings - One Finger: place the first finger on the F# on the E-string; then successively lift and replace the first finger onto the B on the A-string; the E on the D-string; and the A on the G-string. The left elbow will swing “left” to “right” as one swings from the E to the G-string in this progression; and “right” to “left” from the G to the E-string. The trajectory of this arc is different from 1— above.

2a— Four Strings - Different Fingers: repeat trial 2— above, using the other fingers. As we deploy the second, third, and fourth fingers across strings, the locus of the arc shifts “left”. The trick, when playing, will be to combine the different arcs of 1— and 2— above, as we pass from point to point in 3-dimensional space. The left elbow can NOT be flying about madly; but that is rarely a concern. Excess rigidity is the greater, and far more common, problem.

3— Two Strings - Two Fingers:

On the A and E strings, form the following finger pairs
i.e. fingers 1 (A-string) & 2 (E-string), reversing to 2 (A-string) & 1 (E-string); fingers 2 & 3, reversing to 3 & 2; fingers 3 & 4, reversing to 4 & 3. For each reversal, the elbow should swing “left” to “right”, i.e. when the higher of the two fingers is on the lower string, the elbow is more to the “right”. If this test is transposed to the D + A, and then the G + D strings, the overall locus yet again shifts to the “right”, again assuming that the hand and fingers maintain approximately the same configuration for all strings.

4-- Four Strings - Four Fingers:

Form fingerings (x) & (y):

For (x) followed by (y), the elbow swings “left” to “right”. For (y) followed by (x) the swing is “right” to “left”.

Lift the Fingers

When toggling “aggs”, standard technique espouses keeping in place as many fingers as possible (again in the name of purported “efficiency”); but each time we attempt to do so, we must decide which fingers to hold down; which to lift and replace; and what is the preferred finger internal micro-timing for the toggle. This adds to the learning burden, a burden we should accept ONLY IF, for the vast majority of cases, most fingers are best held down, with lifted fingers being the very rare exception.

This approach may have made sense in music where the norm was to stay more or less in the same position, with a harmonic expectation that adjacent chords might very well have common tones, but that is the furthest thing conceivable from the FE, where the hand/arm MUST bounce about almost without cease.

To state the case as briefly as possible, there are 24 possible arrangements (plus “mirrors”), of four fingers, used simultaneously, on four strings.

Table 8. Finger Patterns

Any change within a col. from upper to lower (or the reverse) involves switching all 4 fingers. Within the upper or lower band, any of those 24 possibilities can move to any one of the remaining 23, for a total of 24 X 23 = 552 progressions. Each progression involves switching at least two, and usually more, fingers; and even where some finger(s) could be held down, it is often infinitely preferable to lift all four fingers, due to their non-independence, and a desire for mechanical consistency.¹⁶

For those few cases where we may be able to hold down a finger or two, is it useful to constantly evaluate when, and when not, to lift, which fingers? Is there no more efficient mechanism for changing “agg”s?   

Imagine that each finger arrangement originates from a “neutral”, or “zero”, “stance”. That dramatically reduces the number of one-to-one connections we must learn and store, i.e. if all fingers were always released after each “agg”, and all fingers returned to a “neutral stance”, from whence they could easily reshape into any new arrangement, we only need learn the progressions from “neutral” to whatever different finger arrangements are called for. We must still learn the order of which “agg” follows “agg”; but the mechanical connections between “agg”s have been simplified, and each finger now has far greater capacity to reorient for the next “agg”, since nothing held down can impede fingers that MUST move.

Do we now decide to embrace a fairly uniform system (i.e. “always” lift all fingers); or do we accept a two-class system (lift vs don't lift)? For classical tonality, using a limited number of “positions”, where successive “agg”s with no common tones was mainly the exception, a bias towards a two-class system might have made sense (although I question that); but for music such as the FE, lifting fingers wins, as consistency of thought and mechanics outweighs any (infinitesimal) energy savings provided by exceptional cases.¹⁷ Lifting (perhaps better, releasing) fingers, to return them to “neutral”, does not imply splaying them into the air, as in a Busby Berkeley spectacular. We only require the ability to quickly reconfigure all fingers vis-à-vis the strings. Release duration and height will vary, depending upon specifics; but release there must be!

And for any wishing to argue that fingers must remain in contact with the string in order to sustain the sound, they confuse production with perception.

Miscellany

Within the CC, non-adjacent-strings “aggs” function as progenitors of three or four note “aggs”; but the need to “jump” non-adjacent strings is not new.¹⁸ Much 20th century music requires just such string-jumping, while insisting on the maintenance of a smooth line (Webern springs to mind). The CC non-adjacent-string “aggs” can accustom us to possibilities we hardly know, and far too often ignore; and violinists should accept that “smoothness” is a sonic phenomenon mainly independent of string crossings, i.e. the resonances of the violin, and venue, trick the listener into acceptance of an impression of smoothness; and camouflage the reality of a succession of discrete signals.

If one must engage enormous left-hand stretches (as opposed to jumping strings), never underestimate the fingerboard length that can be spanned by the first two (violin) fingers, a span far larger than that possible between the second & third, or third & fourth, fingers. Classical teaching emphasizes stretching the third & fourth fingers in the direction of the nostrils, ignoring the greater flexibility of backward stretches by the first two fingers. That is a sadly limiting bias.

A simple summary: much standard violin teaching insists upon a mechanically “anchored”, “static” system. Purported security comes from thinking I know “how” to go “where” because I (think I) “know” where I am. A mechanically dynamic system is far more utilitarian. Given the manifold degrees of (anatomical and mechanical) freedom, one can never truly know “where” one is; and using potentially erroneous guesswork as the basis for the next “jump” only compounds “errors”. Let head, neck, and shoulder fix the instrument, leaving the left hand and arm free to perform their ballet in space. Coordinate your left and right elbow motion. Practice your one-finger scales (with each of your four fingers!). Train your proprioceptive senses. Such a system can, at first, be daunting; but the static norm cannot possibly match the ultimate flexibility and speed provided by a more dynamic system.

Utility

The CC can help young composers decide what configurations may be playable. NB: While, as a general rule, I always tried to not “push the envelope” (too far), composers utilizing i-c ranges should (in principle) not increase them, as the additional effort required may be great. Do please reread, and reread yet again, the CAVEAT regarding “snapshots”! 

When confronted with a recalcitrant know-it-all, the young composer may risk disembowelment by pointing out that the CC says such-and-such. That being said, and not to side with the recalcitrant know-it-all, just because one can momentarily approximate a pretzel does not mean one can then instantaneously metamorphose from pretzel to Botticelli’s Venus, i.e. THE major prob of the FE is lack of time between events.

Violinists may use the CC to open their minds to the possibilities of what can actually be done on our instrument; or what we could do were we just a little bit more imaginative, and were our blinders only slightly smaller; and perhaps differently shaped. 

Paul Zukofsky
Oct., 2015
Hong Kong

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