Density in a Musical Context1


E. Michael Harrington

Part One

The following essay will outline this writers attempts at defining density in music, deriving quantitative measurements of density, and using all such information in an analytical and compositional manner. A comprehensive definition and assessment of density, one that will exemplify the interdependence of all of the musical parameters, i.e., pitch compression, range, distribution, dynamics, instrumentation, etc., in a numerical representation, is impossible. Limitations have been imposed from the outset.


This discussion of density will discount the effect of instrumentation upon the sound (i.e., for this paper, how the sound is produced is not of concern to the measurement). Secondly, dynamics will not be of importance in assessing density. The amplitude of a pitch will have no bearing whatsoever upon the density. Lastly, time will not be considered essential in density analysis. It is at this point that I differ most from others who have dealt with density in either a compositional or analytical manner. lannis Xenakis, for example, measures density and utilizes it as a function of time (i.e., the number of attacks/events per unit time). The vertical or concurrent aspects of density (e.g., a chord), are not of the same paramount importance in the compositional process as they are to this writer. Xenakis derives his vertical structures in procedures which are only incidental to considerations of density. Wallace Berry, on the other hand, attempts to measure density in both the concurrent aspect as well as in the linear unfolding and accrual or decrease in the number of parts present in the musical texture. His numerical representations of vertical density (and linear density) fail to differentiate between degrees of consonance or dissonance, the spacing of the chord, the effect of mirror inversion, the register of the event, the effect of transposition upon the chord, and the use of the same intervals in a different order (i.e., permutational variations).


Density, I feel, is a parameter capable of being precisely defined through an accurate and logical means of analysis. Conditional statements regarding analysis of vertical texture in the past have been of a general nature with a minimal distinction afforded between sonorities considerably from each other. For example, it is not enough to state that a chord is a four-note event, or a five-note event, as aspects of register, compression, and chordal quality are not taken into account. Consider the four-note chords below (Example 1). They are obviously representative of disparate constructional features and share in common only the fact that they are all four-voice sonorities. It is clear that the number of voices present in the simultaneity may mean little, if anything, in analysis.


Example 1

An account of pitch-range compression in analysis will yield a more precise description of the chord. For example, the first chord's four elements are distributed over a range of 1512, chord #2 over 1161, chord #3 over 1521, and chord #4 over 1211. A convenient nomenclature for such a density representation is shown in Example 2.

            #1       #2         #3       #4
4:5     4:16     4:52     4:21
Example 2

This quantitative device is used by Wallace Berry and is described in his book Structural Functions of Music.3 Or, two types of fractions can be determined - one that will yield indexes larger than 1, or the reciprocal function which will result in numbers smaller than 1, but greater than zero4 (Example 3).

4/5 = 0.80     4/16 = 0.25     4/52 = 0.08     4/21 = 0.19
5/4 = 1.25    16/4 = 4.00      52/4 = 13.00   21/4 = 5.25

Example 3

The fault of this system lies in the fact that it does not represent a special-case chord, but rather a myriad of chords. To illustrate this weakness, the following example is offered. Constansinarcseccot, by this composer, consists of six chords repeated in various guises. All of the chords are four-voiced and span a 1161, yet this system of density cognition fails to differentiate them (Example 4).



Example 4

Part Two of this paper will attempt solutions to these and other problems. All subsequent definitions of density will involve calculations and treatment of equally-tempered intervals only, and assessment of vertical pitch structures regardless of time consideration, amplitude, and instrumentation, but necessarily involving register, spacing, range, consonance/dissonance level, and pitch-compression.

Part Two

In a vertical structure, frequencies are sounding simultaneously in ratios to one another. The relationships contained within a chord, the sum of the frequencies in vibration, are analogous to the number of

relationships in a star event5 The number of interval relationships which must be assessed in a chord is determined by squaring the number of pitches, subtracting n from this number, and dividing by 2.


(n2 - n)/2 All Self-contained interval relations

n = the number of notes present in the chord

A brief table of sonorities of increasing voices, and their total number of self-contained interval relations (star events), follows in Table 1.

Notes in sonority                 Total interval relations

        2                                          1

        3                                          3

        4                                          6

        5                                         10

        6                                         15

        7                                         21

        8                                         28

        9                                         36

       10                                         45

       11                                         55

       12                                         66

          .                                            .

          .                                            .

       88                                        3828

                                                                                        Table 1

In assessing density, then, more calculations will be involved for a larger chord than for a chord with fewer notes. Because the chord exhibits this number of relationships, which is necessarily more than the number of notes in the chord (unless the chord is of three notes, or a two-note dyad), it is of extreme importance to consider all such elements, as subsequent discussion will prove. In other words, non-adjacency of pitches in a sonority accounts as an integral part of composite analysis.

We are now ready to invoke an operation on the chord which will account for all the relationships. We begin with the division of each frequency by all of the frequencies above it (i.e., higher frequencies are divided into lower frequencies, obtaining numbers less than one). All of the derived quotients are then added together and divided by the number of interval relationships in the sonority to achieve a mean. Symbollically, the equation can be stated as shown in Example 5.                     

                                                                                      n-1     n

                                                                                1   /  (   . .   fj / fi ) /s     = d

                                                                                      j=1   i=j+1



fn-1      is the second highest frequency

fn        is the highest frequency

S       is the number of interval relations (star events)

                Example 5

To consider some chords, we arrive at the indexes shown in Example 6.



Example 6

The density index, at this point, is inverted to yield a number larger than one, and, in fact, larger than 1.0595; the interval of the half-step. The two indexes for Example 6 should be revised - 1.3032 and 1.2558 respectively. More chords and mean densities are shown in Example 7.


Example 7

The measurement of density is such that the chords which span the greatest distance will have a higher index. Conversely, those chords which span a closer, narrow distance, will have a lower index. Two pitches a half-step apart will be 1.0595 according to this index. This is the twelfth root of two and will be the lowest number assigned to a vertical entity because it is the closest possible positioning afforded by equal temperament. Four-note sonorities, which always contain six consequential intervals, will reveal similar features with respect to changing areas of musical space (Example 8).


Example 8

Because the same ratio occurs between each half step, chords with identical spacing will exhibit the same index of density, d, regardless of registral placement. For example, the major triad made up of the contiguous intervals 141 and 131, will have the same density index in any octave in which it appears (Example 9).



Example 9


So far, the chords to which we have ascribed an index corresponded to an auditory experience (i.e., as the range of the three- or four-note chord increased, the numerical representation grew larger, accordingly). And as the chord became more compact, so also did the representative index decrease, reflecting the diminishing area of musical space covered by the chord. What would be the effect, both auditory and analytical, of altering a simple triad by introducing pitches within its extremes (i.e., by changing a three-note chord to a four-or five-note chord of the same intervallic span) (Example 10)?


Example 10

It can be seen from this example what the effect is of this outside-of­time parameter, i.e., the perfect fifth between outermost voices being internally modified. It reveals the same feature as the widely-spaced chord which undergoes a uniformly spatial reduction (i.e., as the pitch compression increases, and/or the overall range decreases, the density index decreases).

What is the effect of interval order permutation upon contiguous intervals of a sonority? For example, would a similarly-voiced major and minor triad have the same density? (Example 11)



Example 11


Reversal of contiguous intervals has no effect upon three-note chords. In a larger chord (i.e., one with more members), will mean densities change when the intervals in a sonority are similarly rearranged? And what is the effect of such dispersion (i.e., permutation through adjacent element order rotation)? If a chord consists of n equal intervals, such as a diminished 7th chord arranged in a series of minor thirds, the density index will be constant provided that the equal interval disposition is never violated (Example 12).     


Example 12

This was realized in the prior analysis of a major chord where the intervallic unity was maintained within various registral contexts. However, will the density index be the same if the pitches within the equal interval chord are dispersed over an enlarged area? Let us examine such a revoicing as a case in point (Example 13).


Example 13

The index increases as the pitch compass increases, as was made evident in the earlier inspection of pitch ambitus effect and function. Now, if the three contiguous intervals within the four-note diminished 7th chord are arranged in a different order (yet maintaining the same span of musical space) from a vertical concurrence of [6], [3], [6] to [3] [6], [6], and [6], [6], [3], will the density index remain constant (Example 14)?


Example 14

This proves that the same intervals rearranged, as in a permutational series, will result in different indexes as well as different quality chords. It further demonstrates that not only must contiguous intervals, but all combinations of the interval events, be calculated, for only in assessing every event do the real analytical, perceptual, and structural differences manifest themselves.

Because of this system's equal treatment of registral location and transposition of an entity, the same density index can represent chords of various guises and locations in musical space. Whether this flexibility of chordal quality, placement, and function stacks up to our perceptual experience on an absolute scale may be irrelevant, especially if the desire is for the compositional generation of a diverse musical system.

The following chords are of a 1.2940 d (Example 15).


Example 15


Only one calculation was necessary for the generation of such a series, for the consequent chords are merely transpositions and permutations of the initial chord's contiguous elements.

If intervallic identity is exactly preserved, the chord may appear in any register and maintain the same a. The following chords (Example 16), exhibit a slightly increasing a between adjacencies. The enormous variety of voicings and registrations attainable through use of this system is of utmost compositional importance. The system can be made as imaginative or as restrictive as the individual composer desires.


Example 16

There are other possible ways to measure density. For example, in the previous equation, the number of notes within each sonority was not necessarily reflected in the density index. A four-note chord could have a higher or lower index than some 2-, 3-, 5-, 6-, or 7-note sonority. An equation which will immediately reflect the number of notes within the sonority (i.e., the weight) has been developed, and is illustrated in Example 17.

                                                                 n-1        n

                                                         1   /  (    .    .    fj / fi ) /s   + number of notes  = d

                                                                 j=1      i=j+1                                  in chords


Example 17

The new equation, da, is similar to the initial measurement, d. To derive da, the initial equation a is performed, to which, the number of notes in the sonority is added. In Example 18, the differences between the two systems are as striking as the conclusions drawn from the two systems.



Example 18

When using d, as the number of notes in a sonority increases (i.e., the weight increases), the density index becomes smaller. However, when using da, as the number of notes in a sonority increases, the index becomes enlarged to reflect this change. Because the original process, a, is intrinsic to, and subsumed within da, interchangeability between indexes is very simply achieved: d to da involves one addition; da to d involves one subtraction.

If we are now to re-evaluate the music of Example 16 according to this new process, da, we will have a different nexus of density. As Example 16 was delineated according to a gradually increasing d, so now is it re­ordered to exemplify an increasing da value between successive simultaneities. The two systems are compared and evaluated in Example 19.

Example 19

The nature of the interval calculations of d is preserved in the da process - chords containing the same number of elements will constitute the same hierarchical order using da as they did when calculated simply by d. The effect of the constant in the process is not significant when dealing with a uniform element-number family (Example 20). da is very similar to d in that the measurement of the density of a sonority will be the same in any registser, provided that intervallic identity be strictly preserved. (Example 21).




Example 20

Example 21


A simpler method may be devised that will account for the number of notes in the sonority - it will reflect the function of increased or decreased weight in the sonority, as did da. The two previously discussed methods of calculation, d and da, encompassed every conceivable interval relationship within the sonority. As was illustrated in Table 1, the amount of star events in a sonority grows larger, necessitating many calculations. For the composer who wishes to attain a density index for chords of a very large size, and does not have the use of a computer to quickly perform the frequency calculations, an easier system is needed. The system will not concern itself with star events, but instead will develop its representative value through a summation of frequencies in the chord. The frequencies are added together, the logarithm of the frequency sum is taken (to reduce the values to a range comparable to the previous indexes), and the number of notes in the chord are then added to this number (Example 22).


log (f ) + n = db


Example 22


To appreciate the practicality of such an equation, one need only consider the number of events which need not be regarded in subsequent calculations. For example, in an eight-note chord, using a or da, 28 intervals must be calculated. Using db, as we will designate it, only eight intervals have to be added. A 20-note chord would necessitate 190 calculations in d or da; only 20 with db. Its operational simplicity is of paramount importance.

In comparing the following chords (Example 23), all triadic with a diminishing span, a decreasing index of density occurs as a result of this summation process (db), as it did in the initial measurements (d and da). This system does, however, distinguish between major and minor chords of the same span.

Example 23

Using the initial density index, d, the lowest possible value is 1.0595. Are we to surmise from the minor second in the previous example that 5.8569 is the minimum possible value in the db scale? This is not so because we are dealing with an arithmetic, linear process, and accordingly, higher pitch groups will have higher indexes than transpositionally equivalent lower pitch groups. The minimum and maximum values for half-steps are listed in Example 24.



Example 24 


This system, then, differs drastically from the previous systems in its treatment of registral placement.  The system considers register and instrinsic entity in the analysis of density. The system does not respect equal-interval sonority transposition as did the other systems (i.e., a sonority will undergo a density modulation as its placement in musical space changes) (Example 25).



Example 25


The chordal series of Example 26 and 27 represent half of the possible permutations of the interval series [1], [2], [4] [7], which occur in the first chord. These chords are now arranged in a specific order - from lowest to highest - according to an inverse mean frequency ratio, d.




Example 26


Example 27

If the system is then analyzed according to db and again arranged in increasing values, a different ordering will necessarily result. This is, of course, due to the inherent differences of the disparate systems involved (Example 28).


Example 28


The four-voice sonorities from Example 1 can now be analyzed according to these various equations. The indexes will necessarily differ due to the disparate considerations of logic and the various definitions ascribed to density (Example 29).




Example 29

Through the use of these density indexes, one can choose to define density by the following means: independent of registral location (d, da); independent of the number of elements contained in the sonority (d); being the result of only its contiguous pitch relations (db); or being independent of consonance/dissonance considerations (db).


1 A paper presented at the American Society of University Composers XIV National Conference at the University of California, San Diego, March 14, 1979.

2 Intervals will often be referred to by the number of halfsteps within their span, e.g., 1121 is an octave, 171 is a perfect fifth, 141 is a major third, etc.

3 Wallace Berry, Structural Functions of Music, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1976.

4 Sole exception occurring in the case where contiguous minor seconds occur and the numerator and denominator exist in an n- 1/n or n/n-1 relationship.

5 Star events is the term used to define relationships between n events taken two at a time. R. Buckminster Fuller, Synergetics, MacMillan Publishing Co., Inc., New York, 1975, p. 60.