Chapter 7
Conditional Selection II: Cumulative Feedback

The typescript for chapter 7 is provided in AutomatedComposition07.pdf.

“Cumulative feedback” is the term my 1984 typescript uses to describe a technique of keeping resources in balance. My 1986 article, “Two Pieces for Amplified Guitar” rechristened the same technique “statistical feedback”. This technique conforms a population of choices to a weighting scheme or distribution. As contrasted with weighted random selection, however, statistical feedback is fundamentally determinate. There are three variants, a special method for single choices (e.g. choosing pitches for notes), a general method for choosing aggregates (e.g. choosing between chords), and a balanced-bit generator designed to be swapped into situations which would otherwise employ standard random number generation.

Special Method

The special method of statistical feedback was first employed in my 1983 computer-composed piece, Undulant for seven instruments. The mechanics of generating this piece were explained in my Interface article, “Notes on Undulant”. The article relates that the each instrument's total playing time is approximately the same, and the profile of which instruments play when (Figure 1 on p. 508) shows that the various ensemble sizes are spaced fairly evenly over the total duration. For example, every fourth section is a trio with just three exceptions — and those exceptions average out. What the article does not explain is that whether a section should be a solo, duet, trio, etc. was decided by which ensemble size had the smallest cumulative duration up to that point (subject to constraints). Which instrument should play during which sections was decided in a second pass, and the criterion was likewise which instrument(s) had the smallest cumulative playing time (again subject to constraints).

General Method

Lejaren Hiller's composition Circus Piece, one of the suite of projects created for the U.S. Pavilion at Expo '85 in Tsukuba, Japan, used general statistical feedback to select musical fragments, which were assembled into a pastiche. To present this, Bob Franki developed animated graphics, which popped up an image of each fragment as it was played in performance. A snapshot of this graphic was reproduced in Perspectives of New Music. Further implementation details about the composing program developed by myself according to Hiller's specifications (only it was my idea to employ statistical feedback in place of free randomness) appear in the 1985 ICMC Proceedings. A Kurzweil 250 realization of Hiller's composition is available on this site in MP3 format.

The general method of statistical feedback was subsequently implemented in COMPOSE, an application in the tradition of Koenig's Project Two which allowed non-programmers to assemble compositional processes from software units. The unit type AGGREGATE applied the minimax principle to selecting collections of elements by favoring collections whose worst-case statistics were minimal. AGGREGATE is presented in “Quantifying Musical Merit” as an example of a “uniform merit function”, and the method's mechanics are explained on p. 66. The general method of statistical feedback is the only heuristic technique explicitly implemented within my 2011 production framework, and it was upon this framework that this site's Statistical Feedback demonstration was built.

Balanced-Bit Generator

The third variant of statistical feedback is the balanced-bit generator. This generator is described on p. 61 of my Catalog of Sequence Generators where its ability to produce uniformity in the short term is illustrated through histograms. I assert here that a sequence of N balanced-bit samples will be as evenly spaced from zero to unity as N samples can be, and that if you space N values from zero to unity and then shuffle them randomly, the result will be no better. Balanced-bit sequences would seem to provide superior performance in applications of fuzzy logic, although I doubt anybody working in that field is aware of this technology.

Commentary

Wikipedia defines a heuristic as a “practical method not guaranteed to be optimal or perfect, but sufficient for the immediate goals.” In my writing about composing programs, the word heuristic refers more specifically to a decision-making technique by which the relative merit (alternatively demerit) of each available option is expressed quantitatively. This makes it possible to rank the options by preference. The options may also be winnowed through the enforcement of constraints. Among the options which survive winnowing, the option with the most merit (or alternatively the least demerit) is the option selected.

Statistical feedback is an exemplar of heuristic selection. It assumes a sequence of decisions which all draw from the same supply of options. Here the merit accorded to an option reflects how far that particular option lags behind its fair share of usage.

The selection model adopted in this chapter resembles Project Two to the extent that decisions pertaining to a specific musical attribute all draw from the same supply of options. However, Project Two has no selection feature implementing statistical feedback.

We return one more time to the pitch-selection scenarios begun in the summary for Chapter 4. Suppose we wish to select from the pitch supply E4, F4, G4, A4, B4, C5, and D5, that we wish the distribution of pitches to conform to the now-familiar values in Table 1, and that we also wish to to continue observing the constraints developed in the Markov scenario for Chapter 6

I have coded a process using the method of statistical feedback to select pitches according to the weights provided by Table 1 and subject to the constraints enumerated in Table 2. Table 3 details quantities calculated during this process, and used to decide which pitch to choose next. The information is limited to the first twelve sequence elements, but that is well enough to get the idea across.

Position	Source	Values	E4	F4	G4	A4	B4	C5	D5	Selection
		Weights	0.085	0.155	0.174	0.173	0.174	0.155	0.085
0		Usage	0	0	0	0	0	0	0	G4
		Preference	12.64	6.723	5.872	5.979	6.310	6.462	12.91
1		Usage	-0.822	-0.822	4.931	-0.822	-0.822	-0.822	-0.822	A4
		Preference	12.06	6.248	11.22	5.194	5.131	5.826	11.55
2	G4 A4	Usage	-1.648	-1.648			-1.648	-1.648	-1.648	B4
		Preference	10.99	4.965			4.471	5.165	10.83
3	A4 B4	Usage	-2.470	-2.470	3.283			-2.470	-2.470	F4
		Preference	9.630	4.071	9.120			4.003	9.950
4	B4 F4	Usage			2.360					G4
		Preference			8.683
5	F4 G4	Usage	-4.215			1.571	1.538	-4.215	-4.215	C5
		Preference	8.441			7.782	7.761	2.785	8.182
6	G4 C5	Usage					0.616			B4
		Preference					6.780
7	C5 B4	Usage	-5.959	0.499	5.547	-0.173			-5.959	A4
		Preference	6.363	6.960	11.34	5.815			6.216
8	B4 A4	Usage	-6.786	-0.328	4.720			-0.328	-6.786	D5
		Preference	5.247	6.682	10.48			6.578	5.899
9	A4 D5	Usage						-2.010		C5
		Preference						4.703
10	D5 C5	Usage	-9.391	-2.933	2.115	2.182	2.115			E4
		Preference	3.312	4.167	8.376	8.066	8.421
11	C5 E4	Usage		-4.615						F4
		Preference		2.394

Table 3: Pitch-selection calculations used by the method of statistical feedback.

• Direct your attention to the row for sequence element #0. This is the topmost data row of Table 3. Notice the set of values labeled “Usage”. These are floating-point statistics, one for each pitch, whose purpose is to track how often the pitch has been used up to the present moment. Since no pitches have been used coming into the decision for element #0, all statistics are initially zero. The formulas which follow will employ the notation S_p to indicate the usage statistic for pitch p.
• Beneath the “Usage” values lie a set of “Preference” values. How these values are calculated will be explained shortly; however, for elements #0 and #1 it doesn't matter because the first two choices, G4 and A4, have been forced upon the process. This fact is acknowledged for element #0 by the fact that the preference value for G4 is highlighted in red.
• Direct your attention to the row for sequence element #1. Notice the set of values labeled “Usage” have changed. Two steps have been undertaken.
  • The first step is to increment S_G4 to record the fact that G4 has now been used. The calculation is:
    

    SG4,1   =   SG4,0   +

    A wG4

    =   0   +

    1 0.174

    =   5.747

    The symbols in this calculation are as follows: A is the “accent” associated with an act of pitch selection. Under the present scenario all acts are equally accented, so A can be fixed at unity. H is the “heterogeneity” factor, whose value is fixed at 0.1. The symbol w_G4 is the weight associated with pitch G4, whose value is provided by Table 1.
  • The second step is to normalize S_E4, S_F4, S_G4, S_A4, S_B4, S_C5, and S_D5 so that these always center around zero. I worked out a complicated way of doing this, but have since adopted the much more direct method suggested by Nick Collins. This direct method calculates the unnormalized (nonzero) average of the usage statistics, which in the present circumstance works out to 0.822. This average is then subtracted from every statistic. Hence usages for unselected pitches change to -0.822, while the usage for G4 changes to 5.747 - 0.822 = 4.931.
• Highlighting the preference value for A4 in red acknowleges the fact that the selection of A4 for element #1 was forced.
• The row for element #2 present neither “Usage” nor “Preference” values for pitches G4 and A4, since Table 2 prevent either of these pitches from following on when the source is G4 A4.
• Since the process actually begins paying attention to “Preference” values from element #2 onward, the time has arrived to explain that the preference value when considering pitch p for sequence element k, the preference value symbolized Θ_p,k is calculated as:
  

  Θp,k   =   Sp,k-1   +

  (A + H*up,k) wp

  Where the symbols A, H and w_p are as previously defined, and u_p,k is a random value which is uniformly distributed between zero and unity. When considering pitch B4 for element #2, with the random value u_B4,2 = 0.647, this calculation works out to:

  ΘB4,2   =   -1.648   +

  (1 + 0.1*0.647) 0.174

  =   4.471

• Highlighting the preference value for B4 in blue acknowleges the fact that 4.471 is the smallest preference value calculated for element #2, and that B4 is indeed the pitch selected for this sequence element.
• The process continues in this manner until a total of 48 pitches have been selected.

Position	Length	Content	Summary	E4	F4	G4	A4	B4	C5	D5
0	48	G4 A4 B4 F4 G4 C5 B4 A4 D5 C5 E4 F4 G4 B4 A4 C5 B4 F4 G4 A4 E4 F4 D5 C5 G4 A4 B4 C5 F4 G4 A4 B4 D5 C5 E4 F4 G4 B4 A4 F4 G4 C5 B4 A4 E4 F4 D5 C5	Counts	4	8	8	8	8	8	4
			Proportions	0.0833	0.167	0.167	0.167	0.167	0.167	0.0833
0	12	G4 A4 B4 F4 G4 C5 B4 A4 D5 C5 E4 F4	Counts	1	2	2	2	2	2	1
			Proportions	0.0833	0.167	0.167	0.167	0.167	0.167	0.0833
6	12	B4 A4 D5 C5 E4 F4 G4 B4 A4 C5 B4 F4	Counts	1	2	1	2	3	2	1
			Proportions	0.0833	0.167	0.0833	0.167	0.250	0.167	0.0833
12	12	G4 B4 A4 C5 B4 F4 G4 A4 E4 F4 D5 C5	Counts	1	2	2	2	2	2	1
			Proportions	0.0833	0.167	0.167	0.167	0.167	0.167	0.0833
18	12	G4 A4 E4 F4 D5 C5 G4 A4 B4 C5 F4 G4	Counts	1	2	3	2	1	2	1
			Proportions	0.0833	0.167	0.250	0.167	0.0833	0.167	0.0833
24	12	G4 A4 B4 C5 F4 G4 A4 B4 D5 C5 E4 F4	Counts	1	2	2	2	2	2	1
			Proportions	0.0833	0.167	0.167	0.167	0.167	0.167	0.0833
30	12	A4 B4 D5 C5 E4 F4 G4 B4 A4 F4 G4 C5	Counts	1	2	2	2	2	2	1
			Proportions	0.0833	0.167	0.167	0.167	0.167	0.167	0.0833
36	12	G4 B4 A4 F4 G4 C5 B4 A4 E4 F4 D5 C5	Counts	1	2	2	2	2	2	1
			Proportions	0.0833	0.167	0.167	0.167	0.167	0.167	0.0833

Table 4: A sequence conforming to the weights detailed Table 1.

Results from this process are presented in Table 4. The first row of this table presents a full sequence of 48 pitches with an accompanying statistical analysis. Subsequent rows analyze 12-member subsequences. The results presented here bear comparison with the most statistically conformant pitch-selection results obtained in previous chapter summaries. This happened in the summary for Chapter 5 covering the method of statistical frames.

The first thing to notice is that the counts for the 12-member subsequences never deviate by more than one from the pattern 1, 2, 2, 2, 2, 2, 1. This was the count pattern that characterized each statistical frame in the Chapter 5 results. But something more subtle is going on here:

• Firstly, the proportions given for the full sequence in the top row exhibit a tapering off toward the boundary pitches. The proportion for F4 is intermediate between the proportions for E4 and G4, while the proportion for C5 is intermediate between the proportions for B4 and D5.
• Secondly, the counts for subsequences beginning in positions 6, 18, and 30 conform as closely to the pattern 1, 2, 2, 2, 2, 2, 1 as do the counts for subsequences beginning in positions 0, 12, 24, and 36. There is no cusp!

Remember that the present process selected pitches while all along observing the constraints enumerated in Table 2, while the process in Chapter 5 was subject to no such rigor.

I regard statistical feedback as an essential technology in the sense that it achieves conformance which is superior even to selection without replacement and does so even under the duress of constraints. Indeed, I have come to regard the method as indispensible. Every composing program I have developed since Undulant makes use of it. Beyond that, I have taken some trouble to promote the method. Not the least of my efforts was to submit the article “Statistics and Compositional Balance” to that bastion of modernist compositional ethic, Perspectives of New Music. I bolstered my case with studies showing how statistical feedback could realize distributions with much smaller population sizes than those required by weighted randomness; these studies appeared a two-part article in the Leonardo Music Journal: “Thresholds of Confidence: An Analysis of Statistical Methods for Composition,” “Part 1: Theory” and “Part 2: Applications”.

Demonstration

The practical programming content of Chapter 7 was provided by Demonstration 5: Cumulative Feedback. The outer loop of this program employs the method of cumulative feedback to select attributes for phrases: phrase duration, average note duration, articulation, prime scale degree, and register. The inner loop of the program uses the method of cumulative feedback to select a chooser for a play-rest trial. It again uses the method of statistical frames to select an unscaled note duration (to be multiplied by the average note duration), and (for notes) a deviation around the central pitch.