Study No. 41

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How did Usher arrive at this approximation of the original?

Mozart - Symphony No. 41 in C, K. 551 [complete] (Jupiter)

Based on personal communication and examination of his rather larger spreadsheet of calculations, Usher proceeded in a fairly intuitive manner, seeking simply to place each note as close as possible to its irrational placement in the original. But why these particular tuplets, and why do they often alternate?

Keep in mind that tuplets are unnecessary where the notation can be simplified. For example, the violin 2 line in measure 55 could have been notated this way: This would have made the underlying structure clearer at the expense of making the notation needlessly complicated. This is possible because any irrational number can be approximated to any degree of accuracy by rational numbers.

As the absolute value of the numerator and denominator increase, the approximation comes ever closer to the original irrational number. When deciding which rational approximation to use the trick is to balance mathematical accuracy with feasibility of performance. A polyrhythm of For any rational tempo ratio there is a higher-level periodicity, what Gann refers to as the convergence period, that results from the potential simultaneous attacks between the voices. For example, in a 7: The shorter this convergence period the easier it is for the listener to perceive, and the perception of any higher-level periodicity between the voices will work against the illusion of irrational tempo ratios, which contain no such convergence periods.

If the rhythms are syncopated, as is often the case with Nancarrow, so that the beginning of the period are seldom articulated by both voices, then this convergence period may not be apparent. It depends on the context. This is a somewhat larger convergence period, but the hypermetric pulse is still fairly prominent. Tempo ratios of In a sense the ratios Even though potential simultaneous attacks occur on beats one and three, the overall convergence period remains at three seconds.

With a variety of rhythmic durations and syncopations this slight unevenness will be extremely difficult to perceive. Passages in the arrangement that do not strictly alternate between these two types of subdivision typically occur for one of two reasons. Most commonly, an While this error 0. Readers may wish to begin by listening to the excerpt from Nancarrow's original player-piano study, given in Example 1. Audio files in Examples 2—5 use the same MIDI piano sound in order to facilitate comparison of the approximations with one another. In this case, the lower and upper ratios are 7: Despite the use of simpler ratios, this is a very convincing approximation in part due to the syncopated rhythmic figures and the fact that there are no simultaneous attacks on the beat.

Audio files of Figure 7 are given in Example 3. Hypothetical version of canon 3, approximating tempo ratio Example 3. Opening of canon 3, hypothetical version approximating tempo ratio a both voices b slower voice only 3. The goal of this approach is to minimize deviations of time points between the original and its approximated version. The main problems with this approach are that 1 the use of many different subdivisions can make performance more difficult, and 2 the approximated voice can often sound uneven, hindering the perception of a steady tempo and tempo ratio.

This unevenness can be heard in the single-voice audio file in Example 4. Hypothetical version of canon 3, approximating beat placement Example 4. Opening of canon 3, hypothetical version approximating beat placement a both voices b slower voice only 3. At the point where canon 3 is recalled system 56 , both voices of the canon are given to the top voice.

Specifically, durations between attack point in the upper part are approximated in the lower part in the follow manner: This approximated recollection is like a strange version of canon 3, one in which all traces of irrationality have been removed and yet the tempo ratio is perceived as roughly the same and the alignment of events between the two parts is undisturbed. Audio files of Nancarrow's approximation are given in Example 5. The first few durations are marked as multiples of a sixteenth note. These are moments of time in which multiple voices moving in different tempos converge on the same position in the canon line, where the echo distance is zero.

The structural properties of a tempo canon are thus determined in large part by the placements of these convergences. The first is a converging-diverging canon, in which the voices indicated by horizontal lines enter staggered from slower to faster, move toward a central convergence point indicated by vertical lines , and then recede from this point, the voices ending the reverse order of their entrance.

The following three structural diagrams are adapted from Gann The second type is a diverging-converging canon, with convergence points at the beginning and end and a tempo exchange indicated by diagonal, crossing lines at or very near the midpoint of the canon. Before this tempo exchange, the voices are diverging from the initial convergence point with the faster voice moving ahead of the slower voice; at the tempo exchange the faster voice becomes the slower and vice versa; after this exchange the voices converge toward the ending convergence with the faster voice catching up to the slower.

Study 33 consists of four canons, with canon one and three of the second type, canon 2 of the first type, and the final canon by far the longest being a combination of both types with a converging-diverging-converging pattern. Specifically, if the tempo ratio is irrational, then the voices cannot change tempo at the same time. For example, in canon 1 each voice has 27 whole-note beats at the faster tempo and 19 beats at the slower tempo. Since the ratio of these beat, Figure 10 Figure Tempo exchange, canon 1, Nancarrow 4.

Choosing the former results in bar lines after the tempo exchange that do not align with the faster voice: Choosing the latter amounts to dropping an eighth of a beat a 32 nd note in this case , yielding a much simpler rhythmic notation after the exchange: The tuplets switching from the lower voice before m.

Performing the Irrational: Paul Usher’s arrangement of Nancarrow’s Study No. 33, Canon 2:√2

In this canon, there are quarter notes in the faster tempo and quarter notes in the slower tempo prior to the tempo exchange. Figure 12 Figure Tempo exchange, canon 3, Nancarrow. In the relevant passage from Usher's arrangement, we can see that the tempo exchange is treated as happening simultaneously, the fourth eighth-note of m. Figure 14 Figure Tempo exchange, canon 4, Nancarrow. With irrational ratios this typically necessitates awkward partial measures. Figure 16 Figure Opening of canon 2, Nancarrow.

Opening of canon 2, Usher 4. Figure 18 Figure Opening of canon 4, Nancarrow. All of the mathematical and notational precision would be pointless if the Arditti was not capable of rendering the rhythmic intricacies with sufficient accuracy. However, we should not be concerned with accuracy for its own sake. Instead, what matters is if the performance of the arrangement is close enough to be perceived as a reasonable facsimile of the original player-piano study.

Thus, the two important questions are: Is each voice perceived as moving in a steady tempo and are these perceived tempos in roughly the right ratio? Do the events occur at roughly the right time and, most importantly, is the order of events preserved? Arditti string quartet recording of the opening of canon 3 5. Superimposed on each of these voice plots are their corresponding linear trend lines.

In the simplest meta-regression models, the only covariate is the time trend.

Studies for Player Piano (Nancarrow)

In later models, we include a more extensive set of predictors to control for other factors that might confound the time trend. To capture sources of variability not covered by the covariates, we use a random effects specification Random effects incorporate a variance component capturing variation in outcomes across studies that are due to unobserved study-level factors Methods and Materials.

Our core analysis focuses on studies that conducted their fieldwork from to , allowing us to observe trends in discrimination over the past 25 years. For some supplementary analyses, we also add four field experiments conducted before , although these studies use less standardized methodologies. For more detailed results, see SI Appendix , section 2 and Figs. Do we find evidence of change over time in rates of hiring discrimination? With respect to African Americans, the answer is no.

The solid line captures the trend since The dashed line extends this time trend back to , adding four resume audits conducted from to The size of the symbol is proportional to the weight it is given in the meta-analysis. The line of best fit for studies since is close to flat, sloping slightly upward, suggesting no change in the rate of discrimination over the past 25 years.

The longer time series includes studies that use a more heterogeneous set of procedures Methods and Materials , but even here we see no clear change over time in the level of hiring discrimination against African Americans. No reduction in hiring discrimination facing African Americans over time. Is there sufficient power based on 21 studies to conclude that discrimination against African Americans did not decline?

Meta-analysis of field experiments shows no change in racial discrimination in hiring over time

The confidence interval of the annual change provides a way to answer this question. If we take this number as the smallest slope consistent with the data based on the confidence interval, this suggests only a slight decline in discrimination each year. We conclude that this evidence rules out all but a slow decline in discrimination—with the most likely estimate being the point estimate, which indicates no decline in discrimination at all. However, sensitivity checks that modified the outcome sample counts slightly result in nonsignificant year coefficients of the difference in proportion or odds ratio, see SI Appendix , section 4 and Table S5.

Modest evidence of a reduction in hiring discrimination facing Latinos over time. Is it possible that key aspects of study design changed over time, influencing our estimates of changes in discrimination? To consider this question, we estimate a meta-regression model of discrimination rates as a function of a time trend plus other study characteristics. The coefficients can be interpreted as the one-year percentage change in the discrimination ratio. The second shows the annual change for the longer time period —, corresponding to the dashed line in Fig.

Alternative estimates of change in hiring discrimination against African-Americans. The next few models alter the dependent variable to see if this changes our results using our base sample of — This makes the outcome variable less uniform across studies, although closer to the outcome of greatest substantive interest, getting a job. This limits the applicant profiles to those with more mainstream job backgrounds and credentials.

The modified results show the trend line slanting slightly more upward, providing less evidence to support a downward trend than the results including a more heterogeneous set of applicant characteristics. A third modification uses only resume audit studies, discarding in-person audits.

The next estimates are based on models that add controls for applicant attributes, region and area unemployment rates, and occupational categories to the baseline time-trend model. Finally we present the coefficient from a trimmed model in which only the predictors with the largest t ratios from prior models are included. In each case, we see coefficients for the time trend that are close to zero—ranging from an estimated increase of 0.

Notably, then, we find evidence of stability, not change, in hiring discrimination against African Americans. Few of the measured covariates in our analysis SI Appendix , Table S4 demonstrate a clear relationship to patterns of discrimination. However, even looking at the point estimates we find no large differences in magnitude across categories. This result is consistent with the findings within individual audit studies that suggest relative stability in measured discrimination across job types, applicant gender, and skill levels e.

We also note that a meta-analysis designed to look specifically at effects of many of these covariates would use within-study variability—such as contrasting male and female auditors in the same study—which could provide more power to discern effects. Within-study variability cannot be applied to understand change over time since studies are generally conducted over the span of just a few months.

As a final check on the influence of covariates, we tested for time trends among our study-level and individual-level characteristics, finding no evidence of systematic change SI Appendix , section 9 and Table S This suggests that covariates are unlikely to influence the observed time trend for discrimination among either the African-American or Latino samples. In relation to our estimate of changes in discrimination over time, the inclusion of study-level and applicant-level characteristics has little impact.

In all models, we see little evidence of a reduction in hiring discrimination against African Americans over time. A potential concern of any meta-analysis is publication bias. In the present case, publication bias may entail studies that show no discrimination being less likely to be published and, thus, included in our study. Their inclusion did little to affect our estimates. Finally, in SI Appendix , section 5 and Table S7 , we show that studies in which racial discrimination was the focus of the analysis and for which there may be more pressure to demonstrate a positive effect show no more discrimination than studies in which other characteristics were the main focus with race included as an secondary or incidental covariate , further reducing concerns over publication bias for our results.

Contrary to widespread assumptions about the declining significance of race, the magnitude and consistency of discrimination we observe over time is a sobering counterpoint. We note that our results do not address the possibility that hiring discrimination may have substantially dropped in the s or early s, during the civil rights era when many forms of direct discrimination were outlawed, as some evidence suggests 1. Further, we note that our results pertain only to discrimination at the point of hire, not at later points in the employment relationship such as in wage setting or termination decisions.

Likewise, from an accountability standpoint, discrimination is less easily detected, and therefore less costly to employers, at the point of hire It may be the case, then, that more meaningful reductions in discrimination have taken place at other points in the employment relationship not measured here.

What our results point to, however, is that at the initial point of entry—hiring decisions—African Americans remain substantially disadvantaged relative to equally qualified whites, and we see little indication of progress over time. These findings lead us to temper our optimism regarding racial progress in the United States.


At one time it was assumed that the gradual fade-out of prejudiced beliefs, through cohort replacement and cultural change, would drive a steady reduction in discriminatory treatment At least in the case of hiring discrimination against African Americans, this expectation does not appear borne out. We find some evidence of a decline in discrimination against Latinos since More evidence is needed to establish the trend in hiring discrimination against Latinos with greater certainty. Our results point toward the need for strong enforcement of antidiscrimination legislation and provide a rationale for continuing compensatory policies like affirmative action to improve equality of opportunity.

Discrimination continues, and we find little evidence in regards to African Americans that it is disappearing or even gradually diminishing. Instead, we find the persistence of discrimination at a distressingly uniform rate. We discuss each of these steps in turn. We aimed to include in our meta-analysis all existing studies, published or unpublished, that use a field experimental method and that provide contrasts in hiring-related outcomes between different race and ethnic groups in the United States.

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This includes both in-person audit studies and resume studies or correspondence studies. We also required that contrasts of hiring outcomes between race or ethnic groups were made for groups that were on average equivalent in their labor market relevant characteristics, since otherwise discrimination estimates are confounded with the difference in nonracial characteristics. We used three methods to identify relevant field experiments: We began with a bibliographic search. Our search covered the following bibliographic databases and working paper repositories: Our second technique for identifying relevant studies relied on citation search.

Working from the initial set of studies located through bibliographic search, we examined the bibliographies of all review articles and eligible field studies to find additional field experiments of hiring discrimination.

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The last technique used was an email request of authors of existing field experiments of discrimination. From our list of audit studies identified by bibliographic and citation search, we compiled a list of email addresses of authors of existing field experiments of discrimination. To this we added the addresses of authors of literature review articles on field experiments.

Our email request asked for citations or copies of field discrimination studies published, unpublished, or ongoing. We also asked that authors refer us to any other researchers who may have recent or ongoing field experiments. The email requests were conducted in two phases. In the initial wave, apparently valid email addresses were contacted. We received 56 responses.

We also sent out a second wave of 68 e-mails which consisted of additional authors identified from the initial wave of surveys and some corrected email addresses. We received 19 responses to this second wave of email surveys. Overall, our search located 34 studies that were US-based field experiments of hiring, included contrasts between white and nonwhite applicant profiles that were on-average equivalent in their labor-market relevant characteristics e. Six studies were excluded for various reasons, as explained in SI Appendix , section 6.

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Our remaining 28 studies yielded 24 estimates of discrimination against African Americans and 9 against Latinos relative to whites. We coded key characteristics of the studies into a database for our analysis. Coding was based on a coding rubric, which listed each potentially relevant characteristic of the research and included coding instructions. To develop the rubric, we initially read several studies and, based on this, developed an initial coding rubric of factors we thought might influence measured rates of discrimination.

The initial rubric was reviewed and updated by all authors of this study for completeness. It was subsequently refined as coding progressed. Each study was coded independently by two raters, with disagreement resolved by the first author. See SI Appendix , section 7 for more discussion of coding procedures. Studies have fieldwork periods range from to for African Americans and to for Latinos. For most analyses in this paper, we focus on the period — We focus on this period because the data are sparse before this period only four studies before and because our reading of the early studies indicates key methodological differences among these early studies that may affect their results.

Resume audits typically signal race by using race-typed names on resumes, but the pre studies either indicated race directly on the resume [McIntyre et al. Excluding the early studies leaves us with 21 estimates of discrimination against African Americans and nine against Latinos from 24 studies six studies include estimates of discrimination against both African Americans and Latinos. A meta-analysis aggregates information from across studies to produce an estimate of an effect of interest In this study, our basic measure of discrimination is the discrimination ratio.

This is the ratio of the percentage of callbacks for interviews received by white applicants to the percentage of callbacks or interviews received by African Americans or Latinos. Ratios above 1 indicate whites received more positive responses than African Americans or Latinos, with the amount above 1 multiplied by indicating the percentage higher callbacks for whites relative to the minority group.

Because audit studies equate groups on their nonracial characteristics either through matching and assignment of characteristics in-person audits or through random assignment most resume audits , no further within-study controls are required. SI Appendix , section 8 discusses potential alternative measures of discrimination using the difference in proportions and the odds ratio, and presents alternative results using these measures. Our basic result—no decline in discrimination against African Americans over time—holds using both of the alternative measures, whereas evidence of a decline in discrimination for Latinos appears somewhat stronger with the difference in proportions or the odds ratio.

The goal of a meta-analysis is to combine information across studies.

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This requires measuring the information each study contains about discrimination against a group. The information each study provides is inversely proportional to the square of the SE of the discrimination ratio. We calculate the SE of the ratio from counts reported in each study, accounting for audit pairs in the design when possible.

In cases where information on paired outcomes is available from the study counts of pairs in which both the white and the nonwhite tester receive a callback, white yes nonwhite no, white no nonwhite yes, neither get a callback , we calculated SEs of discrimination ratios accounting for the pairing see SI Appendix , section 9 for details and formulas. For studies that are not paired between whites or nonwhites or where paired outcomes are not reported, we use formulas for the SE for unpaired groups.

This formula will slightly overestimate the SE of the effect for studies that are paired but we treat as unpaired due to lack of information about the outcomes at the pair level, underweighting these studies a bit in computing the overall effect, and slightly inflating the overall cross-study SE. Of course field experiments vary in their characteristics, such as the geographic area they cover, the exact job sectors covered, and details of their methodology.

To account for this variability in understanding the time trend, we use two procedures. First, we include controls, discussed further below, for many study characteristics. Second, to capture sources of variability not covered by the covariates, we use a random effects specification Random effects incorporate a variance component capturing variation in outcomes across studies that are due to unobserved study-level factors.

Random effects are recommended whenever there is reason to believe that the effect in question is likely to vary as a function of design features of the study, rather than representing a single underlying effect that is constant over the whole population. This is surely the case in our analysis, as we expect that the level of racial discrimination may depend on the year of the study, the situation the study considers e.

The random effect increases the SEs of estimates to correctly account for variabilities among studies in drawing inferences about overall trend. If y i is the discrimination ratio in the i th study, then the meta-analysis model is as follows: Following standard practice in the meta-analysis literature, we log the response ratio to reduce the asymmetry of the ratio.

Meta-regression allows that the rate of discrimination is a function of a vector of k characteristics of the studies and effects, x , plus in the random effects specification residual study-level heterogeneity between study variance not explained by the covariates. The model assumes the study-level heterogeneity follows a normal distribution around the linear predictor: