Undergraduate music majors (N = 27) identified simple musical intervals (m2 through M7), presented at I 0 different pitch levels and in three different presentation modes (ascending, descending , harmonic) . Resulting error matrices were analyzed by direct inspection, repeated measures ANOVA, and multidimensional scaling (MDS). Minor 6ths were the most difficult to identify ; sizes of larger intervals were systematically underestimated; and an interaction of several factors including interval type and acoustical disso nance appeared to shape error rates . ANOV A found no effect for pitch level but a significant effect for presentation mode, with ascending intervals easiest and harmonic intervals the most difficult to identify . A three-dimensional MDS configuration was obtained, indicating an interaction of interval size, interval type , and class of acoustical dissonance. The "classical" interval classes of pitch class set theory can be derived from a particular planar projection of the configuration .

Trained musicians rated the similarity of 24 instances of set classes [0137/0467] and [0157/0267] at three different transposition levels and two different spacing types. Stringent criteria for retention of participants, to ensure greater reliability and predictive power, resulted in a final count of 30 participants. Participants' data were analyzed by using multidimensional scaling and additive tree methods. A three-dimensional multidimensional scaling solution showed clear effects for spacing type and transposition level (Tn). Additive tree analysis showed no grouping according to set-class type, but rather according to pitch height of inner voices or ordered location of semitones. The results are consistent with the hypothesis that even musicians with significant experience of atonal music do not use the equivalence relation TnI in making similarity judgments.

The methodology used in Daniel Perttu's article is analyzed for conformance to several criteria needed in quantitative studies. A number of problems are identified. Some of these appear to be deep structural issues given the nature of the question studied while others are caused by the methodology itself, by both the types of analyses carried out and the nature of the data source. Various suggestions to strengthen the study are made. (2007) takes on one of the "received bits" of (folkloric) music wisdom: that at least since 1600, Western art music has been becoming increasingly chromatic. He takes a large sample of melodic data from Barlow & Morgenstern's (1948; henceforth, B+M) catalogue of themes and carries out several tests to check the proposition on both inter-and intra-composer levels. His use of statistical analysis is most welcome-this is exactly the type of question in historical musicology that is amenable to the use of quantitative methods, and it is delightful to see someone applying them. Unfortunately, an analysis of his methodology shows several problems, so that for the present we must render the Scottish verdict of "not proven." The case is not completely closed, however: a reworking of the methodology might tighten things up sufficiently that we could accept the assertion with confidence. (For the record, I hope that he succeeds, since my own musical intuition agrees with many others that the proposition is true.) REQUIREMENTS FOR QUANTITATIVE ANALYSISA couple of paragraphs on the nature of statistical analysis are needed for readers not well-versed in the subject for them to follow fully much of this critique. Those who already know something of statistics can skip ahead.To successfully apply inferential statistics to a question requires that three criteria be satisfied: 1) Was the question for which data were gathered well-defined? 2) Were there any problems with the data gathered that could bias any test results? and 3) Were appropriate tests applied? Criterion one is straightforward: if your hypothesis is not well-formulated (and by extension, your definition for data), there is no way to check that you are gathering relevant information. Criterion two is more complex in the details but still conceptually straightforward. If one went to take "an opinion poll of Canadians" but only questioned people in Toronto, then the results would almost certainly be unrepresentative of the overall population-a trivial example, but the same principle applies for far less obvious situations. Perttu's article is a case study of some of the subtleties involved.Criterion three is the most technical and which makes many people run in horror at the mere word "statistics," but it is still simple in origin. Inferential statistics is always trying to answer a basic question: do two or more groups differ with respect to some characteristic? That question is very complicated to answer because data can vary so dramatically in make-up, and each individual statistical test must make particular assumptions ...

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