Theoretical and experimental explorations of the Bohlen–Pierce scale

Abstract

A musical scale based on the 3:5:7:9 tetrachord is described. It has clearly audible harmonic properties that are derived from the harmonies of the tetrachord. In its equal-tempered version, the nine tones of the scale are selected from a set of 13 equal steps, which have a frequency ratio equal to 31/13. The tenth note of the scale, here called the tritave, has a 1:3 frequency ratio with respect to the first note and has a role similar to that of the traditional octave. Both ‘‘major’’ and ‘‘minor’’ chords can be formed from the notes of the scale. Musicians and untrained listeners rated the consonance of all possible triads formed from a 1-tritave range of the equal-tempered chromatic scale. A wide range of consonance ratings was observed, and chords were judged as most dissonant when they had 1-step intervals. A critical bandwidth dissonance model fits the data well. The same subjects rated the consonance of a harmonized passage played in different tunings. Both groups judged the equal-tempered version as most consonant, but only the musicians were influenced by previous exposure to these sounds. Subjects also judged the similarity of chords and their inversions, both for chords from the new scale and for traditional chords. Musicians were influenced by key relationships, inversions, and chord type in their ratings of the traditional chord set but judged the new chords only by pitch height. Untrained listeners relied on pitch height for both chord sets. This suggests that the ability to abstract more complex information depends on training.

Humans Rapidly Learn Grammatical Structure in a New Musical Scale

“…This scale was first proposed by Heinz Bohlen in the early 1970s and uses a set of mathematical relationships that are designed to be different from Western music, while still giving rise to a sense of tonality. Thus, it has been adopted by various contemporary composers and has even received some interest in music cognition and psychoacoustics (Krumhansl, 1987; Mathews, Pierce, Reeves, & Roberts, 1988; Sethares, 2004). …”

Construction and interpretation of equal-tempered scales using frequency ratios, maximally even sets, and P-cycles

“…For comparison, the equal-weight generalized desirability for all three ratios is shown. cardinality of dϭ9 ͑in accordance with the modulation properties of unidirectional P-cycles͒ for the sequence of ratios 3:5:7:9 ͑chosen by previous workers 28,35,36 for acoustic and perceptual reasons͒, with chromatic generators of 6 ͑or 7͒, 4 ͑or 9͒, and 3 ͑or 10͒, for an equal-tempered system with base 3 closure, we are ready to construct the ''diatonic'' scale and chord structure for the system. The ME algorithm for c ϭ13, dϭ9, and nϭ4 yields the following for a typical scale: This scale is shown by the filled circles in the first diagram of Fig.…”

“…Over the last 65 years other scale structures that have the properties of or related to unidirectional P-cycles have been constructed by Yasser, 19 Mendalbaum, 12 Chalmers, [20][21][22] Wilson, 23 Gamer, 24 Balzano, 25 Clough and Myerson, 26,27 Mathews, Pierce, Reeves, and Roberts, 28 Agmon, 29 Clampitt, 16 Clough and Douthett, 17 Clough, Cuciurean, and Douthett, 30 and Zweifel. 31 In this discussion we adopt much of the terminology from the seminal works of Clough and Myerson.…”

Expanded chromas and the new frontiers of tonality: An introduction

“…This should also give a hint on why a value of B = 2.10, for instance, would not be likely to produce expanded chroma. Finally, to those readers who are interested in our references, notice how the article by Mathews et al (1988) focuses on consonance (a perceptual attribute) and not in the concept of functional similarity (a cognitive attribute); hence their emphasis on overtone structure concerning the scale A-13, and hence our lack thereof concerning expanded chroma.…”

“…In the first attempt, Bohlen (1978), in Downloaded by [Nanyang Technological University] a study on consonance based on combination tones proposes the scale A-13 and calls the exponential, generating interval Duodezeme. Alternatively, Mathews et al (1988) arrive at the same scale quite independently from Bohlen as a result of their intonation sensitivity studies for chords with frequency ratios 3:5:7:9. They call the generating interval "tritave".…”