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Abstract: Systematic deviations from the equally tempered values were observed while tuning the frequency of a pure tone up and down against the standard frequency, to tune up 12 musical intervals within an octave in randomly selected order. The standard frequencies were 125, 250, 500, and 1000 Hz. Four trained music students served as subjects and the total number of observations per point was not less than 80. It has been shown that small intervals are tuned smaller and large intervals larger than it could be predicted from the equally tempered scale. The observed octave enlargement effect is in agreement with the previously published data.

Abstract: Five performance sections from four different Central Australian songlines with a melodic range larger than an octave have been analysed. Although octave ratios exist, they are limited to one octave interval in each songline: only the finalis has an ‘octave’ counterpart, other intervals in the upper melodic range are shown to be linear shifts of intervals in the lower range. There is evidence that the octave ratio is not treated as an octave in the common musical sense (octave identity) but much more as a specific ratio (as part of the characterization of the songlines). This helps considerably in understanding of how a musical system based on intricate patterns of linear (frequency difference) intervals can, at the same time, contain ratio intervals as essential elements without contradictions: this music does not seem to know a generalized principle of octave identity.

Abstract: A sound composed of the harmonics of 131 Hz produces the note C3 with abuzzy timbre if the components have equal amplitude and are in cosine phase. Notes with the same tone chroma and a tone height between C3 and C4 can be produced by 1) attenuating the lower harmonics of the sound, 2) attenuating the odd harmonics, or 3) shifting the phase of the odd harmonics. The effects of the manipulations were measured in an octave experiment: a note was chosen at random and used to play a brief melody on the notes around C; the octave varied from C1 to C6 and the listeners judged the octave of each melody on a scale from C0 to C7. The results show that waves with the same period can lead to average octave judgements that differ consistently by more than half an octave, and that a substantial component of many timbre differences (e.g. that between a piano and a harpsichord) is actually a tone-height difference. The effects of manipulations 2) and 3) are difficult to explain with traditional hearing theories because ...

Abstract: Using three criteria a desirability function is developed which allows comparison of the reasonableness of equal‐tempered musical scales to approximate just musical intervals. Unlike most other measurements, this scale is not dominated by very fine octave divisions. This feature allows for comparison of systems with few divisions to the octave with those with many divisions to the octave. Based on our measure, an equal‐tempered scale which must best represent the pure fifth requires 12, 41, or 53 notes to the octave. If the major third is to be well represented, equal‐tempered scales of 28 or 59 notes are required. Best representation of the minor third occurs for a 19‐note equal‐tempered scale. These results are shown to be in accordance with previously cited continued fraction results and other published evaluation criteria. The method is generalized to assess the reasonableness of equal‐tempered scales to approximate multiple just intervals simultaneously as well as to account for numerical weighting of intervals. Our results show that an equal‐tempered scale required to approximate the pure fifth, major third, and minor third equally requires 12, 19, 34, or 53 notes. This result is contrary to previously cited results in which 34 notes to the octave is relegated to a relatively minor role.