After producing the interval, they were shown both their produced interval, and the target interval, as two dots on a one-dimensional scale, as well as a numeric representation (e.g. produced interval "2.4 seconds", target interval "2.0 seconds").
Solving the pair of geometric programs produces the interval of the objective value.
Immediately after hearing the tone, participants had to produce the interval relating to the tone (one second, two seconds, or four seconds) (Fig. 6).
The scale that Philolaus adopts is such that the ratio of the highest to the lowest pitch is 2 : 1, which produces the interval of an octave.
Conversely, the algorithm would use 10∶9 rather than 9∶8 to form an interval with a major sixth (5∶3) because this choice produces the interval (3∶2 versus 40∶27) with the higher percentage similarity.
For example, if 9∶8 or 10∶9 represented the second scale degree of a scale being tested (these two intervals are within 22 cents of each other), the algorithm would use 9∶8 rather than 10∶9 to form an interval with a perfect fifth (3∶2) because this choice produces the interval (4∶3 versus 27∶20) with the higher percentage similarity.
Both groups tended to under-produce the 1,000 ms compared to control children (who tended to over-produce the interval).
Ever since the ancient Greeks, scientists and music theorists have been intrigued by the way simple mathematical ratios can produce the intervals that make up the musical scale.
The sixth one produced the comparison interval that was either shorter or longer than the standard (Fig 1B).
[3,4-C]glucose produced the smallest confidence interval, displaying a 5-fold improvement compared to [U-C]glucose.
