Table 1.
Terminology for periodic phenomena.
Table 2.
ASHLE parameters and function.
Fig 1.
Illustration of the musical tasks and corresponding simulation experiments.
(A) The task simulated in experiment 1, in which a musician plays a simple melody with a metronome (top). Illustration of our simulation, in which ASHLE synchronizes with a sinusoidal stimulus (bottom). (B) The task simulated in experiment 2, in which a musician plays a simple melody, without a metronome (top). This specific example shows a performance tempo that periodically became slower due to the musician’s tendency to return to the SMT. Illustration of our simulation, in which ASHLE oscillates without a sinusoidal stimulus and returns to its f0 (bottom). (C) The task stimulated in experiment 3, in which pairs of musicians played a simple melody together after hearing four pacing metronome clicks (top). Illustration of our simulation, in which two ASHLE models synchronize with four cycles of a pacing sinusoidal stimulus (greyed-out blue and red lines), and then stimulate each other without the sinusoidal stimulus (solid blue and red lines) (bottom).
Fig 2.
Simulation of the MA between a musician’s beat and a metronome beat with a period shorter or longer than the musician’s SMP during solo musical performance.
(A) The mean adjusted asynchrony (and standard error; N = 20) between the musician beat and metronome beat during performance of a simple melody in four conditions: metronome period 30% shorter (F30), 15% shorter (F15), 15% longer (S15), and 30% longer (S30) compared to the musician SMP. The x-axis shows “F” and “S” labels as originally used by Scheurich et al. [25] to describe a “faster” and “slower” metronome compared to the SMT. (B) Our simulation results showing the mean adjusted asynchrony (and standard error; N = 20) between ASHLE and a sinusoidal stimulus in six conditions: stimulus period 45% shorter (F45), 30% shorter (F30), 15% shorter (F15), 15% longer (S15), 30% longer (S30), and 45% longer (S45) than the period of ASHLE’s f0. The shaded bars represent predicted measurements for data that has not been collected yet from musicians. (C) Mean adjusted asynchrony predictions when ASHLE models with different f0 periods synchronize with a stimulus period that is 45% shorter (F45), 30% shorter (F30), 15% shorter (F15), 15% longer (S15), 30% longer (S30), or 45% longer (S45).
Fig 3.
Simulation of the slope between consecutive IBIs when an unpaced musician performs a melody starting at a tempo that is different than the SMT.
(A) The mean adjusted slope of consecutive IBIs (and standard error; N = 24) when solo musicians perform a simple melody starting a tempo that is fast, faster, slow, or slower compared to their SMT. (B) Our simulations showing the mean adjusted slope of consecutive IBIs (and standard error; N = 23) when ASHLE oscillates, without a stimulus, starting at a frequency that is fast, faster, slow, or slower compared to its f0. (C) Adjusted slope predictions when different ASHLE models with different f0 oscillate without stimulation, starting with a period that is 45% shorter (F45), 30% shorter (F30), 15% shorter (F15), 15% longer (S15), 30% longer (S30), or 45% longer (S45) compared its to the period of its f0. For consistency with predictions made in experiment 1, here we also use the F and S on the x-axis.
Fig 4.
Simulation of the mean absolute asynchrony between two musicians with matching or mismatching SMTs during duet musical performance.
(A) The mean absolute asynchrony (and standard error; N = 10 per experimental group) between two musicians with matching or mismatching SMTs during performance of a simple melody four consecutive times. (B) Our simulation results showing the mean absolute asynchrony between two synchronizing ASHLE models with f0 values that are close or far from each other. (C) Mean absolute asynchrony predictions when different ASHLE models (with different f0 periods) synchronize with another ASHLE model with a f0 period that is 220ms shorter, 110ms shorter, 10ms shorter, 10ms longer, 110ms longer, and 220ms longer.
Fig 5.
Simulation without noise of the mean absolute asynchrony between two musicians with matching or mismatching SMTs during duet musical performance.
Our simulation results showing the mean absolute asynchrony between two synchronizing ASHLE models with matching or mismatching f0, but no noise added to Eq 3c. The resulting mean absolute asynchronies in this simulation without noise are much smaller compared to the musician data results in Fig 4A. The added noise in Eq 4 improves the model’s results, which are shown in Fig 4B.
Fig 6.
The asynchrony between ASHLE and a sinusoid with period 45% shorter or longer than its spontaneous frequency, as a function of frequency learning and elasticity parameters.
(A) Illustration of the asynchrony between ASHLE’s zm and the sinusoidal stimulus. (B) The asynchrony in milliseconds between ASHLE and a sinusoidal stimulus with a period 45% shorter than ASHLE’s spontaneous frequency, and its change as a function of λ1 and λ2. (C) The same analysis but for a sinusoidal stimulus with a period 45% longer than ASHLE’s spontaneous frequency. Black cells indicate λ1 and λ2 value pairs for simulations where ASHLE could not synchronize.
Fig 7.
The asynchrony between ASHLE and a sinusoid faster or slower than ASHLE’s f0 as a function of a narrower range of values for the frequency learning and elasticity parameters.
Each cell shows the MA between ASHLE and a sinusoidal stimulus with a period 45% shorter, 30% shorter, 15% shorter, 15% longer, 30% longer, and 45% longer than ASHLE’s period, for a pair of values for λ1 and λ2. The pair of λ1 = 4 and λ2 = 2 yield MA values similar to the ones that Scheurich et al. [25] observed in musicians synchronizing with a metronome period 30% shorter, 15% shorter, 15% longer, and 30% longer than the musician’s SMP.
Fig 8.
ASHLE slope values as a function of γ and initial frequency in the absence of a stimulus.
(A) The effect of the γ parameter on the slope values between consecutive period lengths when ASHLE oscillates without a pacing stimulus, starting at a frequency that has a period 45% shorter (), 30% shorter (
), 15% shorter (
), 15% longer (
), 30% longer (
), and 45% longer (
) than ASHLE’s f0 period. (B) the same simulations but with an alternative single-oscillator model, showing a one-order-of-magnitude increase in the slope values.