"""Contains functions for generating and using equal-loudness contours for side-presented steady pure tones according to the ISO/IEC 226 standard. Code from https://gist.github.com/sammosummo/777debf946d0356acada and Cédric Colas """ __author__ = 'Sam Mathias' __version__ = 1.0 import numpy as np from scipy.interpolate import interp1d a_freq = 440 a_pitch = 69 upright_piano_dynamic_range = 60 # in db a_440_db_at_max_velocity = 75 a_440_amplitude_at_max_velocity = 10 ** (a_440_db_at_max_velocity / 20) def iso266(phon, return_freq=False): """Returns an equal-loudness contour evaluated at 29 frequencies between 20 Hz and 12.5 kHz according to the ISO/IEC 226 standard [1]_. Parameters ---------- phon : int or float The phon value represented by the equal-loudness contour, where a value of :math:`x` phon is the loudness of 1-KHz steady pure tone presented at :math:`x` dB SPL. Must be between 0 and 90. return_freq : bool, optional If True, the function returns the frequency values as well as the SPL values of the contour. Default is False. Returns ------- array_like Either a 1-D or a 2-D numpy array, depending on `return_freq`. Reference --------- .. [1] ISO/IEC (2003). ISO/IEC 226:2003 Acoustics -- Normal equal-loudness- level contours. http://www.iso.org/iso/catalogue_detail.htm?csnumber=34222. Example ------- elc = iso266(60, return_freq=True) print elc [[ 20. 25. 31.5 40. 50. 63. 80. 100. 125. 160. 200. 250. 315. 400. 500. 630. 800. 1000. 1250. 1600. 2000. 2500. 3150. 4000. 5000. 6300. 8000. 10000. 12500. ] [ 109.51132227 104.22789784 99.07786826 94.17731862 89.96345731 85.94342131 82.05340072 78.65461863 75.56345314 72.4743448 69.86431929 67.53483532 65.39173983 63.45099627 62.0511792 60.81495942 59.88668375 60.011588 62.1549143 63.18935604 59.96161453 57.25515019 56.42385863 57.56993838 60.8882125 66.36125056 71.66396598 73.15510401 68.63077045]] """ if not 0 <= phon <= 90: raise ValueError('Cannot calculate for this value.') f = np.array([ 20, 25, 31.5, 40, 50, 63, 80, 100, 125, 160, 200, 250, 315, 400, 500, 630, 800, 1000, 1250, 1600, 2000, 2500, 3150, 4000, 5000, 6300, 8000, 10000, 12500 ]) af = np.array([ 0.532, 0.506, 0.480, 0.455, 0.432, 0.409, 0.387, 0.367, 0.349, 0.330, 0.315, 0.301, 0.288, 0.276, 0.267, 0.259, 0.253, 0.250, 0.246, 0.244, 0.243, 0.243, 0.243, 0.242, 0.242, 0.245, 0.254, 0.271, 0.301 ]) Lu = np.array([ -31.6, -27.2, -23.0, -19.1, -15.9, -13.0, -10.3, -8.1, -6.2, -4.5, -3.1, -2.0, -1.1, -0.4, 0.0, 0.3, 0.5, 0.0, -2.7, -4.1, -1.0, 1.7, 2.5, 1.2, -2.1, -7.1, -11.2, -10.7, -3.1 ]) Tf = np.array([ 78.5, 68.7, 59.5, 51.1, 44.0, 37.5, 31.5, 26.5, 22.1, 17.9, 14.4, 11.4, 8.6, 6.2, 4.4, 3.0, 2.2, 2.4, 3.5, 1.7, -1.3, -4.2, -6.0, -5.4, -1.5, 6.0, 12.6, 13.9, 12.3 ]) Ln = phon Af = 4.47e-3 * (10 ** (.025 * Ln) - 1.15) \ + (.4 * 10 ** (((Tf + Lu) / 10.) - 9)) ** af Lp = ((10 / af) * np.log10(Af)) - Lu + 94 spl = Lp freq = f if return_freq is True: return np.array([freq, spl]) else: return spl def equal_loudness(phon, freqs, return_freq=False): """Returns equal-loudness levels for any frequencies between 20 Hz and 12.5 kHz according to the ISO/IEC 226 standard [1]_. Parameters ---------- phon : number The phon value represented by the equal-loudness contour, where a value of :math:`x` phon is the loudness of 1-KHz steady pure tone presented at :math:`x` dB SPL. Must be between 0 and 90. freqs : number or array_like Frequency or frequencies in Hz to be evaluated. Must be between 20 and 12500. return_freq : bool, optional If True, the function returns the frequency values as well as the SPL values of the contour. Default is False. Returns ------- array_like Either a 1-D or a 2-D numpy array, depending on `return_freq`. Reference --------- .. [1] ISO/IEC (2003). ISO/IEC 226:2003 Acoustics -- Normal equal-loudness- level contours. http://www.iso.org/iso/catalogue_detail.htm?csnumber=34222. Example ------- >>> el = equal_loudness(60, [500, 1000, 2000], return_freq=True) >>> print el [[ 500. 1000. 2000. ] [ 62.0511792 60.011588 59.96161453]] """ f = interp1d(*iso266(phon, True), kind='cubic') if return_freq is True: return np.array([freqs, f(freqs)]) else: return f(freqs) def get_loudness(spl, freq): """Returns the approximate loudness level in phons for a side-presented steady pure tone according to the ISO/IEC 226 standard [1]_. This function generates a range of equal-loudness contours and interpolates between them. Therefore it is more efficient to pass many level and frequency values to one function call than it is to make many function calls. Parameters ---------- spl : number or array_like Sound pressure level or levels in dB SPL. freq : number or array_like Frequency or frequencies in Hz. Returns ------- number or array_like Phon value(s). Reference --------- .. [1] ISO/IEC (2003). ISO/IEC 226:2003 Acoustics -- Normal equal-loudness- level contours. http://www.iso.org/iso/catalogue_detail.htm?csnumber=34222. Example ------- phons = get_loudness([50, 60, 70] [500, 500, 500]) print phons [ 47.3 57.8 68.4] """ phons = np.arange(0, 90.1, .1) freqs = np.arange(20, 12501) spls = np.empty((len(phons), len(freqs))) for i, phon in enumerate(phons): spls[i] = equal_loudness(phon, freqs) if not hasattr(spl, '__iter__'): spl = [spl] if not hasattr(freq, '__iter__'): freq = [freq] spls = spls.T results = [] for _spl, _freq in zip(spl, freq): ix = (np.abs(freqs - _freq)).argmin() iy = (np.abs(spls[ix] - _spl)).argmin() results.append(phons[iy]) if len(results) == 1: return results[0] else: return np.array(results) def pitch2freq(pitch): # from https://music.arts.uci.edu/dobrian/maxcookbook/pitch-and-loudness-formulae relative_pitch = pitch - 69 factor = 2 ** (relative_pitch / 12) freq = a_freq * factor return freq def velocity2amplitude(velocity): # from https://www.cs.cmu.edu/~rbd/papers/velocity-icmc2006.pdf r = 10 ** (upright_piano_dynamic_range / 20) b = 127 / (126 * np.sqrt(r)) - (1 / 126) m = (1 - b) / 127 a = (m * velocity + b) ** 2 a *= a_440_amplitude_at_max_velocity # scale amplitudes to get realistic perceived loudness return a def amplitude2db(amplitude): power_db = 20 * np.log10(amplitude) return power_db def get_db_of_equivalent_loudness_at_440hz(freqs, db): phons = get_loudness(db, freqs) equal_dbs = [] for p in phons: equal_dbs.append(equal_loudness(p, [440])[0]) return np.array(equal_dbs) def compute_total_loudness(eq_amplitudes_440hz, onsets, offsets): # Compute the instantaneous amplitude, turn it back to dbs, then to perceived loudness with unique freq 440 Hz # model amplitude as square function, loudness = peak amplitude from onset to offset, 0 afterwards. # an exponential model might be better assert all([len(values) == len(onsets) for values in [eq_amplitudes_440hz, offsets]]) timepoints = np.array(sorted(onsets + offsets)) amplitudes_per_time = np.zeros(len(timepoints)) # on each segment, compute the total amplitude # amplitudes are not just summed: p1+p2 = sqrt(p1**2 + p2**2) # ref: https://erlend-viggen.no/acoustic-quantities-1/ for i_n in range(len(onsets)): indexes = np.where(np.logical_and(timepoints >= onsets[i_n], timepoints < offsets[i_n])) amplitudes_per_time[indexes] += eq_amplitudes_440hz[i_n] ** 2 for i in range(len(amplitudes_per_time)): amplitudes_per_time[i] = np.sqrt(amplitudes_per_time[i]) # compute power power_per_time = amplitude2db(amplitudes_per_time) power_per_time[np.where(power_per_time == -np.inf)] = 0 # compute loudness loudness_per_time = get_loudness(power_per_time, [440] * len(power_per_time)) # amplitudes at 440hz, they were computed to make same loudness as original amplitudes at original F. # now integrate total_loudness = 0 for i_t in range(len(timepoints) - 1): total_loudness += loudness_per_time[i_t] * (timepoints[i_t + 1] - timepoints[i_t]) return total_loudness / (timepoints[-1] - timepoints[0]), np.std(loudness_per_time) if __name__ == '__main__': pass