#!bin/bash
import numpy as np
import matplotlib.pyplot as plt
import sys
from biotuner.biotuner_utils import (
nth_root,
rebound,
NTET_ratios,
scale2frac,
findsubsets,
scale_from_pairs,
ratio2frac,
)
from biotuner.metrics import (
ratios2harmsim,
euler,
dyad_similarity,
metric_denom,
tuning_cons_matrix,
consonant_ratios,
tuning_to_metrics
)
from pytuning import create_euler_fokker_scale
import itertools
from collections import Counter
from numpy import linspace, empty, concatenate, log2
from scipy.signal import argrelextrema
from fractions import Fraction
from scipy.stats import norm
import contfrac
from math import log2
import math
sys.setrecursionlimit(120000)
[docs]def oct_subdiv(ratio, octave_limit=0.01365, octave=2, n=5):
"""
N-TET tuning from Generator Interval.
This function uses a generator interval to suggest numbers of steps to divide the octave.
Parameters
----------
ratio : float
Ratio that corresponds to the generator_interval.
For example, by giving the fifth (3/2) as generator interval,
this function will suggest to subdivide the octave in 12, 53, etc.
octave_limit : float, default=0.01365
Approximation of the octave corresponding to the acceptable distance
between the ratio of the generator interval after multiple iterations
and the octave value.
The default value of 0.01365 corresponds to the Pythagorean comma.
octave : int, default=2
Value of the octave.
n : int, default=5
Number of suggested octave subdivisions.
Returns
-------
Octdiv : List of int
List of N-TET tunings according to the generator interval.
Octvalue : List of float
List of the approximations of the octave for each N-TET tuning.
Examples
--------
>>> oct_subdiv(3/2, n=3)
([12, 53, 106], [1.0136432647705078, 1.0020903140410862, 1.0041849974949628])
"""
Octdiv, Octvalue, i = [], [], 1
ratios = []
while len(Octdiv) < n:
ratio_mult = ratio**i
while ratio_mult > octave:
ratio_mult = ratio_mult / octave
rescale_ratio = ratio_mult - round(ratio_mult)
ratios.append(ratio_mult)
i += 1
if -octave_limit < rescale_ratio < octave_limit:
Octdiv.append(i - 1)
Octvalue.append(ratio_mult)
else:
continue
return Octdiv, Octvalue
[docs]def compare_oct_div(Octdiv=12, Octdiv2=53, bounds=0.005, octave=2):
"""
Function that compares steps for two N-TET tunings
and returns matching ratios and corresponding degrees
Parameters
----------
Octdiv : int, default=12
First N-TET tuning number of steps.
Octdiv2 : int, default=53
Second N-TET tuning number of steps.
bounds : float, default=0.005
Maximum distance between one ratio of Octdiv and one ratio of Octdiv2
to consider a match.
octave : int, default=2
Value of the octave
Returns
-------
avg_ratios : numpy.ndarray
List of ratios corresponding to the shared steps in the two N-TET tunings
shared_steps : List of tuples
The two elements of each tuple corresponds to the tuning steps
sharing the same interval in the two N-TET tunings
Examples
--------
>>> ratios, shared_steps = compare_oct_div(Octdiv=12, Octdiv2=53, bounds=0.005, octave=2)
>>> ratios, shared_steps
([1.124, 1.187, 1.334, 1.499, 1.78, 2.0],
[(2, 9), (3, 13), (5, 22), (7, 31), (10, 44), (12, 53)])
"""
ListOctdiv = []
ListOctdiv2 = []
OctdivSum = 1
OctdivSum2 = 1
i = 1
i2 = 1
while OctdivSum < octave:
OctdivSum = (nth_root(octave, Octdiv)) ** i
i += 1
ListOctdiv.append(OctdivSum)
while OctdivSum2 < octave:
OctdivSum2 = (nth_root(octave, Octdiv2)) ** i2
i2 += 1
ListOctdiv2.append(OctdivSum2)
shared_steps = []
avg_ratios = []
for i, n in enumerate(ListOctdiv):
for j, harm in enumerate(ListOctdiv2):
if harm - bounds < n < harm + bounds:
shared_steps.append((i + 1, j + 1))
avg_ratios.append((n + harm) / 2)
avg_ratios = [np.mean(x, 3) for x in avg_ratios]
return avg_ratios, shared_steps
[docs]def multi_oct_subdiv(
peaks,
max_sub=100,
octave_limit=0.01365,
octave=2,
n_scales=10,
cons_limit=0.1
):
"""
Determine optimal octave subdivisions based on consonant peaks ratios.
This function takes the most consonant peaks ratios and uses them as input for
the oct_subdiv function. Each consonant ratio generates a list of possible
octave subdivisions. The function then compares these lists and identifies
optimal octave subdivisions that are common across multiple generator intervals.
Parameters
----------
peaks : List of float
Peaks represent local maximum in a spectrum.
max_sub : int, default=100
Maximum number of intervals in N-TET tuning suggestions.
octave_limit : float, default=0.01365
Approximation of the octave corresponding to the acceptable distance
between the ratio of the generator interval after
multiple iterations and the octave value.
octave : int, default=2
value of the octave
n_scales : int, default=10
Number of N-TET tunings to compute for each generator interval (ratio).
cons_limit : float, default=0.1
Limit for the consonance of the peaks ratios.
Returns
-------
multi_oct_div : List of int
List of octave subdivisions that fit with multiple generator intervals.
ratios : List of float
List of the generator intervals for which at least 1 N-TET tuning
matches with another generator interval.
Examples
--------
>>> peaks = [2, 3, 9]
>>> oct_divs, x = multi_oct_subdiv(peaks, max_sub=100)
>>> oct_divs, x
([53], array([1.125, 1.5 ]))
"""
ratios, cons = consonant_ratios(peaks, cons_limit)
list_oct_div = []
for i in range(len(ratios)):
list_temp, _ = oct_subdiv(ratios[i], octave_limit, octave, n_scales)
list_oct_div.append(list_temp)
counts = Counter(list(itertools.chain(*list_oct_div)))
oct_div_temp = []
for k, v in counts.items():
if v > 1:
oct_div_temp.append(k)
oct_div_temp = np.sort(oct_div_temp)
multi_oct_div = []
for i in range(len(oct_div_temp)):
if oct_div_temp[i] < max_sub:
multi_oct_div.append(oct_div_temp[i])
return multi_oct_div, ratios
[docs]def harmonic_tuning(list_harmonics, octave=2, min_ratio=1, max_ratio=2):
"""
Generates a tuning based on a list of harmonic positions.
Parameters
----------
list_harmonics : List of int
harmonic positions to use in the scale construction
octave : int
value of the period reference
min_ratio : float, default=1
Value of the unison.
max_ratio : float, default=2
Value of the octave.
Returns
-------
ratios : List of float
Generated tuning.
Examples
--------
>>> list_harmonics = [3, 5, 7, 9]
>>> harmonic_tuning(list_harmonics, octave=2, min_ratio=1, max_ratio=2)
[1.125, 1.25, 1.5, 1.75]
"""
list_harmonics = np.abs(list_harmonics)
ratios = []
for i in list_harmonics:
ratios.append(rebound(1 * i, min_ratio, max_ratio, octave))
ratios = list(set(ratios))
ratios = list(np.sort(np.array(ratios)))
return ratios
[docs]def euler_fokker_scale(intervals, n=1, octave=2):
"""
Function that takes as input a series of intervals
and derives a Euler Fokker Genera scale. Usually,
Parameters
----------
intervals : List of int
List of integers that represent the intervals.
n : int, default=1
number of times the interval is used in the scale generation
Returns
-------
ratios : List of float
Generated tuning.
Examples
--------
>>> intervals = [5, 7, 9]
>>> euler_fokker_scale(intervals, n=1, octave=2)
[1, 35/32, 9/8, 315/256, 5/4, 45/32, 7/4, 63/32, 2]
"""
multiplicities = [n for x in intervals] # Each factor is used once.
scale = create_euler_fokker_scale(intervals, multiplicities, octave=octave)
return scale
def generator_interval_tuning(interval=3/2, steps=12, octave=2, harmonic_min=0):
"""
Function that takes a generator interval and
derives a tuning based on its stacking.
Parameters
----------
interval : float
Generator interval
steps : int, default=12
Number of steps in the scale
When default, 12-TET for interval 3/2
octave : int
Defaults to 2
Value of the octave
Returns
-------
tuning : List of float
Generated tuning.
Examples
--------
>>> tuning = generator_interval_tuning(interval=3/2, steps=12, octave=2, harmonic_min=0)
>>> tuning
[1.0,
1.06787109375,
1.125,
1.20135498046875,
1.265625,
1.3515243530273438,
1.423828125,
1.5,
1.601806640625,
1.6875,
1.802032470703125,
1.8984375]
"""
tuning = []
for s in range(steps):
degree = interval**harmonic_min
while degree > octave:
degree = degree / octave
while degree < octave / 2:
degree = degree * octave
tuning.append(degree)
harmonic_min += 1
tuning = sorted(list(set(tuning)))
return tuning
[docs]def convergents(interval):
"""
Return the convergents of the log2 of a ratio.
The second value represents the number of steps to divide the octave
while the first value represents the number of octaves up before
the stacked ratio arrives approximately to the octave value.
For example, the output of the interval 1.5 will includes [7, 12],
which means that to approximate the fifth (1.5) in a NTET-tuning, you can
divide the octave in 12, while stacking 12 fifth will lead to the 7th octave up.
Parameters
----------
interval : float
Interval to find convergent.
Returns
-------
convergents : List of lists
Each sublist corresponds to a pair of convergents.
Examples
--------
>>> convergents(3/2)
[(0, 1),
(1, 1),
(1, 2),
(3, 5),
(7, 12),
(24, 41),
(31, 53),
(179, 306),
(389, 665),
(9126, 15601),
(18641, 31867)]
"""
value = np.log2(interval)
convergents = list(contfrac.convergents(value))
return convergents
# Dissonance curves
[docs]def dissmeasure(fvec, amp, model="min"):
"""
Given a list of partials (peak frequencies) in fvec, with amplitudes in amp,
this routine calculates the dissonance by summing the roughness of
every sine pair based on a model of Plomp-Levelt's roughness curve.
The older model (model='product') was based on the product of the two
amplitudes, but the newer model (model='min') is based on the minimum
of the two amplitudes, since this matches the beat frequency amplitude.
Parameters
----------
fvec : List
List of frequency values
amp : List
List of amplitude values
model : str, default='min'
Description of parameter `model`.
Returns
-------
D : float
Dissonance value
"""
# Sort by frequency
sort_idx = np.argsort(fvec)
am_sorted = np.asarray(amp)[sort_idx]
fr_sorted = np.asarray(fvec)[sort_idx]
# Used to stretch dissonance curve for different freqs:
Dstar = 0.24 # Point of maximum dissonance
S1 = 0.0207
S2 = 18.96
C1 = 5
C2 = -5
# Plomp-Levelt roughness curve:
A1 = -3.51
A2 = -5.75
# Generate all combinations of frequency components
idx = np.transpose(np.triu_indices(len(fr_sorted), 1))
fr_pairs = fr_sorted[idx]
am_pairs = am_sorted[idx]
Fmin = fr_pairs[:, 0]
S = Dstar / (S1 * Fmin + S2)
Fdif = fr_pairs[:, 1] - fr_pairs[:, 0]
if model == "min":
a = np.amin(am_pairs, axis=1)
elif model == "product":
a = np.prod(am_pairs, axis=1) # Older model
else:
raise ValueError('model should be "min" or "product"')
SFdif = S * Fdif
D = np.sum(a * (C1 * np.exp(A1 * SFdif) + C2 * np.exp(A2 * SFdif)))
return D
[docs]def diss_curve(
freqs,
amps,
denom=1000,
max_ratio=2,
euler_comp=True,
method="min",
plot=True,
n_tet_grid=None,
):
"""
This function computes the dissonance curve and related metrics for
a given set of frequencies (freqs) and amplitudes (amps).
Parameters
----------
freqs : List (float)
list of frequencies associated with spectral peaks
amps : List (float)
list of amplitudes associated with freqs (must be same lenght)
denom : int, default=1000
Highest value for the denominator of each interval
max_ratio : int, default=2
Value of the maximum ratio
Set to 2 for a span of 1 octave
Set to 4 for a span of 2 octaves
Set to 8 for a span of 3 octaves
Set to 2**n for a span of n octaves
euler : bool, default=True
When set to True, compute the Euler Gradus Suavitatis
for the derived scale.
method : str, default='min'
Refer to dissmeasure function for more information.
- 'min'
- 'product'
plot : bool, default=True
Plot the dissonance curve.
n_tet_grid : int, default=None
When an integer is given, dotted lines will be add to the plot
at steps of the given N-TET scale
Returns
-------
intervals : List of tuples
Each tuple corresponds to the numerator and the denominator
of each scale step ratio
ratios : List of floats
list of ratios that constitute the scale
euler_score : int
value of consonance of the scale
diss : float
value of averaged dissonance of the total curve
dyad_sims : List of floats
list of dyad similarities for each ratio of the scale
"""
freqs = np.array(freqs)
r_low = 1
alpharange = max_ratio
method = method
n = 1000
diss = empty(n)
a = concatenate((amps, amps))
for i, alpha in enumerate(linspace(r_low, alpharange, n)):
f = concatenate((freqs, alpha * freqs))
d = dissmeasure(f, a, method)
diss[i] = d
diss_minima = argrelextrema(diss, np.less)
intervals = []
for d in range(len(diss_minima[0])):
frac = Fraction(
diss_minima[0][d] / (n / (max_ratio - 1)) + 1
).limit_denominator(denom)
frac = (frac.numerator, frac.denominator)
intervals.append(frac)
intervals.append((2, 1))
ratios = [i[0] / i[1] for i in intervals]
dyad_sims = ratios2harmsim(ratios[:-1])
a = 1
ratios_euler = [a] + ratios
ratios_euler = [int(round(num, 2) * 1000) for num in ratios]
euler_score = None
if euler_comp is True:
euler_score = euler(*ratios_euler)
euler_score = euler_score / len(diss_minima)
else:
euler_score = "NaN"
if plot is True:
plt.figure(figsize=(14, 6), facecolor='white')
plt.plot(linspace(r_low, alpharange, len(diss)), diss)
plt.xscale("linear")
plt.xlim(r_low, alpharange)
try:
plt.text(
1.9,
1.5,
"Euler = " + str(int(euler_score)),
horizontalalignment="center",
verticalalignment="center",
fontsize=16,
)
except:
pass
for n, d in intervals:
plt.axvline(n / d, color="silver")
# Plot N-TET grid
if n_tet_grid is not None:
n_tet = NTET_ratios(n_tet_grid, max_ratio=max_ratio)
for n in n_tet:
plt.axvline(n, color="red", linestyle="--")
# Plot scale ticks
plt.minorticks_off()
plt.xticks(
[n / d for n, d in intervals],
["{}/{}".format(n, d) for n, d in intervals],
fontsize=13,
)
plt.xlabel('Frequency ratio', fontsize=14)
plt.ylabel('Dissonance', fontsize=14)
plt.yticks(fontsize=13)
plt.tight_layout()
plt.show()
return intervals, ratios, euler_score, np.average(diss), dyad_sims
"""Harmonic Entropy"""
[docs]def compute_harmonic_entropy_domain_integral(
ratios, ratio_interval, spread=0.01, min_tol=1e-15
):
"""
Computes the harmonic entropy of a list of frequency ratios for a given set of possible intervals.
Parameters
----------
ratios : List of floats
List of frequency ratios.
ratio_interval : List of floats
List of possible intervals to consider.
spread : float, default=0.01
Controls the width of the Gaussian kernel used to smooth the probability density function of the ratios.
min_tol : float, default=1e-15
The smallest tolerance value for considering the probability density function.
Returns
-------
weight_ratios : ndarray
Sorted ratios.
HE : ndarray
Harmonic entropy values for each interval in `ratio_interval`.
Notes
-----
Harmonic entropy is a measure of the deviation of a set of frequency ratios from the idealized harmonics
(integer multiples of a fundamental frequency) and is defined as:
HE = - sum_i(p_i * log2(p_i))
where p_i is the probability of a given ratio in a smoothed probability density function.
The `ratio_interval` defines a range of possible intervals to consider. The algorithm computes the harmonic entropy
of each possible interval in `ratio_interval` and returns an array of HE values, one for each interval.
"""
# The first step is to pre-sort the ratios to speed up computation
ind = np.argsort(ratios)
weight_ratios = ratios[ind]
centers = (weight_ratios[:-1] + weight_ratios[1:]) / 2
ratio_interval = np.array(ratio_interval)
N = len(ratio_interval)
HE = np.zeros(N)
for i, x in enumerate(ratio_interval):
P = np.diff(
concatenate(([0], norm.cdf(log2(centers), loc=log2(x), scale=spread), [1]))
)
ind = P > min_tol
HE[i] = -np.sum(P[ind] * log2(P[ind]))
return weight_ratios, HE
[docs]def compute_harmonic_entropy_simple_weights(
numerators, denominators, ratio_interval, spread=0.01, min_tol=1e-15
):
"""
Compute the harmonic entropy of a set of ratios using simple weights.
Parameters
----------
numerators : array-like
Numerators of the ratios.
denominators : array-like
Denominators of the ratios.
ratio_interval : array-like
Interval to compute the harmonic entropy over.
spread : float, default=0.01
Spread of the normal distribution used to compute the weights.
min_tol : float, default=1e-15
Minimum tolerance for the weights.
Returns
-------
weight_ratios : ndarray
Sorted weight ratios.
HE : ndarray
Harmonic entropy.
"""
# The first step is to pre-sort the ratios to speed up computation
ratios = numerators / denominators
ind = np.argsort(ratios)
numerators = numerators[ind]
denominators = denominators[ind]
weight_ratios = ratios[ind]
ratio_interval = np.array(ratio_interval)
N = len(ratio_interval)
HE = np.zeros(N)
for i, x in enumerate(ratio_interval):
P = norm.pdf(log2(weight_ratios), loc=log2(x), scale=spread) / np.sqrt(
numerators * denominators
)
ind = P > min_tol
P = P[ind]
P /= np.sum(P)
HE[i] = -np.sum(P * log2(P))
return weight_ratios, HE
[docs]def harmonic_entropy(
ratios, res=0.001, spread=0.01,
plot_entropy=True, plot_tenney=False, octave=2
):
"""
Harmonic entropy is a measure of the uncertainty in pitch perception,
and it provides a physical correlate of tonalness,one aspect of the
psychoacoustic concept of dissonance (Sethares). High tonalness corresponds
to low entropy and low tonalness corresponds to high entropy.
Parameters
----------
ratios : List of floats
Ratios between each pairs of frequency peaks.
res : float, default=0.001
Resolution of the ratio steps.
spread : float, default=0.01
Spread of the normal distribution used to compute the weights.
plot_entropy : bool, default=True
When set to True, plot the harmonic entropy curve.
plot_tenney : bool, default=False
When set to True, plot the tenney heights (y-axis)
across ratios (x-axis).
octave : int, default=2
Value of reference period.
Returns
----------
HE_minima : List of floats
List of ratios corresponding to minima of the harmonic entropy curve
HE : float
Value of the averaged harmonic entropy
"""
fracs, numerators, denominators = scale2frac(ratios)
ratios = numerators / denominators
bendetti_heights = numerators * denominators
tenney_heights = log2(bendetti_heights)
ind = np.argsort(tenney_heights) # sort by Tenney height
bendetti_heights = bendetti_heights[ind]
tenney_heights = tenney_heights[ind]
numerators = numerators[ind]
denominators = denominators[ind]
if plot_tenney is True:
fig = plt.figure(figsize=(10, 4), dpi=150)
ax = fig.add_subplot(111)
# ax.scatter(ratios, 2**tenney_heights, s=1)
ax.scatter(ratios, tenney_heights, s=1, alpha=0.2)
# ax.scatter(ratios[:200], tenney_heights[:200], s=1, color='r')
plt.show()
# Next, we need to ensure a distance `d` between adjacent ratios
M = len(bendetti_heights)
delta = 0.00001
indices = np.ones(M, dtype=bool)
for i in range(M - 2):
ind = abs(ratios[i + 1:] - ratios[i]) > delta
indices[i + 1:] = indices[i + 1:] * ind
bendetti_heights = bendetti_heights[indices]
tenney_heights = tenney_heights[indices]
numerators = numerators[indices]
denominators = denominators[indices]
ratios = ratios[indices]
M = len(tenney_heights)
x_ratios = np.arange(1, octave, res)
_, HE = compute_harmonic_entropy_domain_integral(ratios,
x_ratios,
spread=spread)
# HE = compute_harmonic_entropy_simple_weights(numerators,
# denominators,
# x_ratios, spread=0.01)
ind = argrelextrema(HE, np.less)
HE_minima = (x_ratios[ind], HE[ind])
if plot_entropy is True:
fig = plt.figure(figsize=(10, 4), dpi=150)
ax = fig.add_subplot(111)
ax.plot(x_ratios, HE)
ax.scatter(HE_minima[0], HE_minima[1], color="k", s=4)
ax.set_xlim(1, octave)
plt.xlabel('Frequency ratio')
plt.ylabel('Harmonic entropy')
plt.show()
return HE_minima, np.average(HE)
"""Scale reduction"""
[docs]def tuning_reduction(tuning, mode_n_steps, function, rounding=4, ratio_type="pos_harm"):
"""
Function that reduces the number of steps in a scale according
to the consonance between pairs of ratios.
Parameters
----------
tuning : List (float)
scale to reduce
mode_n_steps : int
number of steps of the reduced scale
function : function, default=compute_consonance
function used to compute the consonance between pairs of ratios
Choose between:
- :func:`compute_consonance <biotuner.metrics.compute_consonance>`
- :func:`dyad_similarity <biotuner.metrics.dyad_similarity>`
- :func:`metric_denom <biotuner.metrics.metric_denom>`
rounding : int
maximum number of decimals for each step
ratio_type : str, default='pos_harm'
Choose between:
- 'pos_harm':a/b when a>b
- 'sub_harm':a/b when a<b
- 'all': pos_harm + sub_harm
Returns
-------
tuning_consonance : float
Consonance value of the input tuning.
mode_out : List of floats
List of mode intervals.
mode_consonance : float
Consonance value of the output mode.
Examples
--------
>>> tuning = [1, 1.21, 1.31, 1.45, 1.5, 1.7, 1.875]
>>> harm_tuning, mode, harm_mode = tuning_reduction(tuning, mode_n_steps=5, function=dyad_similarity, rounding=4, ratio_type="pos_harm")
>>> print('Tuning harmonicity: ', harm_tuning, '\nMode: ', mode, '\nMode harmonicity: ', harm_mode)
Tuning harmonicity: 9.267212965965944
Mode: [1.5, 1, 1.875, 1.7, 1.45]
Mode harmonicity: 17.9500338066261
"""
tuning_values = []
mode_values = []
for index1 in range(len(tuning)):
for index2 in range(len(tuning)):
if tuning[index1] != tuning[index2]: # not include the diagonale
if ratio_type == "pos_harm":
if tuning[index1] > tuning[index2]:
entry = tuning[index1] / tuning[index2]
mode_values.append([tuning[index1], tuning[index2]])
tuning_values.append(function(entry))
if ratio_type == "sub_harm":
if tuning[index1] < tuning[index2]:
entry = tuning[index1] / tuning[index2]
mode_values.append([tuning[index1], tuning[index2]])
tuning_values.append(function(entry))
if ratio_type == "all":
entry = tuning[index1] / tuning[index2]
mode_values.append([tuning[index1], tuning[index2]])
tuning_values.append(function(entry))
if function == metric_denom:
cons_ratios = [x for _, x in sorted(zip(tuning_values, mode_values))]
else:
cons_ratios = [x for _, x in sorted(zip(tuning_values, mode_values))][::-1]
i = 0
mode_ = []
mode_out = []
while len(mode_out) < mode_n_steps:
cons_temp = cons_ratios[i]
mode_.append(cons_temp)
mode_out_temp = [item for sublist in mode_ for item in sublist]
mode_out_temp = [np.round(x, rounding) for x in mode_out_temp]
mode_out = sorted(set(mode_out_temp), key=mode_out_temp.index)[0:mode_n_steps]
i += 1
mode_metric = []
for index1 in range(len(mode_out)):
for index2 in range(len(mode_out)):
if mode_out[index1] > mode_out[index2]:
entry = mode_out[index1] / mode_out[index2]
mode_metric.append(function(entry))
tuning_consonance = np.average(tuning_values)
mode_consonance = np.average(mode_metric)
return tuning_consonance, mode_out, mode_consonance
[docs]def create_mode(tuning, n_steps, function):
"""Create a mode from a tuning based on the consonance of
subsets of tuning steps.
Parameters
----------
tuning : List of floats
scale to reduce
n_steps : int
number of steps of the reduced scale
function : function, default=compute_consonance
function used to compute the consonance between pairs of ratios
Choose between:
- :func:`compute_consonance <biotuner.metrics.compute_consonance>`
- :func:`dyad_similarity <biotuner.metrics.dyad_similarity>`
- :func:`metric_denom <biotuner.metrics.metric_denom>`
Returns
-------
mode : List of floats
Reduced tuning.
Examples
--------
>>> tuning = [1, 1.21, 1.31, 1.45, 1.5, 1.7, 1.875]
>>> create_mode(tuning, n_steps=5, function=dyad_similarity)
[1, 1.45, 1.5, 1.7, 1.875]
"""
sets = list(findsubsets(tuning, n_steps))
metric_values = []
for s in sets:
_, met = tuning_cons_matrix(s, function)
metric_values.append(met)
idx = np.argmax(metric_values)
mode = list(sets[idx])
return mode
[docs]def pac_mode(pac_freqs, n, function=dyad_similarity, method="subset"):
"""
Compute the pac mode of a set of frequency pairs.
Parameters
----------
pac_freqs : List of tuples
List of frequency pairs (f1, f2) representing phase-amplitude coupling.
n : int
Number of steps in the tuning system.
function : function, default=dyad_similarity
A function that takes two frequencies as input and returns a similarity score.
method : str, default='subset'
The method used to compute the pac mode.
Possible values:
- 'pairwise'
- 'subset'
Returns
-------
List
The pac mode as a list of frequencies.
"""
if method == "pairwise":
_, mode, _ = tuning_reduction(
scale_from_pairs(pac_freqs), n_steps=n, function=function
)
if method == "subset":
mode = create_mode(
scale_from_pairs(pac_freqs), n_steps=n, function=function
)
return sorted(mode)
"""--------------------------MOMENTS OF SYMMETRY---------------------------"""
import sympy as sp
[docs]def tuning_range_to_MOS(frac1, frac2, octave=2, max_denom_in=100, max_denom_out=100):
"""
Compute the Moment of Symmetry (MOS) signature for a range of ratios defined by two input fractions,
and compute the generative interval for that range.
The MOS signature of a ratio is a tuple of integers representing the number of equally spaced intervals
that can fit into an octave when starting from the ratio, going in one direction, and repeating the
interval until the octave is filled. For example, the MOS signature of an octave is (1,0) because there
is only one interval that fits into an octave when starting from the ratio of 1:1 and going up.
The MOS signature of a perfect fifth is (0,1) because there are no smaller intervals that fit into an
octave when starting from the ratio of 3:2 and going up, but there is one larger interval that fits,
which is the octave above the perfect fifth.
The generative interval is the interval that corresponds to the mediant of the two input fractions.
The mediant is the fraction that lies between the two input fractions and corresponds to the interval
where small and large steps are equal.
Parameters
----------
frac1 : str or float
First ratio as a string or float.
frac2 : str or float
Second ratio as a string or float.
octave : float, default=2
The ratio of an octave.
max_denom_in : int, default=100
Maximum denominator to use when converting the input fractions to rational numbers.
max_denom_out : int, default=100
Maximum denominator to use when approximating the generative interval as a rational number.
Returns
-------
tuple
A tuple containing:
- the mediant as a float,
- the mediant as a fraction with a denominator not greater than `max_denom_out`,
- the generative interval as a float,
- the generative interval as a fraction with a denominator not greater than `max_denom_out`,
- the MOS signature of the generative interval as a tuple of integers,
- the MOS signature of the inverse of the generative interval as a tuple of integers.
"""
a = Fraction(frac1).limit_denominator(max_denom_in).numerator
b = Fraction(frac1).limit_denominator(max_denom_in).denominator
c = Fraction(frac2).limit_denominator(max_denom_in).numerator
d = Fraction(frac2).limit_denominator(max_denom_in).denominator
#print(a, b, c, d)
mediant = (a + c) / (b + d)
mediant_frac = sp.Rational((a + c) / (b + d)).limit_denominator(max_denom_out)
gen_interval = octave ** (mediant)
gen_interval_frac = sp.Rational(octave ** (mediant)).limit_denominator(
max_denom_out
)
MOS_signature = [d, b]
invert_MOS_signature = [b, d]
return (
mediant,
mediant_frac,
gen_interval,
gen_interval_frac,
MOS_signature,
invert_MOS_signature,
)
[docs]def stern_brocot_to_generator_interval(ratio, octave=2):
"""
Converts a fraction in the stern-brocot tree to
a generator interval for moment of symmetry tunings
Parameters
----------
ratio : float
stern-brocot ratio
octave : float, default=2
Reference period.
Returns
-------
gen_interval : float
Generator interval
"""
gen_interval = octave ** (ratio)
return gen_interval
[docs]def gen_interval_to_stern_brocot(gen):
"""
Convert a generator interval to fraction in the stern-brocot tree.
Parameters
----------
gen : float
Generator interval.
Returns
-------
root_ratio : float
Fraction in the stern-brocot tree.
"""
root_ratio = log2(gen)
return root_ratio
[docs]def horogram_tree_steps(ratio1, ratio2, steps=10, limit=1000):
ratios_list = [ratio1, ratio2]
s = 0
while s < steps:
ratio3 = horogram_tree(ratio1, ratio2, limit)
ratios_list.append(ratio3)
ratio1 = ratio2
ratio2 = ratio3
s += 1
frac_list = [ratio2frac(x) for x in ratios_list]
return frac_list, ratios_list
[docs]def horogram_tree(ratio1, ratio2, limit):
"""
Compute the next step of the horogram tree.
Parameters
----------
ratio1 : float
First ratio input.
ratio2 : float
Second ratio input.
limit : int
Limit for the denominator of the fraction.
Returns
-------
next_step : float
Next step of the horogram tree.
"""
a = Fraction(ratio1).limit_denominator(limit).numerator
b = Fraction(ratio1).limit_denominator(limit).denominator
c = Fraction(ratio2).limit_denominator(limit).numerator
d = Fraction(ratio2).limit_denominator(limit).denominator
next_step = (a + c) / (b + d)
return next_step
[docs]def phi_convergent_point(ratio1, ratio2):
"""
Compute the phi convergent point of two ratios.
Parameters
----------
ratio1 : float
First ratio input.
ratio2 : float
Second ratio input.
Returns
-------
convergent_point : float
Phi convergent point of the two ratios.
"""
Phi = (1 + 5**0.5) / 2
a = Fraction(ratio1).limit_denominator(1000).numerator
b = Fraction(ratio1).limit_denominator(1000).denominator
c = Fraction(ratio2).limit_denominator(1000).numerator
d = Fraction(ratio2).limit_denominator(1000).denominator
convergent_point = (c * Phi + a) / (d * Phi + b)
return convergent_point
[docs]def Stern_Brocot(n, a=0, b=1, c=1, d=1):
"""
Compute the Stern-Brocot tree of a given depth.
Parameters
----------
n : int
Depth of the tree.
a, b, c, d : int
Initial values for the Stern-Brocot recursion. Default is a=0, b=1, c=1, d=1.
Returns
-------
list
List of fractions in the Stern-Brocot tree.
"""
if a + b + c + d > n:
return 0
x = Stern_Brocot(n, a + c, b + d, c, d)
y = Stern_Brocot(n, a, b, a + c, b + d)
if x == 0:
if y == 0:
return [a + c, b + d]
else:
return [a + c] + [b + d] + y
else:
if y == 0:
return [a + c] + [b + d] + x
else:
return [a + c] + [b + d] + x + y
[docs]def generator_interval_tuning (interval = 3/2, steps = 12, octave = 2, harmonic_min = 0):
'''
Function that takes a generator interval and derives a tuning based on its stacking.
interval: float
Generator interval
steps: int, default=12
Number of steps in the scale.
When set to 12 --> 12-TET for interval 3/2
octave: int, default=2
Value of the octave
'''
scale = []
for s in range(steps):
degree = interval**harmonic_min
while degree > octave:
degree = degree/octave
while degree < octave/2:
degree = degree*octave
scale.append(degree)
harmonic_min += 1
return sorted(scale), scale
[docs]def interval_exponents(interval, n_steps):
list_intervals = []
for n in range(n_steps):
n+=1
list_intervals.append(interval**n)
return list_intervals[:-1]
[docs]def interval_to_radian(interval):
degree = 360*(log2(interval))
#print(degree)
return math.radians(degree), degree
[docs]def tuning_to_radians(interval, n_steps):
_, tuning = generator_interval_tuning (interval=interval, steps=n_steps, octave=2, harmonic_min=1)
radians = []
degrees = []
#print(tuning)
for step in tuning:
rad, deg = interval_to_radian(step)
radians.append(rad)
degrees.append(deg)
return radians, degrees
[docs]def tuning_MOS_info (interval=3/2, steps=12, octave=2):
tuning, _ = generator_interval_tuning (interval = interval, steps = steps, octave = octave, harmonic_min=0)
tuning = tuning+[2]
tuning = np.round(tuning, 10)
tuning = np.sort(list(set(tuning)))
intervals = []
intervals_frac = []
for i in range(len(tuning)):
try:
interval_ = np.round((1200*log2(tuning[i+1])-1200*log2(tuning[i])), 3)
intervals.append(interval_)
intervals_frac.append(Fraction(tuning[i+1]-tuning[i]).limit_denominator(100))
except:
pass
#print(distance)
#print(intervals_frac)
distances = list(Counter(intervals).keys()) # equals to list(set(words))
steps = list(Counter(intervals).values())
sL = [steps for _,steps in sorted(zip(distances,steps))]
if len((set(intervals))) == 1:
#print('Large and small steps are equal')
Large=sL[0]
small=sL[0]
else:
Large=sL[1]
small=sL[0]
#print(sL)
return len(set(intervals)), Large, small, tuning, sorted(distances)[::-1]
[docs]def find_MOS (interval, max_steps=53, octave=2):
steps=2
MOS = {"steps":[],"sig":[], 'tuning':[], 'distances':[], 'distances_frac':[], 'NTET':[], 'harmsim':[],
'matrix_harmsim':[], 'stern_brocot_fracs':[]}
while steps<max_steps:
steps+=1
n_gaps, L, s, tuning, distances = tuning_MOS_info (interval, steps, octave)
if n_gaps==2 or n_gaps==1:
stern = Fraction(log2(interval)).limit_denominator(steps)
stern = [stern.numerator, stern.denominator]
MOS['stern_brocot_fracs'].append(stern)
MOS['steps'].append(steps)
MOS['sig'].append([L, s])
MOS['tuning'].append(tuning)
MOS['distances'].append(distances)
if n_gaps==2:
MOS['NTET'].append(False)
if n_gaps==1:
MOS['NTET'].append(True)
#MOS['distances_frac'].append(distances_frac)
MOS_metrics = []
MOS_harmsim = []
for tuning in MOS['tuning']:
dict_metrics = tuning_to_metrics(tuning)
MOS_metrics.append(dict_metrics)
MOS['harmsim'].append(dict_metrics['harm_sim'])
MOS['matrix_harmsim'].append(dict_metrics['matrix_harm_sim'])
return MOS
[docs]def MOS_metric_harmonic_mean(MOS_dict, metric='harmsim'):
total_steps = np.sum(MOS_dict['steps'])
harm_mean = []
for step, harmsim in zip(MOS_dict['steps'], MOS_dict[metric]):
harm_mean_ = step*harmsim
harm_mean.append(harm_mean_)
harm_mean = np.sum(harm_mean)/total_steps
return harm_mean
[docs]def generator_to_stern_brocot_fractions (gen, limit):
stern_fraction = []
for i in range(1, limit):
stern = Fraction(log2(gen)).limit_denominator(i)
stern = [stern.numerator, stern.denominator]
stern_fraction.append(stern)
stern_fraction = sorted(list(set(tuple(row) for row in stern_fraction)))
return stern_fraction
[docs]def measure_symmetry(generator_interval, max_steps=20, octave=2):
"""
Measure the maximum deviation in symmetry for a given generator interval.
This function calculates the MOS scales for the given generator interval and determines
the maximum deviation in symmetry for the scales.
Parameters
----------
generator_interval : int or float
The generator interval for which MOS scales will be calculated.
max_steps : int, default=20
The maximum number of steps to consider for each MOS scale calculation.
octave : int, default=2
The octave size for which the MOS scales will be calculated.
Returns
-------
float
The maximum deviation in symmetry for the given generator interval.
Examples
--------
>>> generator_interval = 3/2
>>> measure_symmetry(generator_interval)
"""
MOS = find_MOS(generator_interval, max_steps=max_steps, octave=octave)
deviations = []
for sig in MOS['sig']:
deviations.append([abs(s - np.mean(sig)) for s in sig])
avg_deviations = []
for i in range(max([len(sig) for sig in MOS['sig']])):
deviations_i = [d[i] for d in deviations if i < len(d)]
if deviations_i:
avg_deviations.append(np.mean(deviations_i))
norm_deviations = [d / len(MOS['sig']) for d in avg_deviations]
max_deviation = max(norm_deviations)
return max_deviation
