Number Theory¶

While this part of the package isn’t particularly fleshed out yet, there are a few number-theoretic functions for the analysis of scales.

Odd Limits¶

PyTuning contains functions for finding the odd Limit for both intervals and scales.

We can define and interval – say, $\frac{45}{32}$ , and find its odd-limit with the following:

from pytuning.number_theory import odd_limit

interval = sp.Rational(45,32)
limit = odd_limit(interval)

which yields and answer of 45.

One can also find the odd limit of an entire scale with the find_odd_limit_for_scale() function:

from pytuning.scales import create_euler_fokker_scale
from pytuning.number_theory import find_odd_limit_for_scale

scale = create_euler_fokker_scale([3,5],[3,1])
limit = find_odd_limit_for_scale(scale)

which yields 135. (Examining the scale:

$\left [ 1, \quad \frac{135}{128}, \quad \frac{9}{8}, \quad \frac{5}{4}, \quad \frac{45}{32}, \quad \frac{3}{2}, \quad \frac{27}{16}, \quad \frac{15}{8}, \quad 2\right ]$

you will see that this is the largest odd number, and is found in the second degree.)

pytuning.number_theory.odd_limit(i)¶

Find the odd-limit of an interval.

Parameters:	i – The interval (a `sympy.Rational`)
Returns:	The odd-limit for the interval.

pytuning.number_theory.find_odd_limit_for_scale(s)¶

Find the odd-limit of an interval.

Parameters:	s – The scale (a list of `sympy.Rational` values)
Returns:	The odd-limit for the scale.

Prime Limits¶

One can also compute prime limits for both scales and intervals. Extending the above example, one would assume that the Euler-Fokker scale would have a prime-limit of 5, since that’s the highest prime used in the generation, and in fact:

from pytuning.scales import create_euler_fokker_scale
from pytuning.number_theory import find_prime_limit_for_scale

scale = create_euler_fokker_scale([3,5],[3,1])
limit = find_prime_limit_for_scale(scale)

will return 5 as the limit.

pytuning.number_theory.prime_limit(i)¶

Find the prime-limit of an interval.

Parameters:	i – The interval (a `sympy.Rational`)
Returns:	The prime-limit for the interval.

pytuning.number_theory.find_prime_limit_for_scale(s)¶

Find the prime-limit of an interval.

Parameters:	s – The scale (a list of `sympy.Rational` values)
Returns:	The prime-limit for the scale.

Prime Factorization of Degrees¶

PyTuning has functions for breaking a ratio down into prime factors, and the inverse of reassembling them.

pytuning.number_theory.prime_factor_ratio(r, return_as_vector=False)¶

Decompose a ratio (degree) to prime factors

Parameters:	r – The degree to factor (`sympy.Rational`) return_as_vector – If `True`, return the factors as a vector (see below)
Returns:	The factoring of the ratio

By default this function will return a dict, with each key being a prime, and each value being the exponent of the factorization.

As an example, the syntonic comma, $\frac{81}{80}$ , can be factored:

r = syntonic_comma
q = prime_factor_ratio(r)
print(q)
{2: -4, 3: 4, 5: -1}

which can be interpreted as:

$\frac{3^4}{2^{4} \cdot 5^{1}}$

If return_as_vector is True, the function will return a tuple, with each position a successive prime. The above example yields:

r = syntonic_comma
q = prime_factor_ratio(r, return_as_vector=True)
print(q)
(-4, 4, -1)

Note that zeros will be included in the vector, thus:

r = sp.Rational(243,224)
q = prime_factor_ratio(r, return_as_vector=True)
print(q)
(-5, 5, 0, -1)
q = prime_factor_ratio(r)
print(q)
{2: -5, 3: 5, 7: -1}

pytuning.number_theory.create_ratio_from_primes(factors)¶

Given a prime factorization of a ratio, reconstruct the ratio. This function in the inverse of prime_factor_ratio().

Parameters:	factors – The factors. Can be a `dict` or a `tuple` (see `prime_factor_ratio` for a discussion).
Returns:	The ratio (`sympy` value).