The PyTuning package contains some utilities which may be useful. In general these tend to be smaller utilities and tasks that are useful in the analysis of musical scales, but they are not full-featured “things” in and of themselves.

Interval Normalization

pytuning.utilities.normalize_interval(interval, octave=2)

Normalize a musical interval

  • interval – The interval to normalize. Should be a frequency ratio, most usefully expressed as a sympy.Rational or related data item
  • octave – The formal octave. Defaults to 2

The interval, normalized

Note that any formal octave can be used. In normal usage a 2 will be used (i.e., a doubling of frequency is an octave).

Normalization works by finding the smallest power of two (or octave) that when multiplied by the interval (in the case of an interval less than 1) or divided into the interval (for intervals greater than 2) will bring the interval into the target range of 1 \le i \le 2.

As an example, the interval 9 would normalize to \frac{9}{8}, because 9 needs to be scaled down by three octaves to fall within the limit of 1 and 2:

\frac{9}{8} = \frac{9}{2^3}

ni = normalize_interval(sp.Integer(9))

One can also normalize on a non-standard interval, for example, 3:

ni = normalize_interval(sp.Integer(34), octave=3)

Distinct Intervals


Find the distinct intervals in a scale, including inversions

Parameters:scale – The scale to analyze
Returns:A list of distinct intervals

The scale should be specified as a list of sympy numerical values (Rational or Integer). Note that the convention adopted in this code is that scale[0] is a unison and scale[-1] is the formal octave (often 2).

As an example of a valid scale, a standardized Pythagorean tuning could be passed into the function:

\left [ 1, \frac{256}{243}, \frac{9}{8}, \frac{32}{27}, 
\frac{81}{64}, \frac{4}{3}, \frac{1024}{729}, \frac{3}{2}, 
\frac{128}{81}, \frac{27}{16}, \frac{16}{9}, 
\frac{243}{128}, 2\right ]

If one were hand-crafting this scale, it would look something like:

import sympy as sp
scale = [sp.Integer(1), sp.Rational(256,243), sp.Rational(9,8), ...]

The function returns a list in rational/symbolic terms. If numerical values are needed, one can, for example, map ratio_to_cents to obtain it:

di = distinct_intervals(scale)
di_in_cents = [ratio_to_cents(x) for x in di]

distinct_intervals() returns all the distinct intervals within a musical scale. Note, though, that it does not include the unison (or the octave) in the results, as all scales contain those intervals by definitions.

As an example, if we were to take a Pythagorean scale and find the intervals that exist within it:

pythag = create_pythagorean_scale()
di = distinct_intervals(pythag)

we end up with:

\left [ \frac{2187}{2048}, \quad \frac{256}{243}, \quad \frac{8192}{6561}, \quad \frac{262144}{177147},
\quad \frac{4096}{2187}, \quad \frac{3}{2}, \quad \frac{243}{128}, \quad \frac{1024}{729},
\quad \frac{19683}{16384}, \quad \frac{729}{512},
\quad \frac{6561}{4096},
\quad \frac{65536}{59049}, \quad \frac{177147}{131072}, \quad \frac{59049}{32768}, \quad \frac{81}{64}, \quad
\frac{32}{27}, \quad \frac{27}{16},
\quad \frac{4}{3}, \quad \frac{9}{8}, \quad \frac{32768}{19683},
\quad \frac{16}{9}, \quad \frac{128}{81}\right ]

Converting a Ratio to a Cent Value


Convert a scale degree to a cent value

Parameters:ratio – The scale degree (sympy value)
Returns:The scale degree in cents


\sqrt[2^{\left[ \frac{1}{1200} \right]}]{\text{degree}}

Note that this function returns a floating point number, not a sympy ratio.

This function is useful if you have a symbolic value (a rational or transcendental, for example) and you want to see its value in cents (a logarithmic scale in which there are 1200 steps in a factor of two). For example:

interval = sp.Rational(3,2) # A perfect fifth
cents = ratio_to_cents(interval)

Converting a Cent Value to a Ratio


Convert a cent value to a ratio

Parameters:cents – The degree value in cents
Returns:the frequency ratio

This function takes a cent value and returns it as a frequency ratio (a sympy floating point number).


(In other words, the 12-EDO fifth (700 cents) is very close to that of the Pythagorean fifth (\frac{3}{2}, or 1.5).)

Converting a Note Number to a Frequency

pytuning.utilities.note_number_to_freq(note, scale=None, reference_note=69, reference_frequency=440.0)

Convert a note number (MIDI) to a frequency (Hz).

  • note (reference) – The note number (0<=note<=127)
  • scale – The scale. If none it assume EDO 12.
  • note – The conversions reference note
  • reference_frequency – The frequency of the reference note

The frequency of the note in Hertz

The default values for reference_note and reference_frequency correspond to standard orchestral tuning, a4 = 440 Hz.

With this function we can calculate the frequency of any note number. If defaults to the MIDI standard, which pegs note number 69 to 440 Hz and uses a 12-EDO scale.

As an example, MIDI note 60 (Middle-C):


But if, for example, we wanted to use a different pitch standard, we could peg A to 444 Hz.

print(note_number_to_freq(60, reference_frequency=444.0))

You can also pass in a non-EDO tuning if you’re converting a different kind of scale to frequencies. This is used often in the code associated with the tuning tables.

Naming A Ratio


Convert a scale degree to a name

Parameters:ratio – The input scale degree (a sympy value)
Returns:The degree name if found, None otherwise

This function will look up the name of a ratio and return it (returning None) if it is not found.

As an example:

pythag = create_pythagorean_scale()
names = [ratio_to_name(x) for x in pythag]

names now contains:

'Pythagorean Minor Second',
'Pythagorean Major Second',
'Pythagorean Minor Third',
'Pythagorean Major Third',
'Perfect Fourth',
'Pythagorean Diminished Fifth',
'Perfect Fifth',
'Pythagorean Minor Sixth',
'Pythagorean Major Sixth',
'Pythagorean Minor Seventh',
'Pythagorean Major Seventh',

There are currently about 260 intervals in the internal catalog, so while not complete, the database is fairly extensive.

Comparing Two Scales

pytuning.utilities.compare_two_scales(scale1, scale2, reference_freq=220.0, title=['Scale1', 'Scale2'])

Compare two scales

param scale1:The first scale (list of sympy values)
param scale2:The second scale (list of sympy values)
param reference_freq:
 The frequency (Hz) of the first degree
param title:The scale names (list of strings with len = 2)
returns:None, (ie nothing)

This function will produce a simple textual representation of the difference between two scales. As an example, comparing the 12-EDO and Pythagorean scales:

from pytuning.scales import create_edo_scale, create_pythagorean_scale
from pytuning.utilities import compare_two_scales

scale_1 = create_edo_scale(12)
scale_2 = create_pythagorean_scale()

compare_two_scales(scale_1, scale_2, title=['12-TET', 'Pythagorean'])


          12-TET              Pythagorean
    Cents       Freq      Cents       Freq  Delta(Cents)
=========  =========  =========  =========  ============
   0.0000   220.0000     0.0000   220.0000        0.0000
 100.0000   233.0819    90.2250   231.7695        9.7750
 200.0000   246.9417   203.9100   247.5000       -3.9100
 300.0000   261.6256   294.1350   260.7407        5.8650
 400.0000   277.1826   407.8200   278.4375       -7.8200
 500.0000   293.6648   498.0450   293.3333        1.9550
 600.0000   311.1270   588.2700   309.0261       11.7300
 700.0000   329.6276   701.9550   330.0000       -1.9550
 800.0000   349.2282   792.1800   347.6543        7.8200
 900.0000   369.9944   905.8650   371.2500       -5.8650
1000.0000   391.9954   996.0900   391.1111        3.9100
1100.0000   415.3047  1109.7750   417.6562       -9.7750
1200.0000   440.0000  1200.0000   440.0000        0.0000