opycleid.musicmonoids

This module includes musical monoids and groups commonly encountered in TMT.

Noll_Monoid

opycleid.musicmonoids.Noll_Monoid()

This monoid of 8 elements acts on the set of the twelve pitch classes $\{C,C\sharp,D,E\flat,E,F,F\sharp,G,G\sharp,A,B\flat,B\}$ encoded with the usual semi-tone encoding, i.e. the set $\mathbb{Z}_{12}$ with $C=0, C\sharp=1, \text{etc.}$ .

It is defined by Thomas Noll as the monoid generated by the two transformations $f \colon \mathbb{Z}_{12} \to \mathbb{Z}_{12}$ and $g \colon \mathbb{Z}_{12} \to \mathbb{Z}_{12}$ such that $f(x)= 3x+7,$ and $g(x)= 8x+4.$

References

Thomas Noll: The topos of triads, In: Colloquium on Mathematical Music Theory, Volume 347 of Grazer Math. Ber. Karl-Franzens-Univ. Graz, pp. 103–135, (2005).

TI_Group_PC

opycleid.musicmonoids.TI_Group_PC()

This group is the commonly named $T\text{/}I$ group which acts on the set of the twelve pitch classes $\{C,C\sharp,D,E\flat,E,F,F\sharp,G,G\sharp,A,B\flat,B\}$ encoded with the usual semi-tone encoding.

It is isomorphic to the dihedral group $D_{24}$ of order 24, and is generated by the following two transformations.

The transformation $T_1 \colon \mathbb{Z}_{12} \to \mathbb{Z}_{12}$ is called the transposition operation, and is such that $T_1(x)=x+1$
The transformation $I_0 \colon \mathbb{Z}_{12} \to \mathbb{Z}_{12}$ is called the inversion operation, and is such that $I_0(x)=-x$

Example

from opycleid.musicmonoids import TI_Group_PC

my_group = TI_Group_PC()
print(my_group.get_operation("D","A")) ## ['T7', 'I11']
print(my_group.apply_operation("I11","B")) ## ['C']
print(my_group.mult("I11","T3")) ## "I8"

References

Alissa S. Crans, Thomas M. Fiore & Ramon Satyendra: Musical Actions of Dihedral Groups, The American Mathematical Monthly, 116:6, pp. 479-495, (2018).

TI_Group_Triads

opycleid.musicmonoids.TI_Group_Triads()

This group is the commonly named $T\text{/}I$ group which acts simply transitively on the set $\{n_M, n=0\ldots11\} \cup \{n_m, n=0\ldots11\}$ of the 24 major and minor triads, where $n_M$ (resp. $n_m$ ) represents a major (resp. minor) triad with root $n$ in the usual semi-tone encoding of pitch classes.

It is isomorphic to the dihedral group $D_{24}$ of order 24, and is generated by the following two transformations.

The transformation $T_1 \colon \mathbb{Z}_{12} \to \mathbb{Z}_{12}$ is called the transposition operation, and is such that $T_1(n_M)=(n+1)_M, T_1(n_m)=(n+1)_m$
The transformation $I_0 \colon \mathbb{Z}_{12} \to \mathbb{Z}_{12}$ is called the inversion operation, and is such that $I_0(n_M)=(5-n)_m$

In effect, the $T_1$ and $I_0$ operations act element-wise on the pitch classes constituting the chords.

Example

from opycleid.musicmonoids import TI_Group_Triads

my_group = TI_Group_Triads()
print(my_group.get_operation("E_M","Gs_m")) ## ['I7']
print(my_group.apply_operation("I7","B_M")) ## ['Cs_m']
print(my_group.mult("I7","T4")) ## "I3"

References

Alissa S. Crans, Thomas M. Fiore & Ramon Satyendra: Musical Actions of Dihedral Groups, The American Mathematical Monthly, 116:6, pp. 479-495, (2018).

PRL_Group

opycleid.musicmonoids.PRL_Group()

This group is the $\text{PRL}$ group commonly used in neo-Riemannian theory, which acts simply transitively on the set $\{n_M, n=0\ldots11\} \cup \{n_m, n=0\ldots11\}$ of the 24 major and minor triads, where $n_M$ (resp. $n_m$ ) represents a major (resp. minor) triad with root $n$ in the usual semi-tone encoding of pitch classes.

It is isomorphic to the dihedral group $D_{24}$ of order 24, and is generated by the following two transformations.

The transformation $L \colon \mathbb{Z}_{12} \to \mathbb{Z}_{12}$ is called the leading-tone operation, and is such that $L(n_M)=(n+4)_m$
The transformation $R \colon \mathbb{Z}_{12} \to \mathbb{Z}_{12}$ is called the relative operation, and is such that $R(n_M)=(n+9)_m$

Though not a generator, the operation $P = (RL)^3R$ , called the parallel operation, is often considered, and is such that $P(n_M)=n_m$ .

Example

from opycleid.musicmonoids import PRL_Group

my_group = PRL_Group()
print(my_group.get_operation("E_M","Gs_m")) ## ['L']
print(my_group.apply_operation("LPR","B_M")) ## ['C_m']
print(my_group.mult("R","LPR")) ## "PL"

References

Alissa S. Crans, Thomas M. Fiore & Ramon Satyendra (2018): Musical Actions of Dihedral Groups, The American Mathematical Monthly, 116:6, 479-495.
Neo-Riemannian theory.

UTT_Group

opycleid.musicmonoids.UTT_Group()

This group is called the group of Uniform Triadic Transformations (UTT) and was introduced by Julian Hook in his Ph.D. dissertation. It acts on the set $\{n_M, n=0\ldots11\} \cup \{n_m, n=0\ldots11\}$ of the 24 major and minor triads, where $n_M$ (resp. $n_m$ ) represents a major (resp. minor) triad with root $n$ in the usual semi-tone encoding of pitch classes.

The UTT group is isomorphic to the wreath product $\mathbb{Z}_{12} \wr \mathbb{Z}_2$ of order 288 and its elements are usually notated as $(p,q,\sigma)$ , with $p=0\ldots11$ , $q=0\ldots11$ , and $\sigma \in \{+,-\}$ . Their action is as follows.

The UTT $(p,q,+)$ sends the major triad $n_M$ to $(n+p)_M$ , and the minor triad $n_m$ to $(n+q)_m$ .
The UTT $(p,q,-)$ sends the major triad $n_M$ to $(n+p)_m$ , and the minor triad $n_m$ to $(n+q)_M$ .

Example

from opycleid.musicmonoids import UTT_Group

my_group = UTT_Group()
print(my_group.get_operation("E_M","Gs_m"))
## Returns
## ['<4,1,->', '<4,3,->', '<4,9,->', '<4,7,->', '<4,2,->',
##  '<4,11,->', '<4,6,->', '<4,10,->', '<4,4,->', '<4,0,->',
##  '<4,5,->', '<4,8,->']
print(my_group.apply_operation("<7,6,->","A_M")) ## ['E_m']
print(my_group.mult("<3,5,->","<2,9,->")) ## "<7,0,+>"

References

Julian Hook: Uniform triadic transformations, Journal of Music Theory, 46(1/2), pp. 57–126, (2002).

Left_Z3Q8_Group

opycleid.musicmonoids.Left_Z3Q8_Group()

Like the $T\text{/}I$ group and the $\text{PRL}$ group, this group is an extension of $\mathbb{Z}_{12}$ by $\mathbb{Z}_2$ which acts simply transitively on the set $\{n_M, n=0\ldots11\} \cup \{n_m, n=0\ldots11\}$ of the 24 major and minor triads as defined above.

It is generated by the following two transformations.

The transformation $T_1 \colon \mathbb{Z}_{12} \to \mathbb{Z}_{12}$ such that $T_1(n_M)=(n+1)_M, T_1(n_m)=(n+1)_m.$
The transformation $J_0 \colon \mathbb{Z}_{12} \to \mathbb{Z}_{12}$ such that $J_0(n_M)=(-n)_m, J_0(n_m)=(-n+6)_M.$

Example

from opycleid.musicmonoids import Left_Z3Q8_Group

my_group = Left_Z3Q8_Group()
print(my_group.get_operation("E_M","Gs_m")) ## ['J0']
print(my_group.apply_operation("J0","Gs_m")) ## ['Bb_M']
print(my_group.mult("J3","T2")) ## "J1"

References

Alexandre Popoff: Building Generalized Neo-Riemannian Groups of Musical Transformations as Extensions, Journal of Mathematics and Music, 7(1), pp. 55–72, (2013).

Right_Z3Q8_Group

opycleid.musicmonoids.Right_Z3Q8_Group()

This group is the right version of the 'Left_Z3Q8_Group' above. It also acts simply transitively on the set of major and minor triads, and is generated by the following two transformations.

The transformation $T_1 \colon \mathbb{Z}_{12} \to \mathbb{Z}_{12}$ such that $T_1(n_M)=(n+1)_M, T_1(n_m)=(n-1)_m.$
The transformation $J_0 \colon \mathbb{Z}_{12} \to \mathbb{Z}_{12}$ such that $J_0(n_M)=n_m, J_0(n_m)=(n+6)_M.$

Example

from opycleid.musicmonoids import Right_Z3Q8_Group

my_group = Right_Z3Q8_Group()
print(my_group.get_operation("E_M","Gs_m")) ## ['J8']
print(my_group.apply_operation("J0","Gs_m")) ## ['D_M']
print(my_group.mult("J3","T2")) ## "J1"

References

Alexandre Popoff: Building Generalized Neo-Riemannian Groups of Musical Transformations as Extensions, Journal of Mathematics and Music, 7(1), pp. 55–72, (2013).

UPL_Monoid

opycleid.musicmonoids.UPL_Monoid()

A monoid which acts on the set $\{n_M, n=0\ldots11\} \cup \{n_m, n=0\ldots11\} \cup \{n_{\text{aug}}, n=0\ldots3\}$ of the 28 major, minor, and augmented triads, where $n_M$ (resp. $n_m$ , $n_{\text{aug}}$ ) represents a major (resp. minor, augmented) triad with root $n$ in the usual semi-tone encoding of pitch classes.

It is generated by the following three relations.

The relation $\mathcal{P}$ is the symmetric relation such that we have $n_M \mathcal{P} n_m$ for $0 \leq n \leq 11$ , and $n_\text{aug} \mathcal{P} n_\text{aug}$ for $0 \leq n \leq 3$ . This is the relational analogue of the neo-Riemannian $P$ operation.
The relation $\mathcal{L}$ is the symmetric relation such that we have $n_M \mathcal{L} (n+4)_m$ for $0 \leq n \leq 11$ , and $n_\text{aug} \mathcal{L} n_\text{aug}$ for $0 \leq n \leq 3$ . This is the relational analogue of the neo-Riemannian $L$ operation.
The relation $\mathcal{U}$ is the symmetric relation such that we have $n_M \mathcal{U} (n \pmod 4)_\text{aug}$ for $0 \leq n \leq 11$ , and $n_m \mathcal{U} ((n+3) \pmod 4)_\text{aug}$ for $0 \leq n \leq 11$ . It relates major and minor triads to augmented triads.

Note that for any of these generators, two triads are related if they differ by a single semi-tone move.

Example

from opycleid.musicmonoids import UPL_Monoid

my_group = UPL_Monoid()
print(my_group.get_operation("E_M","F_aug"))
## Returns ['PUPUU', 'UPUU', 'UPUPU', 'PUPUPU']
print(my_group.apply_operation("U","F_aug"))
## Returns ['Cs_M', 'F_M', 'A_M', 'D_m', 'Fs_m', 'Bb_m']
print(my_group.mult("U","UUPUU")) ## "UPUU"

References

Alexandre Popoff, Moreno Andreatta, Andrée Ehresmann: Relational PK-Nets for Transformational Music Analysis, arXiv:1611.02249.

S_Monoid

opycleid.musicmonoids.S_Monoid()

A monoid which acts on the set $\{n_M, n=0\ldots11\} \cup \{n_m, n=0\ldots11\} \cup \{n_{\text{aug}}, n=0\ldots3\}$ of the 28 major, minor, and augmented triads, where $n_M$ (resp. $n_m$ , $n_{\text{aug}}$ ) represents a major (resp. minor, augmented) triad with root $n$ in the usual semi-tone encoding of pitch classes.

It is generated by the $\mathcal{S}$ relation defined as the symmetric relation such that we have $n_M \mathcal{S} n_m$ , $n_M \mathcal{S} (n+4)_m$ , $n_M \mathcal{S} (n \pmod{4})_\text{aug}$ , and $n_m \mathcal{S} ((n+3) \pmod{4})_\text{aug}$ for $0 \leq n \leq 11$ . This relations is the same as the $\mathcal{P}_{1,0}$ relation introduced by Douthett in his work on parsimonious graphs. Two triads are related by $\mathcal{S}$ if they differ by a single semi-tone move.

Example

from opycleid.musicmonoids import S_Monoid

my_group = S_Monoid()
print(my_group.get_operation("E_M","F_aug"))
## Returns ['SSSS', 'SSSSSS']
print(my_group.apply_operation("S","E_M"))
## Returns ['E_m', 'Gs_m', 'C_aug']

References

Douthett, J., Steinbach, P.: Parsimonious Graphs: A Study in Parsimony, Contextual Transformations, and Modes of Limited Transposition, Journal of Music Theory, 42(2), pp. 241-263, (1998).

T_Monoid

opycleid.musicmonoids.T_Monoid()

A monoid which acts on the set $\{n_M, n=0\ldots11\} \cup \{n_m, n=0\ldots11\} \cup \{n_{\text{aug}}, n=0\ldots3\}$ of the 28 major, minor, and augmented triads, where $n_M$ (resp. $n_m$ , $n_{\text{aug}}$ ) represents a major (resp. minor, augmented) triad with root $n$ in the usual semi-tone encoding of pitch classes.

It is generated by the $\mathcal{T}$ relation defined as the symmetric relation such that we have, for $0 \leq n \leq 11$ ,

$n_M \mathcal{T} (n+4)_M,$
$n_M \mathcal{T} (n+8)_M,$
$n_m \mathcal{T} (n+4)_m,$
$n_m \mathcal{T} (n+8)_m,$
$n_M \mathcal{T} (n+1)_m,$
$n_M \mathcal{T} (n+5)_m,$
$n_M \mathcal{T} ((n+3) \pmod{4})_\text{aug},$ and
$n_m \mathcal{T} (n \pmod{4})_\text{aug}.$

This relation is the same as the $\mathcal{P}_{2,0}$ relation introduced by Douthett in his work on parsimonious graphs. Two triads are related by $\mathcal{T}$ if they differ by two semi-tone moves.

Example

from opycleid.musicmonoids import T_Monoid

my_group = T_Monoid()
print(my_group.get_operation("E_M","F_aug"))
## Returns ['TT', 'TTT']
print(my_group.apply_operation("T","E_M"))
## Returns ['C_M', 'Gs_M', 'F_m', 'A_m', 'G_aug']

References

Douthett, J., Steinbach, P.: Parsimonious Graphs: A Study in Parsimony, Contextual Transformations, and Modes of Limited Transposition, Journal of Music Theory, 42(2), pp. 241-263, (1998).

K_Monoid

opycleid.musicmonoids.K_Monoid()

A monoid which acts on the set $\{n_M, n=0\ldots11\} \cup \{n_m, n=0\ldots11\} \cup \{n_{\text{aug}}, n=0\ldots3\}$ of the 28 major, minor, and augmented triads, where $n_M$ (resp. $n_m$ , $n_{\text{aug}}$ ) represents a major (resp. minor, augmented) triad with root $n$ in the usual semi-tone encoding of pitch classes.

It is generated by the $\mathcal{K}$ relation. This is the same as the $\mathcal{P}_{2,1}$ relation introduced by Douthett in his work on parsimonious graphs. Two triads are related by $\mathcal{K}$ if they differ by the movement of two notes by a semitone each, and the remaining note by a tone.

References

Douthett, J., Steinbach, P.: Parsimonious Graphs: A Study in Parsimony, Contextual Transformations, and Modes of Limited Transposition, Journal of Music Theory, 42(2), pp. 241-263, (1998).

W_Monoid

opycleid.musicmonoids.W_Monoid()

A monoid which acts on the set $\{n_M, n=0\ldots11\} \cup \{n_m, n=0\ldots11\} \cup \{n_{\text{aug}}, n=0\ldots3\}$ of the 28 major, minor, and augmented triads, where $n_M$ (resp. $n_m$ , $n_{\text{aug}}$ ) represents a major (resp. minor, augmented) triad with root $n$ in the usual semi-tone encoding of pitch classes.

It is generated by the $\mathcal{K}$ relation. This is the same as the $\mathcal{P}_{1,2}$ relation introduced by Douthett in his work on parsimonious graphs. Two triads are related by $\mathcal{K}$ if they differ by the movement of a single note by a semitone, and the remaining notes by a tone each.

References

Douthett, J., Steinbach, P.: Parsimonious Graphs: A Study in Parsimony, Contextual Transformations, and Modes of Limited Transposition, Journal of Music Theory, 42(2), pp. 241-263, (1998).

ST_Monoid

opycleid.musicmonoids.ST_Monoid()

A monoid which acts on the set $\{n_M, n=0\ldots11\} \cup \{n_m, n=0\ldots11\} \cup \{n_{\text{aug}}, n=0\ldots3\}$ of the 28 major, minor, and augmented triads, where $n_M$ (resp. $n_m$ , $n_{\text{aug}}$ ) represents a major (resp. minor, augmented) triad with root $n$ in the usual semi-tone encoding of pitch classes.

It is generated by the two relations $\mathcal{S}$ and $\mathcal{T}$ described above.