Database on tone rows and tropes -- Help on chord diagrams

Help on chord diagrams

Let f: {1,…,12}→ Z₁₂ be a tone row. The preimages of the six tritone pairs {0,6},{1,7},…,{5,11} partition the domain {1,…,12} into six 2-sets. There are 554 D₁₂-orbits of these partitions which correspond to Gauss words. Gauss words can be represented as functions of restricted growth from {1,…,12} to {1,…,6} where each element of the range occurs exactly twice. Franck Jedrzejewski computed the serial groups of tone rows. All tone rows of the orbit (D₁₂ × D₁₂)(f) determine the same Gauss word and the same serial group as f. Hence, these are properties of the orbit of f. According to F. Jedrzejewski there are 26 non-isomorphic serial groups and their orders lie between 24 and 12!.

Using this program it is possible to

to determine the chord diagrams of all tone rows given by the number of their D₁₂ × D₁₂-orbit,
display all D₁₂ × D₁₂-orbits of tone rows which yield a given chord diagram,
display all D₁₂ × D₁₂-orbits of tone rows the serial group of which has a given order. There are 3 non-isomorphic serial groups of order 384, which are indicated as 384a, 384b, and 384c, and 3 non-isomorphic groups of order 768, indicated as 768a, 768b, and 768c.

The following tables were found in a paper by Franck Jedrzejewski: They show the GAP-index of the serial group in the list of all 301 transitive groups on {1,…,12}, the order of the group, the numbers of chord diagrams which yield this serial group, and some information on the block types of these groups.

r	\|G\|	Chord diagrams	Blocks
12	24	358,554	B1
28	48	444,509	B1
54	96	186	B2
81	144	1,381, 553	B3
86	192	491,549	B3
118	216	50,359, 474	B4
125	288	487	B3
151	384a	414	B6
152	384b	383,497,507,508	B6
154	384c	136,536	B2
156	432	248	B4
185	768a	157,552	B6
186	768b	348,530	B6
193	768c	490	B2
217	1296	247,516	B4
218	1320	103,161,184,241,395,417,448,510
240	2304	43,351,440,537	B5
248	2592	364	B4
260	4608	544	B5
267	5184	11,282,303	B8
270	7680	42,61,111,347,355,382,412,441,504,538,551	B9
288	28800	24,51,56,150,163,183,252,332,371,376,419,420,439,481,514	B7
293	46080	109,185,346,350,357,488,531,535,546,550	B9
294	82944	170,174,285,289,313,321	B8
299	1036800	3,6,7,26,31,34,36,60,137,142,153,155,168,181,188,190,207,209,281,	B7
		290,295,305,334,352,362,418,434,482,494,511,532
301	479001600	431 remaining chord diagrams

The Block Systems for the serial groups are

B1	=	[ 1, 3, 5, 7, 9, 11 ], [ 1, 4, 7, 10 ], [ 1, 5, 9 ], [ 1, 7 ]
B2	=	[ 1, 3, 5, 7, 9, 11 ], [ 1, 4, 7, 10 ], [ 1, 7 ]
B3	=	[ 1, 3, 5, 7, 9, 11 ], [ 1, 5, 9 ], [ 1, 7 ]
B4	=	[ 1, 3, 5, 7, 9, 11 ], [ 1, 5, 9 ]
B5	=	[ 1, 3, 5, 7, 9, 11 ], [ 1, 7 ]
B6	=	[ 1, 4, 7, 10 ], [ 1, 7 ]
B7	=	[ 1, 3, 5, 7, 9, 11 ]
B8	=	[ 1, 4, 7, 10 ]
B9	=	[ 1, 7 ]

This is the lattice of the 26 serial groups computed by Franck Jedrzejewski. The groups are represented by their order. A red cross on a line connecting two groups indicates that the smaller group is a normal subgroup in the bigger one.

Database on tone rows and tropes
harald.fripertinger "at" uni-graz.at
November 29, 2022