How to read Csgrouper’s Metana()’s output

Below is the analysis of the row 1978624305 (in base 10).

S is the original Series (or “row”), I the schoenbergian Inverse, R the schoenbergian Reverse, O the schoenbergian Opposite and M the Mapped Opposite (see below).

In such a data sheet, the indices of characters in sequences are often used to denote cardinalities.

Simple Interval Sets (ict – for “intervals count”) are made with the cardinality of each one of the intervals (int) that one can pick in a row without octave shift: 0231110010 means that there are 2 intervals of 1 semitone, 3 of 2 semitones, 1 of 3 semitones, 1 of 4 semitones, 1 of 5 semitones, etc.

Extended Interval Sets (cyc – for “cyclic intervals”) are made the same way with the cardinality of the smallest intervals (cyn) that one can make in a row with octave shift.

Partition (par) is made with the n-uple subsets in the permutation. In the sequence “19527380 64”, there are 2 independant subsets, one of 8 permuting cyphers, and the other is a pair. Singletons, being on their own indices, do not permute.

Types are the partition type of a permutation: the cardinality of n-tuple subsets. In the preceding example 8-tuple is at 1 and 2-tuple is at 1, everything else being at 0.

For series (ser) “9530476812” which is the power 2 of ser “19527380” (ind = 2), the Degree of the row (deg) is 4 and the degree of its Opposite mapped onto “9530476812” (mdeg) is 10. The degree is the number of times one can apply a permutation till it comes to the “identity” (which is always “0123456789” in base 10).

The Gradual Suite is the list of permutations starting with S and to which S is applied till it comes to the identity.

The Opposite Suite is the list of permutations starting with O and to which the opposite is applied till it comes to the original permutation.

The Mapped Opposite is the operation of taking the Opposite and permuting it with the ultimate power of S (the last permutation in the gradual suite before reaching identity). This operation has much importance since it allows to separate the set of all permutations into smaller subsets that support transposition : transposing the original row and mapping its Opposite onto it, will yield the same permutation that is obtained by this process on the untransposed row.

The Unmapped Opposite Gradual Suite is the demapping of each row of the Gradual Suite made out of the Mapped Opposite.

Reading “P” as “permuted by”, the permutational notation of mapping O on S is :

(O) P (S^-1) 

Data sheet sample for the row 1978624305:

	Metana(1978624305)	:	
S: 1978624305 I: 1354608927 R: 5034268791 O: 7298064531 M: 2513846970
Gradual Suite:
ind ser int ict cyn cyc deg odg mod typ opt mot par
1 1978624305 821242135 0231110010 821242135 0231110010 8 10 8 0100000100 0110100000 2000000100 19527380 64
2 9530476812 423431271 0222200100 423431271 0222200100 4 10 10 2002000000 1001100000 0000000001 9230 5781 4 6
3 5281634097 367531492 0112111101 367531492 0112111101 8 10 10 0100000100 2000000100 0000000001 53128970 64
4 2709486153 579542542 0020230101 579542542 0020230101 2 10 10 2400000000 0000000001 0000000001 20 71 93 4 85 6
5 7315604928 424164576 0110312100 424164576 0110312100 8 10 10 0100000100 1001100000 0000000001 79821350 64
6 3892416570 517235127 0221020200 517235127 0221020200 4 10 10 2002000000 2000000100 0000000001 3290 8751 4 6
7 8057694231 852135212 0231020010 852135212 0231020010 8 10 10 0100000100 0110100000 0000000001 83725910 64
Opposite Suite:
ind ser int ict cyn cyc deg odg mod typ opt mot par
1 7298064531 571862122 0230011110 571862122 0230011110 30 10 4 0110100000 2100010000 2201000000 75640 291 83
2 3190846527 289842135 0121110021 289842135 0121110021 6 10 8 2100010000 2120000000 2000000100 30 1 975482 6
3 9410286753 531262122 0241011000 531262122 0241011000 6 10 4 2120000000 2010100000 2201000000 930 421 85 6 7
4 5312068749 221262135 0241011000 221262135 0241011000 15 10 8 2010100000 1100001000 2000000100 56840 321 7 9
5 1632408975 531248122 0231110010 531248122 0231110010 14 10 4 1100001000 0010001000 2201000000 1687950 32 4
6 7534280961 221268935 0131011011 221268935 0131011011 21 10 8 0010001000 0102000000 2000000100 7915860 342
7 3854620197 531242182 0231110010 531242182 0231110010 4 10 4 0102000000 2100010000 2201000000 3460 8971 52
8 9756402183 221242175 0240110100 221242175 0240110100 6 10 8 2100010000 1001100000 2000000100 936250 71 4 8
9 5076842319 571242128 0230110110 571242128 0230110110 20 10 4 1001100000 0100000100 2201000000 54810 7362 9
10 1978624305 821242135 0231110010 821242135 0231110010 8 10 8 0100000100 0110100000 2000000100 19527380 64
Gradual Omap Suite (unmapped):
ind ser int ict cyn cyc deg odg mod typ opt mot par
1 7298064531 571862122 0230011110 571862122 0230011110 30 10 4 0110100000 2100010000 2201000000 75640 291 83
2 9628304157 346534342 0013311000 346534342 0013311000 9 10 10 1000000010 2000000100 0000000001 971643850 2
3 2068534719 262321368 0132002010 262321368 0132002010 8 10 10 2000000100 1000000010 0000000001 26453810 7 9
4 6308154972 338741525 0112120110 338741525 0112120110 9 10 8 1000000010 1100001000 2000000100 641387920 5
5 0538714296 525163273 0122021100 525163273 0122021100 14 10 10 1100001000 2010100000 0000000001 0 51 3896472
6 3158974620 243123242 0142200000 243123242 0142200000 9 10 8 1000000010 0000000001 2000000100 382576490 1
7 5718294063 267675463 0011113200 267675463 0011113200 10 10 10 0000000001 0100000100 0000000001 5938642170

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