# 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
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