In the Saṅgītaratnākara, a thirteenth-century musical text by Śārṅgadeva, listed under svaraprastāra (lit. extension of notes) is a complete enumeration of all the possible combinations of the 7 notes of the Indian musical scale. This enumeration begins with the single note (ārcika) and is followed by all the possible combinations of two notes (gāthika), three notes (sāmika), four notes (svarāntara), five notes (auḍuva), six notes (ṣāḍava), and seven notes (pūrṇa). Each of these series of kūṭatānas (series of notes in which the continuity of the sequence is broken) develop in the same logical order based on the precedence of the ascending line over the descending line. In Śārṅgadeva's arrangement the first of the 7 note series is the straight ascending line, sa ri ga ma pa dha ni, which, for easy comprehension will be rendered as l 2 3 4 5 6 7 in this paper; and the last of the 7 note series is the straight descending line, ni dha pa ma ga ri sa, rendered 7 6 5 4 3 2 1 here. The changes in the order of the notes take place from the beginning of the series, at first involving only the first two notes, then the first three notes, then the first four notes, and so on. In fact, the progression for the 7 note series includes the progressions for all the smaller series within it. Thus the 7th note of the 7 note series remains constant until the progressions of one, two, three, four, five, and six notes have been exhausted. Only then is the 7th note replaced by the 6th. The chart on p. 308 is an abbreviation showing the nature of the progression. The 2 and 3 note series involving the first 2 and 3 notes respectively are complete. Beginning from the 4 note series, the chart is abbreviated as follows. The 4 note series is divided into four groups determined by the terminal note, each involves change in the first 3 notes, and each of these groups corresponds to the previous 3 note series, which is in fact the first group of the 4 note series. Of the remaining groups only the first and last sequences are given with an indication as to the number of sequences comprising that group. Similar abbreviations are used in the longer series that follow. Commas have been placed to indicate that the preceding numbers now replace the original ascending sequence (mūlakrama) and that the progressions which follow in that group involve change in only these preceding numbers. For example, if one wishes to determine the complete series from 1 2 4,3 5 6 7 to the end of its particular group 4 2 1 3 5 6 7 the comma after 4 indicates that only the first three numbers change.
On the Complexity of Computing the Topology of Real Algebraic Space Curves