Friday, October 7, 2016

The shape of the music in the Psalms

I got a question today for my database:  How many verses of the Psalms follow, say, [silluq]-munah-tifha-munah-atnah-merkha-silluq? And list them in order, with the most frequent sequences at the top and the least frequent at the bottom.

Well this is far from a trivial problem. Here is a cross-tab of how a verse approaches the mid point rest on the subdominant and how it returns. There are 24 distinct ways in which the verses of the Psalms return to the tonic. There are 278 distinct ways in which the verses get to the subdominant. Many of these are unique.

Without the cross-tab, we would have 24*278 (6,672) distinct patterns. Even with the tables, it is tricky to assess common patterns.

These first 50 rows show the approaches to the subdominant that occur more than three times. There are another 50 or so that occur more than once, then the remaining 175+ are each unique approaches to the subdominant. This does not include those verses than have no atnah. And I have not considered the ornamentation here.

The table is too wide for a blog, so I have split it in two. If I were to use accent names rather than the tonal abbreviations, the headings would be wider and I think less suggestive of tonality. But I have simplified my query and the table has 706 lines so in the next post I have reduced it to the top 40, only those that happen in more than ten verses.

So for example below: There are 307 verses that have the shape do-sol-fa, of these 58 return directly to the tonic, 4 return do-sol-do, 1 has the shape do-re-sol-do, 28 have the shape do, re, do. and so on reading across the table row by row. So the most common pattern is in the first row and the column f e with the value 102. (in red)

Statistical tables are of some value (but not much). Perhaps this one is not perfect at revealing shape. It omits the common ornament revia-mugrash (r-m) and as such we cannot see how the return to the tonic (for r-m only occurs in the second half of a verse after the atnah) may be ornamented on the subdominant or on another reciting note.

There are some anomalies (headings in bold): the column headed f should not occur. This verse in the data is missing a final silluq. The column g B, similarly highlights verses that are missing a final silluq. I have looked at some of these in the Leningrad codex and they are indeed wrong. But tanach.us notes them. They probably need repairing. I found 11 verses in my data for the Psalms that do not end on the tonic and 12 others in the other books.

These numbers are suggestive of common patterns, but there are many possible patterns in music.

Approach to Returning to the Tonic via
the Subdominant Frequency e e B e e f B e e f e e f e f e e g B e f f B e f C e f e f e B e f e f e f e g B e
e B ^A  307 58 4 1 28 0 10 1 10 0 102 0 2 0
g B ^A  287 57 2 3 20 0 4 0 8 0 97 0 1 0
e f ^A  280 57 4 3 28 2 8 0 15 0 92 0 2 0
e B g B ^A  193 20 0 2 18 0 3 1 14 0 73 0 3 1
e g B ^A  172 27 1 0 11 0 1 0 7 0 63 0 2 0
e B g ^A  170 42 2 0 13 0 6 0 5 0 55 0 1 0
B g B ^A  56 6 0 1 3 0 0 0 1 0 28 0 1 0
C B ^A  39 4 1 1 5 0 2 0 1 0 10 0 0 0
e f e ^A  37 3 0 0 6 0 1 0 1 0 11 0 0 0
B ^A  35 3 2 0 1 0 0 0 1 0 13 0 0 0
B g ^A  29 7 0 0 1 0 1 0 1 0 9 1 0 0
e C B ^A  28 1 0 0 2 0 1 0 0 0 13 0 1 0
C B g B ^A  25 3 0 1 1 0 0 1 2 0 6 0 0 1
e C B g B ^A  23 2 0 0 1 0 1 0 0 0 9 0 3 0
e g ^A  22 5 0 0 1 0 0 0 3 0 6 0 0 0
e B g e ^A  20 1 0 0 3 0 1 0 1 0 9 0 0 0
e C f ^A  20 3 0 0 3 0 1 0 0 0 3 1 0 0
e C B g ^A  19 2 0 0 2 1 0 0 0 0 10 0 0 0
f ^A  18 5 0 1 0 0 0 0 2 0 6 0 1 0
e C g B ^A  16 4 0 0 0 0 0 0 1 0 6 0 0 0
e f g B ^A  15 7 1 0 0 0 1 0 0 0 4 0 0 0
e g f ^A  14 3 0 0 1 0 1 0 2 0 5 0 0 0
C B g ^A  13 0 0 0 2 0 0 0 0 1 7 0 0 0
e C e f ^A  12 2 0 0 1 0 0 0 1 0 7 0 0 0
e C e g B ^A  12 1 1 1 1 0 0 0 1 0 3 0 0 0
g e B ^A  11 0 1 0 1 0 0 0 0 0 7 0 0 0
C f ^A  10 2 0 0 1 0 1 0 0 0 4 0 0 0
e d f g B ^A  10 2 0 0 1 0 0 0 1 0 2 0 0 0
C g B ^A  9 3 0 0 1 0 0 0 1 0 3 0 0 0
e B e B ^A  9 1 0 1 2 0 0 0 1 0 2 0 0 0
e f e f ^A  9 2 0 0 1 0 0 0 0 0 1 0 0 0
e B e g ^A  8 1 0 0 1 0 0 0 1 0 3 0 0 0
e f B ^A  8 1 0 0 1 0 0 0 0 0 3 0 0 0
C e f ^A  7 0 0 0 0 0 0 0 2 0 2 0 0 0
e B e g B ^A  7 3 0 0 0 0 0 0 0 0 3 0 0 0
e C e B ^A  7 1 0 0 1 0 0 0 0 0 4 0 0 0
e f B g B ^A  7 1 0 0 0 0 0 0 1 0 1 0 0 0
e g e ^A  7 1 0 0 1 0 0 0 0 0 2 0 0 0
g B e ^A  7 1 0 0 1 0 0 0 0 0 3 0 0 0
B g e ^A  6 1 0 0 1 0 0 0 0 0 2 0 0 0
e d f ^A  6 1 0 0 0 0 0 0 0 0 2 0 0 0
e B d f ^A  5 1 0 0 2 0 0 0 0 0 0 0 0 0
e C e d f ^A  5 0 0 0 0 0 0 0 1 0 2 0 0 0
e d f B g ^A  5 2 0 0 0 0 0 0 0 0 1 0 0 0
e f B g ^A  5 0 0 0 0 0 0 0 0 0 5 0 0 0
f e f ^A  5 2 0 0 1 0 0 0 1 0 1 0 0 0
C B d f ^A  4 1 0 0 0 0 0 0 0 0 2 0 0 0
C d f g B ^A  4 1 0 0 0 0 0 0 0 0 1 0 0 0
C e g B ^A  4 1 0 0 0 0 0 0 2 0 0 0 0 0
e B d f B g B ^A  4 2 0 0 0 0 0 0 0 0 1 0 0 0
Approach to
the Subdominant Frequency f g B e g B g B e g e B e g f e B e C e C e f e C e g B e C f g B e C g B e
e B ^A  307 0 2 52 2 0 34 1 0 0 0 0
g B ^A  287 0 0 61 5 0 27 1 0 0 0 0
e f ^A  280 1 1 42 0 1 14 6 1 0 0 3
e B g B ^A  193 1 0 33 0 0 18 2 0 1 0 2
e g B ^A  172 0 2 39 1 0 9 3 0 2 1 1
e B g ^A  170 0 0 25 3 0 13 1 0 3 0 0
B g B ^A  56 0 0 15 0 0 1 0 0 0 0 0
C B ^A  39 0 0 8 0 0 7 0 0 0 0 0
e f e ^A  37 0 0 7 0 0 3 0 0 0 2 2
B ^A  35 0 0 11 0 0 3 0 0 0 0 1
B g ^A  29 0 0 7 0 0 1 0 0 0 0 1
e C B ^A  28 0 0 7 0 1 1 0 0 0 0 1
C B g B ^A  25 0 0 7 0 0 1 1 0 0 0 1
e C B g B ^A  23 0 0 5 0 0 1 0 0 0 0 1
e g ^A  22 0 0 6 0 0 0 0 0 0 0 1
e B g e ^A  20 0 0 2 0 0 3 0 0 0 0 0
e C f ^A  20 0 0 7 0 0 1 0 0 0 0 1
e C B g ^A  19 0 0 1 0 0 1 0 0 1 0 1
f ^A  18 0 0 2 0 0 1 0 0 0 0 0
e C g B ^A  16 0 0 2 0 0 2 1 0 0 0 0
e f g B ^A  15 0 0 0 0 0 2 0 0 0 0 0
e g f ^A  14 0 0 0 0 0 1 0 0 0 0 1
C B g ^A  13 0 0 2 0 0 1 0 0 0 0 0
e C e f ^A  12 0 0 1 0 0 0 0 0 0 0 0
e C e g B ^A  12 0 0 1 0 0 3 0 0 0 0 0
g e B ^A  11 0 0 1 0 0 1 0 0 0 0 0
C f ^A  10 0 0 1 0 0 0 0 0 0 0 1
e d f g B ^A  10 0 0 3 0 0 1 0 0 0 0 0
C g B ^A  9 0 0 1 0 0 0 0 0 0 0 0
e B e B ^A  9 0 0 2 0 0 0 0 0 0 0 0
e f e f ^A  9 0 0 4 0 0 1 0 0 0 0 0
e B e g ^A  8 0 0 1 0 0 1 0 0 0 0 0
e f B ^A  8 0 0 2 0 0 1 0 0 0 0 0
C e f ^A  7 0 0 2 1 0 0 0 0 0 0 0
e B e g B ^A  7 0 0 0 0 0 0 0 0 0 0 0
e C e B ^A  7 0 0 1 0 0 0 0 0 0 0 0
e f B g B ^A  7 0 0 2 0 0 2 0 0 0 0 0
e g e ^A  7 0 0 2 0 0 0 0 0 0 0 1
g B e ^A  7 0 0 2 0 0 0 0 0 0 0 0
B g e ^A  6 0 0 1 0 0 1 0 0 0 0 0
e d f ^A  6 0 0 1 0 0 2 0 0 0 0 0
e B d f ^A  5 0 0 1 0 0 1 0 0 0 0 0
e C e d f ^A  5 0 0 2 0 0 0 0 0 0 0 0
e d f B g ^A  5 0 0 1 0 0 1 0 0 0 0 0
e f B g ^A  5 0 0 0 0 0 0 0 0 0 0 0
f e f ^A  5 0 0 0 0 0 0 0 0 0 0 0
C B d f ^A  4 0 0 0 0 0 1 0 0 0 0 0
C d f g B ^A  4 0 0 2 0 0 0 0 0 0 0 0
C e g B ^A  4 0 0 0 0 0 0 0 0 0 0 1
e B d f B g B ^A  4 0 0 0 0 0 1 0 0 0 0 0



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