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Wednesday, 22 January 2020

More statistics

I read in Burn's work on the accents that he counts roughly 30 te'amim and roughly seventy or eighty different sequences. I give my conclusion at the bottom of this post.

Accent count
I count 26 accents in the table below. There are 31 noted in the Unicode table here. 4 are functionally the same, and one, the circle, is not a musical symbol.

Many of these 26 are very rarely used, kefulah for instance, only 14 times. qarne only 16 times.

The Accents of the Hebrew Bible relating to the Music. Updated table here.
Below the text
Above the text
Reciting
Note

Accent name

Ornament relative
to the reciting note
Full name of the ornament
c
֧
darga
֝֗
ger-rev
revia-mugrash
d
֢
galgal, tevir
֨
pas
pashta
e
ֽ
silluq
֜
ger
geresh
f
֥
merkha
֞
tar
tarsin
g
֭
tifha, (d'khi)
֡
paz
pazer
A
֑
atnah
֔
z-q
zaqef-qatan
B
֣
munah
֕
z-g
zaqef-gadol
C
֚
mahpakh, yetiv
֙
qad
qadma
dm
֦
double merkha, kefulah
֒
seg
segol



֘
zar
zarqa, tsinnor



֩
t-q
telisha qetana



֠
t-g
telisha gedolah



֟
qar
qarne farah



֓
shl
shalshalet



֬
ill
illuy



֫
ole
ole



֗
rev
revia
Sequence count
There are 1513 sequences used more than twice and 14,790 unique sequences. So Burns must be using sequence in a different sense if he numbers them in the 70-80 range. I wonder if I can discover why he says this.

Perhaps he is decomposing the sequences of approach to the Atnah and return from there to the silluq in the nested hierarchies that accent scholars are so fond of.

Can I think of an easy way to disentangle the music and find subsets of the sequences approaching the Atnah? If I remove the ornaments, I will not miss things that Burns would find significant because his hierarchy is entirely composed of accents below the text, i.e. the substance of the musical line. (That alone is significant).

So there are a limited number of ways to get from the start of the verse to A. The note just prior to A could be c, d, e, f, g, B, or C. We know A only occurs once per verse. So 7 possibilities here, C A, B A, g A, etc. If we take just one of them, g A, it happens in 9,256 verses. There are 2,229 distinct phrases with g A as part of them (tifha-atnah).

A little more poking about forces me to consider up to four notes at a time. There are 83 such combinations that occur in the text as music to approach the atnah. If I extend it to five notes, there are 201 combinations used. And bear in mind that I am ignoring the effects of all accents above the text.

Back up a bit. Obviously the combinations increase as I extend the number of notes. If I try just the last three notes before the rest, there are 27 ways actually used out of 42 (7x6) possibilities. How many of each, I wonder?

These are the counts of the approach to the atnah in all the books together, just counting the two, three, and four note sequences. So out of 210 possible sequences, we have 97 used. For the unique ones, I have put a reference in. For the common ones, there is too much data to list without writing a program to aggregate them. This is still oversimplifying the problem since there are 939 distinct sequences to the mid point rest where it exists.

Sequence of accents prior to the atnah, Number of verses
f g B A f-mer g-tif B-mun A-atn1894...
B g B A B-mun g-tif B-mun A-atn1768...
e g B A e-sil g-tif B-mun A-atn1177...
e f g A e-sil f-mer g-tif A-atn926...
e B g A e-sil B-mun g-tif A-atn832...
B f g A B-mun f-mer g-tif A-atn823...
d f g A d-gal f-mer g-tif A-atn633...
g B A g-tif B-mun A-atn615...
e B A e-sil B-mun A-atn602...
e g A e-sil g-tif A-atn498...
e f A e-sil f-mer A-atn478...
C B g A C-mph B-mun g-tif A-atn394...
d g B A d-gal g-tif B-mun A-atn377...
C g B A C-mph g-tif B-mun A-atn341...
C f g A C-mph f-mer g-tif A-atn305...
f e g A f-mer e-sil g-tif A-atn213...
g B e A g-tif B-mun e-sil A-atn198...
B g e A B-mun g-tif e-sil A-atn196...
f g e A f-mer g-tif e-sil A-atn194...
c d g A c-dar d-gal g-tif A-atn186...
B e g A B-mun e-sil g-tif A-atn169...
e C g A e-sil C-mph g-tif A-atn164...
f d g A f-mer d-gal g-tif A-atn122...
C e g A C-mph e-sil g-tif A-atn109...
e g e A e-sil g-tif e-sil A-atn105...
B C g A B-mun C-mph g-tif A-atn89...
B g A B-mun g-tif A-atn70...
g e B A g-tif e-sil B-mun A-atn69...
C B A C-mph B-mun A-atn66...
e f e A e-sil f-mer e-sil A-atn66...
e C B A e-sil C-mph B-mun A-atn51...
f B g A f-mer B-mun g-tif A-atn48...
f g A f-mer g-tif A-atn46...
f e f A f-mer e-sil f-mer A-atn45...
d g e A d-gal g-tif e-sil A-atn35...
C e f A C-mph e-sil f-mer A-atn30...
e A e-sil A-atn28...
e C f A e-sil C-mph f-mer A-atn27...
C g e A C-mph g-tif e-sil A-atn26...
g A g-tif A-atn25...
B A B-mun A-atn20...
B e B A B-mun e-sil B-mun A-atn20...
e g f A e-sil g-tif f-mer A-atn20...
d f B A d-gal f-mer B-mun A-atn19...
e d f A e-sil d-gal f-mer A-atn19...
e f B A e-sil f-mer B-mun A-atn19...
C g A C-mph g-tif A-atn18...
e d g A e-sil d-gal g-tif A-atn17...
C e B A C-mph e-sil B-mun A-atn16...
d e g A d-gal e-sil g-tif A-atn16...
f C B A f-mer C-mph B-mun A-atn16...
f A f-mer A-atn14...
C f A C-mph f-mer A-atn12...
B d f A B-mun d-gal f-mer A-atn11...
B g f A B-mun g-tif f-mer A-atn10...
C f e A C-mph f-mer e-sil A-atn9...
e B e A e-sil B-mun e-sil A-atn9...
f e A f-mer e-sil A-atn9...
f e B A f-mer e-sil B-mun A-atn9...
c B g A c-dar B-mun g-tif A-atn6...
f C f A f-mer C-mph f-mer A-atn6...
C g f A C-mph g-tif f-mer A-atn5...
g f A g-tif f-mer A-atn5...
g f e A g-tif f-mer e-sil A-atn5...
d e f A d-gal e-sil f-mer A-atn4...
f d f A f-mer d-gal f-mer A-atn4...
C d f A C-mph d-gal f-mer A-atn3...
d C f A d-gal C-mph f-mer A-atn3...
d e B A d-gal e-sil B-mun A-atn3...
f C e A f-mer C-mph e-sil A-atn3...
g e A g-tif e-sil A-atn3...
g e f A g-tif e-sil f-mer A-atn3...
B d g A B-mun d-gal g-tif A-atn2...
B g C A B-mun g-tif C-mph A-atn2...
C C B A C-mph C-mph B-mun A-atn2...
C f B A C-mph f-mer B-mun A-atn2...
e d B A e-sil d-gal B-mun A-atn2...
f C g A f-mer C-mph g-tif A-atn2...
g f B A g-tif f-mer B-mun A-atn2...
B C f A B-mun C-mph f-mer A-atn1Psalms 24:10
B e f A B-mun e-sil f-mer A-atn1Psalms 140:4
B f e A B-mun f-mer e-sil A-atn1Psalms 15:5
C A C-mph A-atn1Proverbs 1:9
C B e A C-mph B-mun e-sil A-atn1Proverbs 1:32
C B f A C-mph B-mun f-mer A-atn1Psalms 32:4
C d B A C-mph d-gal B-mun A-atn1Psalms 11:2
C e A C-mph e-sil A-atn1Proverbs 16:10
d C B A d-gal C-mph B-mun A-atn1Job 16:4
d e C A d-gal e-sil C-mph A-atn1Proverbs 6:3
e B f A e-sil B-mun f-mer A-atn1Job 30:1
e C A e-sil C-mph A-atn1Proverbs 6:27
e C e A e-sil C-mph e-sil A-atn12 Kings 5:27
f B A f-mer B-mun A-atn1Psalms 88:1
f g C A f-mer g-tif C-mph A-atn11 Kings 6:2
g B C A g-tif B-mun C-mph A-atn1Psalms 137:7
g B f A g-tif B-mun f-mer A-atn1Deuteronomy 5:6
g C e A g-tif C-mph e-sil A-atn1Proverbs 24:29

Similarly we can do a subset of the patterns of returning from the atnah to the silluq (there are 1349 patterns if we try to do them all). Here are 70 from 1 to 4 notes following the subdominant. Here I let those with just 2 examples show their reference, so a few lines are partially repeated. 64 examples in this table, including anomalies pointed out before, and one that I hadn't noted, a verse ending on the dominant.

Every note is represented within either the approach to the atnah (subdominant) or the return to the silluq (tonic). The theory hangs together well. The sequences of notes are all singable, memorable, and readable. There is still ample room for melismas in the ornaments, but the whole line is not a continuous stream of ornamentation. It has harmonic shape.

^A ^A-atn 2Numbers 25:19
^A ^A-atn 2Psalms 37:32
^A B ^A-atn B-mun 1Psalms 31:20
^A B C g e ^A-atn B-mun C-mph g-tif e-sil 42...
^A B d g e ^A-atn B-mun d-gal g-tif e-sil 3...
^A B d g f ^A-atn B-mun d-gal g-tif f-mer 1Deuteronomy 10:15
^A B e ^A-atn B-mun e-sil 388...
^A B e g e ^A-atn B-mun e-sil g-tif e-sil 107...
^A B f e ^A-atn B-mun f-mer e-sil 3...
^A B f g e ^A-atn B-mun f-mer g-tif e-sil 551...
^A B g e ^A-atn B-mun g-tif e-sil 730...
^A B g f e ^A-atn B-mun g-tif f-mer e-sil 754...
^A C B g e ^A-atn C-mph B-mun g-tif e-sil 263...
^A C e ^A-atn C-mph e-sil 25...
^A C e f e ^A-atn C-mph e-sil f-mer e-sil 2Psalms 146:3
^A C e f e ^A-atn C-mph e-sil f-mer e-sil 2Psalms 18:1
^A C e g e ^A-atn C-mph e-sil g-tif e-sil 96...
^A C e g f ^A-atn C-mph e-sil g-tif f-mer 1Deuteronomy 12:2
^A C f e ^A-atn C-mph f-mer e-sil 1Proverbs 27:1
^A C f g e ^A-atn C-mph f-mer g-tif e-sil 271...
^A C g ^A-atn C-mph g-tif 1Leviticus 26:28
^A C g B e ^A-atn C-mph g-tif B-mun e-sil 37...
^A C g e ^A-atn C-mph g-tif e-sil 235...
^A C g f e ^A-atn C-mph g-tif f-mer e-sil 282...
^A c B g e ^A-atn c-dar B-mun g-tif e-sil 22 Chronicles 34:3
^A c B g e ^A-atn c-dar B-mun g-tif e-sil 22 Samuel 3:32
^A c d g e ^A-atn c-dar d-gal g-tif e-sil 148...
^A d ^A-atn d-gal 1Judges 13:18
^A d f g e ^A-atn d-gal f-mer g-tif e-sil 427...
^A d g e ^A-atn d-gal g-tif e-sil 4...
^A d g f e ^A-atn d-gal g-tif f-mer e-sil 4...
^A e ^A-atn e-sil 670...
^A e B e ^A-atn e-sil B-mun e-sil 46...
^A e B g e ^A-atn e-sil B-mun g-tif e-sil 92...
^A e C e ^A-atn e-sil C-mph e-sil 1Proverbs 19:10
^A e C g e ^A-atn e-sil C-mph g-tif e-sil 30...
^A e d g e ^A-atn e-sil d-gal g-tif e-sil 11 Chronicles 10:9
^A e f ^A-atn e-sil f-mer 1Proverbs 24:15
^A e f B e ^A-atn e-sil f-mer B-mun e-sil 26...
^A e f e ^A-atn e-sil f-mer e-sil 339...
^A e f g e ^A-atn e-sil f-mer g-tif e-sil 239...
^A e f g f ^A-atn e-sil f-mer g-tif f-mer 1Deuteronomy 23:18
^A e g ^A-atn e-sil g-tif 2Genesis 32:24
^A e g ^A-atn e-sil g-tif 2Isaiah 13:16
^A e g B e ^A-atn e-sil g-tif B-mun e-sil 86...
^A e g e ^A-atn e-sil g-tif e-sil 210...
^A e g f e ^A-atn e-sil g-tif f-mer e-sil 178...
^A f ^A-atn f-mer 4...
^A f B e ^A-atn f-mer B-mun e-sil 225...
^A f B g e ^A-atn f-mer B-mun g-tif e-sil 7...
^A f C e ^A-atn f-mer C-mph e-sil 4...
^A f d g e ^A-atn f-mer d-gal g-tif e-sil 89...
^A f e ^A-atn f-mer e-sil 1429...
^A f e B e ^A-atn f-mer e-sil B-mun e-sil 7...
^A f e C e ^A-atn f-mer e-sil C-mph e-sil 1Psalms 127:1
^A f e f e ^A-atn f-mer e-sil f-mer e-sil 26...
^A f e g e ^A-atn f-mer e-sil g-tif e-sil 126...
^A f g ^A-atn f-mer g-tif 2Isaiah 13:7
^A f g ^A-atn f-mer g-tif 2Numbers 27:9
^A f g B e ^A-atn f-mer g-tif B-mun e-sil 2Psalms 47:5
^A f g B e ^A-atn f-mer g-tif B-mun e-sil 2Psalms 89:2
^A f g e ^A-atn f-mer g-tif e-sil 1576...
^A f g f e ^A-atn f-mer g-tif f-mer e-sil 1012...
^A g B ^A-atn g-tif B-mun 6...
^A g B e ^A-atn g-tif B-mun e-sil 893...
^A g e ^A-atn g-tif e-sil 1306...
^A g e B e ^A-atn g-tif e-sil B-mun e-sil 15...
^A g e f e ^A-atn g-tif e-sil f-mer e-sil 39...
^A g f ^A-atn g-tif f-mer 1Hosea 11:7
^A g f e ^A-atn g-tif f-mer e-sil 1212...

I didn't know what Burns means by sequence because my stats are significantly different from his. He uses sequence on page 55 and defines it as the music leading to the atnah and from the atnah to return to the tonic (silluq). But there are far more than seventy or eighty such musical sequences. His sequence is merkha-tifha-munah-atnah merkha-tifha-merkha-sof-pasuq. Sof-pasuq is not a note in this music. It really is punctuation. His sequence translates to (e) f g B ^A then from the A, f g f e.

His sequence occurs 3146 times and 1012 times. As you can see, using this example is an oversimplification of the problem of describing music. It's no wonder that this thinking does not lead to a breakthrough.

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