Forbes - sketching his overall divisions of the Psalter.

What has Forbes done in dividing the Psalms into their 7's, 10's, 12's, 20's, and 22's? 

I am using my cleaned-up copy of his treatise here. Help yourself to a copy. I update this copy as I discover more typos from the OCR process. [Updated Feb 5 to reduce my Editorial notes - I will put my impressions on the blog and only use editorial notes to clarify the few typographical errors I have found in the text.] This is also available on the Internet archive. I had trouble figuring out how to update it so the link has changed since the first one. I think I can keep them in sync by delete and replace now. The archive copy has more potential longevity, but my copy has immediate update.

Forbes was also a mathematician and 150 is a number with many possibilities for subdivisions: 150 = 3*7**2+3. Its prime factors are even simpler in base 10 thinking: 2 x 3 x 5 x 5. It is also the sum of the 7 consecutive primes from 7 to 31 - but that is not an attractive design principle. 

The number lends itself to endless divisions by 7. All such divisions have a centre. Centres are beloved by those who design and think in circular structures. Circular structures, also called chiastic, allow for writers to frame what seems important to them in what we otherwise might read only as sequential, or in the case of a book of poetry, random. It is a credible construction process that can be discovered by a sequential reader. There are many today who construct these when reading. All a human reader requires is a little bit of short-term memory (3 bits actually - binary 111 = 7 - each bit carrying a 1 or a 0, so 3 bits can carry 7 simple things or 3 complex ones in your short-term memory). No magic needed, but a pencil might be useful. The design should also be tested by the music embedded in the hand-signals in the text of the Tiberian tradition.

One of his divisions is: isolate Psalms 1 and then allow 2-22 (21 psalms) to be 3 'heptades' (that's his name) and counting back from 41, consider the series 20-40 as a second overlapping alphabet-1 of psalms, i.e. 3 heptades. The last of these has Psalms 37 as its centre. (Every 7 has a centre 3+1+3.) Psalms 37 is indeed an important psalm in the overall structure of the Psalter as I have noted before, and there are hints in the acrostics that the Psalmists might have considered 22 an 'important' structural number.

Several parts of the book point to Forbes' numerical thinking. E.g. this bit:

Ps. xxii., as has been already observed (see p. 82), would seem to be the termination of such an arrangement, being the concluding member of the trilogy xx.-xxii., which forms the centre of Book 1, there being 19 Psalms on either side of it. [Forbers-1888 in the section on Psalm iv.]

Book 1 is three short of 2 alphabets, so the trio 20-22 is included in each and so the book defines a triptych. I also note his division of the first 89 psalms into 44 + 1 + 44:  2 equal alphabets on either side of Psalms 45. (Nice touch). The final Book 5 is 44 psalms = 2 alphabets worth of psalms (but I don't think he mentions this). This book he divides into the three hallelujah books, creating a sevenfold structure for the psalter in the shape of a menorah.

The place and number of the Amens and Hallelujahs are found to be adjusted with remarkable precision, two Amens closing each of the three Amen Books, and forming, with the single Amen of the Central Book, in all seven Amens; while the Hallelujah Books, beginning each with the words, “O give thanks unto the Lord, for He is good: for His mercy endureth for ever,” close with Hallelujah Psalms, the whole number of Hallelujahs in the Psalter being twenty-four (in reference to the twenty-four orders of Levites and singers), allotted in significant proportions to each of the several books. [Forbers-1888 in the section on Book 5.]

Here's a grid with foreground and background colours to show the groupings that Forbes suggests. Of course he substantiates these numerical divisions with analysis - and at least to some extent he does, using recurring key words as his mechanism. I have not yet followed his full arguments to see if I concur. 

I	{{1	2	3	4	5	6	7	8
		9	10	11	12	13	14	15
		16	17	18	19	{20	21	22}
		23	24	25	26
		27	28	29	30	31	32	33
		34	35	36	37	38	39	40	41}
II	42	43	44	}45{	46	47	48
	49	50	51	52	53	54	55
	56	57	58	59	60	61	62
	63	64	65	66	67	68	69	70	71	72
III	73	74	75	76	77	78	79
	80	81	82	83	84	85	86	87	88	89}
IV			90	91	92	93	94	95	96	97	98	99
			100	101	102	103	104	105	106
V	107	108	109	110	111	112	113	114	115	116	117
VI	118	119	120	121	122	123	124	125	126	127
			128	129	130	131	132	133	134	135
VII	136	137	138	139	140	141	142	143
	144	145	146	147	148	149	150

There is not much analysis of the substance of Books 2 and 3 in his volume, though there is considerable detail on psalms 72 and 89. Nor have I found yet much on the last 15 psalms, the last 5 being obviously tied together in a crescendo of praise.