Matrix of Embedded Rhythms –
Dotted Values
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Matrix of Embedded Rhythms –
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The above matrix (hereinafter EM2) provides patterns for the exact placement for 49 varieties of embedded rhythms. It is similar to Embedded Matrix #1 (hereinafter EM1) except that in this instance, it is the Below are two charts (A & B) of subdivisions. They show, in musical notation, the subdivisions from 1 to 8 for five different note values. The five notes that are subdivided appear at the top, in row 1, one note per column. At the left of the chart are the subdivision fractions. Chart (A) shows the subdivisions for non-dotted values.
All notes in row 1 are my choices. Dividing the values of row 1 by 1/2, 1/4, and 1/8 results in the usual given symbols that are found respectively on the rows labeled 1/2, 1/4, and 1/8. The logic dictating what note value(s) to use between these geometric divisions is: RULE 1: use the symbol for the larger value until the next rhythmic notation level has been reached, and the new symbol for that new level has been used. This explains (as an example — for the There is some dispute regarding the notation for division by seven, i.e. whether, for example, one seventh of a The two most frequently proffered defenses for this shorter-value alternative are: a) that 1/7 (= .143) of a beat is closer in duration to 1/8 (.125) than to 1/4 (.250), or in musical values (for a b) as discussed briefly in a footnote to chart (B) below, there exists a minor inconsistency which may be minimized by notating non-dotted-rhythm septuplets using the smaller values. Neither of these concerns have been sufficient to cause the majority of people to use the shorter-value alternative, so I have stayed with the more usual notation. Let us now consider the relationships across each row. The CRUCIAL point is: RULE 2: that the geometric relations across and for each row ARE CONSERVED either for (a) rows without superscripts, or (b) across values with the SAME superscripts or note suffixes (dots, double dots, etc). In other words, just as there are 2 In other words, the geometric ratios 1:2:4:8:16 etc remain inviolate, as long as all the note values are members of the same superscript i.e. all triplets, or all quintuplets, or septuplets etc. This frequently overlooked fact is probably responsible for more miscalculation and misunderstanding of "complex" rhythmic notation than just about anything else. Let us now turn to chart (B), which provides information similar to (A), but for dotted values.
As before, row 1 are my choices. The symbols for rows 1/2, 1/4, and 1/8 are those of chart (A), but with the addition of a dot next to all notes, so as to be consistent with the dot that has been added to the whole note. This addition of the dot across values may appear to be merely a convention, but is not. By definition a One can perform similar calculations for both 12 One can also derive the values for 1/3 and 1/6: i.e. 1/3 of a This leaves remaining the rows marked 1/5 and 1/7th. In regards 1/5, as this value occurs between 1/4 and 1/6, whose musical symbols are already given, one uses the symbol for the longer value and adds the number 5 as a superscript — in the specific case of the An additional reason for using the In regards 1/7, the use of As regards the EM2 itself, meters appear on the left, descending the page, and the As with EM1, in order to find the exact placement-pattern for a specific X:Y ("X in the space of Y"), find the row labeled X , and the column labeled Y, and locate the cell which exists at the intersection of that row and column. The method(s) of practicing remain the same. Indeed, unless the tempo is slow, and the The RULE for finding other exemplars remains the same, except that one must be comfortable with the subdivisions for dotted values found in chart (B) above, and one must be aware that, when calculating a pattern that does not exist in EM2, (or proofing a pattern which does!) the choice of an appropriate B (in using the formula X:YB) is particularly important. To pursue this, consider 5:21
Both patterns are arithmetically correct, and are congruent with each other, but provide different metrical information, and which pattern is more appropriate and/or efficacious depends on various factors. As an example of one of these factors — at a slow tempo, with the The discussion of EM1 regarding handedness and the psychology of production remains valid and applicable, as do the instructions regarding performance. Indeed, if you are comfortable with all the elements of EM1, you need only recognize the dotted notation of EM2 for the phantasmagoria it is. Paul Zukofsky |
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that is the basis for all given meters. The differences between EM1 and EM2 are primarily notational, and everything said in the EM1 explanation still holds; however, the specifics of precisely locating embedded rhythms within a dotted note context are far more irritating than for EM1, so it is appropriate to review and compare the differences between the systems that underlie EM1 and 2.
).
column) the use of the 
between 1/2 and 1/4; and the use of the 
, between 1/4 and 1/8 until one reaches the 
; etc. However, in order to distinguish the intermediary divisions from the original value (for example, to distinguish between a regular 
, one adds a superscript to indicate the actual arithmetic and durational value.
or 
. On the grounds of both efficacy and internal consistency, certain composers (Carter especially) recommend the shorter-value alternative, and under those circumstances the 1/7 row of this chart would read:
, 
, 
, 
.
.
in 1 
; 2 
in 1 
; 2 
= 6 
.
, (both of which = 1 
.
is the result of a process of elimination. One seventh of a 
as the progression of the duration-symbols within chart (B) would then be non-monotonic, given that we would have already used a shorter value (the 
) and then take 21 of them at a time, resulting in the pattern that may be found within EM2 — (also see ex. (a) 5:21 below). For 5:7 
, which we then take 7 at a time, resulting in the pattern of ex. (b) 5:7 below.
