The last syllable of each stanza, the seimur, stands out for several reasons: It is usually rendered more softly than the rest of the stanza, it is ornamented with dillandi more often than any other part, it is held relatively long, and its pitch is stable. To hold it is called a draga seiminn (lit. to draw the seimur out). As mentioned earlier it was a common practice for at least one or two persons in the audience to take part in this task, according to Thórur and Gunnar so that the performer could relax. That it helped the performer not to fall in pitch, as the physician Jón Jónsson has suggested, seems plausible since stability of pitch is clearly an important attribute to seimur;1 in fact it seems to be the only pitch which should ideally remain constant from one stanza to the next. Thus the seimur of Margrét in a performance of seven stanzas (beginning with No. 13a) and of Gunnar in a performance of ten stanzas (beginning with No. 14) does not fall more than one-eighth of a tone. Kristján does not fall at all in his performance of seven stanzas of which No. 1 is the first, but he ends the fourth approximately a semitone higher than the others, and this happens too in the first stanza of No. 7 and on the second stanza of Jónas' No 24b. The higher pitch in such cases must be regarded as a legitimate alternative since it occurs frequently in the kvæaskapur of the whole Breiafjörur area and even as a rule in the performances of Thórur and Thorgils.
Possibly more than one function could be ascribed to the seimur, but with regard to melodic motion it seems primarily to function as an intermittent drone, i.e., the pitch against which the pitches of the melody are measured. Otherwise there would hardly be any reason to keep it constant, as the performers evidently tried to do. Gumunder is the only exception. His seimur falls rapidly in pitch, and when it reaches a certain level (usually near "d"), he transposes an interval of a second or a third upwards anywhere in the performance. In the second ríma from Rimur af Svoldarbardaga (see p. 13) this happens six times, the first time already after the first two stanzas, since he starts low and reaches his limit at the end of the second stanza.
Melodic motion is dominated by seconds of various sizes. Thirds are relatively few, and intervals larger than a tritone do never occur. Whereas the whole pitch continuum of the melodic range generally seems to be used, the question arises whether this continuum can be subdivided into a limited number of pitch classes or tone units. The pitch of such tones would necessarily vary above the threshold of perception if the number of tones is supposed to be within reasonable limits.
Svend Nielsen states that a range of three tones is primarily in Thórur's I kvæaskapur and that it can be extended upwards by adding one, two, or three tones and downwards by adding one tone.The distance between the tones are variable but approach the intervals of the major scale.2 The range of approximately a major third is certainly of primary importance in Breiafjörur. In the most usual range for the A contours, the C contour, the first and last line of D , and the second half of F . If this range contains only three tones it cannot be traversed in more than two steps, but since three steps are frequently found as an alternative, at least four tones must be involved. The following figure (Fig. 5), where 1 and 4 represent the limit of the range and 2 and 3 are two tones dividing it, illustrates both cases (a, b, c and d, e, f respectively):

Fig. 5

Tone 2 falls into the middle of the range in e and 3 in f, while the other tone remains near the limit. In c, on the other hand, where the range is divided into two equal intervals, it is impossible to tell if it is divided by 2 or 3. If an additional tone x between 2 and 3 were supposed, progressions in four small seconds (roughly a semitone each) involving 2, x, and 3 could be expected, but they never occur. The range of approximately a minor third is usually traversed in two intervals according to a, b, or c of Fig. 5, but three small seconds are also possible. Fourths are traversed in two or three intervals like the thirds but tritones in two, three, or four intervals.
Ranges greater than a tritone are usually divided into smaller ranges by changes of direction, reiterations, or both, and not traversed directly. The I stemma of Jónas (No. 24) is the most important exception in this respect with its range of a seventh frequently swept through without any detour. Parts of this example (especially No. 24a and c) and several of the others (for instance the first half of No. 22a2, the second half of No. 33, and the second half of No. 36) seem to be based on a scale of eight tones where small and large seconds alternate:

Major thirds and fourths are divided into three seconds, while tritones are divided into four.
At least one tune based on this scale has apparently been known in Breiafjörur and Vestfirir for at least two hundred years. It was first published in the year 1780 in the second volume of Laborde's Essai sur la musique ancienne et moderne and taken down in Copenhagen by J.E. Hartmann probably shortly before. According to Jón Helgason the performer was Jón Ólafsson from Svefneyjar in Breiafjörur.3 The text is a stanza from the religious poem Lilja composed in the 14th century in the skaldic metre hrynhent . Half a stanza in this metre with regard to the number of feet and lines is the same as a stanza in the rímur-metre n´ylanghenda (IIg). Since the melody is repeated for the second half of the stanza, it could easily be used for IIg, and this seems actually to have happened. Below, the first couplet of the Lilja stanza transposed a tritone upwards (a) is compared with a IIg stanza from Núma rímur (b) performed in 1967 by Gisli Jónsasson, who learned it in his youth in Vestfirir:

The pause in the third measure of b is due to lack of breath, and there is a slight fall in pitch at the end, but no major deviation from the scale apart from the "b-flat"at the beginning of the second measure. The manner of performance is more like singing of hymns than kvæaskapur, and lengthening the last syllable of third line is extraordinary for a II metre, so b is apparently a song (sönglag) rather than a stemma. In spite of considerable differences a and b have basic contour and scale in common and look like two versions of the same tune.
Even if the eight-tone scale is a relatively stable element in a tune intended to be sung, it is hardly so in kvæaskapur, where it is especially contradicted by the frequent division of the major third into two and the tritone into thr ee large seconds. One of the reasons why this scale seems to be present so often in kvæaskapur could be its structure of interlocking thirds.
After the initial ascent in the Lilja melody (a) moves within a major third subdivided into two interlocking minor thirds until the third line. This line has no parallel in kvæaskapur any more than the others, but if the two high pitches, "c" and "b," are lowered a semitone each, the resulting progression of pitches ("a," "b-flat," "b," "b-flat," "g-sharp," "a," "f," "e") would be almost the same as in Thóur's third line of No. 22a1, which in turn resembles Karl's second line of No. 31b. In both cases the range of a tritone is divided into a minor third containing four pitches ("b," "b-flat," "g-sharp," "a") and an interlocking major third ("a," "g-sharp," "f") with two pitches in common. In Karl's last line of No. 30b the same progression again recurs except now the highest pitch is about a semitone higher, and in the lower range of a third a fourth pitch ("f-sharp") had been added to the three in the preceding examples. In this case the pitches fit the eight-tone scale except the "b-flat". But rather than regard the the second line of No. 31b as a contradiction of this one it should be judged normal and the second an expanded variant, as the range of a tritone is more usual in this place in the contour than the range of a fifth. The third line of No. 31b is essentially a repetition of the second and the second line of No. 31b resembles it still more. But now the same range is divided into two major thirds a major second apart, the higher divided in the middle and the lower as before ("a," "g-sharp," "f"). In the second line of No. 30a no elements of the eight-tone scale are present, as both thirds are divided into two large seconds, but in other respects the two lines are almost identical. The first line of stanza 2b also closely resembles the first line of 1a except that it is a small second higher as a whole. The same happens in Thórur's No.22. The first line of a3 is higher but otherwise exactly the same as the first line of b, whereas in the preceding stanza (a2) the line as a whole is not higher but its range extended upwards to a major third. Similar traits characterize the second and the last lone of a2 and the third line of a3 when compared with b.
From the examples cited melodic subdivisions clearly can be expanded or contracted while other parts remain within the same range. In such cases the total range is not fixed but results from the combination of smaller ranges variable in relation to each other no less than the tones dividing them. At the same time it is difficult to assign the pitches to definite scales or combinations of scales unless one is ready to admit that there is no unified system and only a collection of odds and ends.
Above, variable tones have been considered only within short segments. They will now be examined in a wider context but within a range permitting at most four tones, according to the limits discussed earlier.
In Pétur's No. 4,1 the main tones seem to be two ("f" and "g-flat") and the distance between them to expand from approximately a minor second in the first line to about a major second in the third measure. At the end of the measure a third tone ("e") extends the range less than a minor second downwards. The range contracts again, but the tones are still three at the end of the couplet. In the first half of the second stanza, on the other hand, the interval between the two tones is close to a major second most of the time. At the beginning of the second couplet it is traversed in two steps involving a tone ("f-sharp") which seems to be the third main tone, since the stanza ends on it. If the tone below the main one ("e") is numbered 1 and the main tones 2, 3, and 4, the last line of both stanzas would be 44221244 followed by a trill between 3 and 2. At the end of the first stanza (432) the interval of approximately a neutral second (150 cents) between a slightly flat "g" and "f" is divided near the middle by a trilled "g-flat." The same interval also divided in the middle occurs at the end of the last syllable of the first couplet, but now it is possible to interpret the progression in two ways. If the third measure is 44221 in accordance with the seventh then the fourth begins with 1, but it is not clear whether it is followed by 321 or 432, especially since the pitch of the slightly heightened "e" and the lowered "f" is approximately the same. Also it is questionable whether the stanza begins with 23, 34, or even 24 as the second stanza.
Apparently it is impossible to decide with certitude what pitches represent each of the variable tones and consequently their range of variability with regard to each other and the pitch of the seimur. And since a variable tone exists only as a range of variability, no valid system of such tones for the entire example can be defined. If on the other hand the pitches are simply regarded as the boundaries of the intervals but not as the representatives of particular tone units, an altogether different approach is possible. In the following equations expressing a progression of n pitches, p1, p2, p3...pn represent the pitches as distances from P0, the pitch of the seimur:

(1) (p1-p0) + (p2-p1) + (p3-p2)... + (pn-pn-1) = pn-p0

The value for P0 will be put at zero for all performances since the absolute pitch does not matter in this context. Pitches above the seimur then are positive and those below negative. Each bracket can be replaced by the interval size (i) if the same measuring unit is used as for the pitches. The nth pitch would then be expressed as the sum of the intervals leading up to is as in (2). Values for ascending intervals are evidently positive and for descending intervals negative while reiterations are zeros.

(2) i1 + i2 + i3 ... + in = pn

If the intervals, treated as distances only, actually combine according to a definable pattern, then this pattern could be expected to reflect the tone system of kvæaskapur with intervals rather than undefinable pitch-classes as the basic elements.
Returning to Pétur's No. 4, the neutral second is the smallest divisible interval in the whole performance (16 stanzas). Half this interval (ca. 60-80 cents) will be regarded as the smallest second possible and smaller differences between adjacent pitches as zeros (reiterations). The largest interval is approximately midway between a major second and a minor third (ca. 250 cents) and occurs in two stanzas where the range is expanded still further upwards. If 1 stands for seconds close to a tempered semitone (100 cents) those close to a whole-tone are 2, the neutral seconds 1.5, the smallest 0.7, and the largest 2.5. Not only Pétur's performances but generally in the other examples where relatively accurate measurements have been possible, the greater part of the seconds may be divided into 1.2 and 1.5 if the range of variability is not less than ±25 cents. At the same time seconds between 1.5 and 2.5 are divisible while those between 0.7 and 1.5 are not. This distinction between two classes of seconds, small and large, seems therefore to be justified if 1.5 is classified with either depending on the context.
Since any pitch delimiting a melodic range is not fixed in advance but results from a progression of intervals, ranges larger than 2 will be considered to be built up from the smallest ranges, the seconds. The variability of seconds will be disregarded for the moment and consequently the variability of larger elements, and it will be assumed that the mean values for small and large seconds will never be far from 1 and 2 in the long run.
If 1 and 2 are the only primary elements, secondary elements resulting from their additions would be +1+1, +1+2 (or +2+1), and +2+2 upwards and -1-1, etc. downwards. Column a of Table 1 shows the primary elements and column b the secondary elements, with pluses and minus omitted (11, etc.). The elements in column c are obtained by adding a to b elements (placed in brackets) and by adding the second b element (12...) to itself, an operation analogous to adding the a elements to themselves producing 11 and 22 in b. Other additions within b would amount to the same as adding a and c, but further extensions of c elements (11) is not only an addition but also 2 divided, since it sums up to 2. Summing up 12... and 2 on the other hand results in the two new elements 3 and 4 (the small and the large third) which will be considered as reductions of b and symbolized with b'. Summing up the brackets of c yields c' elements, the elements in the first and the fifth line of c being excepted since they sum up the b elements 12 and 22 respectively.

Table 1

It is possible to regard c' as additions of a and b' as well as reductions of c (except 33) and consequently related to b' in a similar way as b is related to a. The relationship between the five classes (the columns of the table) is shown in the following diagram, where the reductions are placed vertically below the unreduced elements and extensions (resulting from addition of intervals) are arranged horizontally from left to right. Classes containing elements made up of the same number but different sizes of intervals are then connected diagonally and expansion of intervals indicated by downward displacement of the classes from left to right:

Fig. 6


Extension and reduction could obviously also be applied to melodies based on scales, extension increasing the range by adding a tone (for instance "a" to "f-g" resulting in "f-g-a") and reduction omitting a tone (for instance "g" from "f-g-a" resulting in "f-a") without changing the range. Both operations suggest in fact scale degrees. Expansion (for instance of "f-g" to "f-a") changing the range without increasing or decreasing the number of intervals means, on the other hand, widening or contraction of intervals not resulting necessarily from a definite number of added or omitted steps. In the following, extension and reduction are only a first attempt to approach expansion, which is really continuous.
This investigation will at first be limited to the progressions through the ranges 1-6, extracted from the material according to the expression (2) and classified as elements in Table 1. (The interval sequence 0+2+1+1 broken down into elements +0+3-2+0-1+3-3-2 broken down into the elements 2(11), 2, 3, 2, 3, 32). The greater part of the melodic progression of the contours A-G can be described in terms of the tabulated elements, and at the same time there is not a significant element in Table 1 not found in the examples. Nos. 33 and 24 are the only examples based on the contours with several progressions not accounted for in the table.
The distinct elements in Table 1 are considered to be 29 in number. The second line of b is then regarded as a single element (indicated by 12...). Its form is aa for both the variants 12 and 21 frequently replaced by 1.5 1.5, as neutral seconds are found more often in this connection than any other. Thus the element appears as simultaneous combination of 1 and 2 rather than an extension from either. The same applies to either half of the only element composed of four intervals (the fourth line of c) produced by adding the second b element to itself, resulting in four variants ((12)(12), (21)(21), (12)(21), (21)(12)) besides the additional ones containing neutral seconds. It seems right, on the other hand, to keep apart c elements with identical successions of intervals, 2(22) and (22)2, and 1(12) and (11)2 for example, according to whether or not they are ab or ba. This distinction expresses different relations with the b elements included, and consequently different c' elements (ab', b'a), which can hardly be confused, result from their reduction. It is clear that elements appearing for the second time (in square brackets) belong to two classes if they are reductions of 11 and 11 extensions in c, and this view would be more in keeping with the general order of the table than to consider them as divisions.
In order to clarify the relations suggested so far the elements are arranged in Fig. 7 within classes a-c' according to Fig. 6. Elements with the same range and number of intervals then necessarily fall on the same spot, as they do not relate as extensions to each other and are therefore represented with the common range only; c4 thus stands for all c elements with the range of 4. The encircled 4 stands both for b4 and c'4, and c6 stands for the fourth line of c in Table 1, the only element consisting of four intervals. Numbers representing extensions from more than one element in the adjacent diagonal to the left are placed between them (c4 between b2 and b3 and c5 between b3 and 4 for instance).



Besides the central circled 4, all of the numbers 1, 2, 3, and 6 fall simultaneously into horizontal and diagonal rows except c6 which is connected only horizontally with b3; whereas the two remaining 4's and both 5's have only diagonal but no horizontal connections. If the intervals 4 and 6 (the tritone) are included with the elements connected both horizontally and diagonally a scheme of two interlocking triangles is formed where the horizontal and diagonal rows of each are 1:2:3 and every point in the higher is related to a corresponding point in the lower according to the ratio 1:2.

Fig. 8


The elements of the second (the lower) triangle represent the classes of Fig. 6 expanded to cÿ, and additional class obtained by summing up the c' elements and represented here by the tritone. Since the structure of the first triangle is the same, the corresponding elements of both are of the same order. Accordingly, 2 is b' in relation to 1 and b2 but a at the outset of the triangle, and therefore in this context it is the upper limit for the class a and the lower for b'. Similarly b3 is the lower limit for c' and 3 for cÿ (if Table 1 contained cÿ, 3 and 4 would appear there in square brackets between 5 and 6 without brackets). Thus the elements of the first triangle represent the system at its lower limit. The basic structure then is already possible within the range 3, the range of the narrowest examples (Nos. 3-6).
Since 2 is a with regard to 4, b4 and c6, it seems reasonable to consider 3 primary in relation to c'6 and its extension to 9 (in brackets in Fig. 8) which is the widest range of three-interval progressions in the examples. Then 3 may be primary in addition to 1 and 2 at the higher limit, whereas only 1 can be admitted at the lower limit. Hence it is only possible to fix the hierarchical order of the small second, as the rank of other intervals varies according to context. From this viewpoint the system is an expansion in two stages from the lower limits (the first triangle of Fig. 8) producing the remaining elements of Table 1 at the first stage (accompanied with a single extension, b3 to c6) and the widest progressions at the second stage. The three divisions of the system may conveniently be called narrow (1 primary), medium (1-2 primary), and wide (1-3 primary).
Obviously the first horizontal alone (1-b2-c3) could be taken as the starting point for the expansion. But if the interval zero instead of seconds is considered the basis of the system the horizontal in turn appears as a gradual expansion from an element of three zeros (not necessarily articulated by reiterations) when the elements are arranged as in Fig. 8.

Fig. 9

In this connection 11 and 2 are the two possibilities of expanding 1 and two zeros (the order of zeros determines three variants): Then 11 leads to c3 and b3, and 2 to 3 but also to b3 if one of its zeros is replaced with 1. Similarly, it is possible to derive the c4 elements from b3 as well as c3. Continuing in this way, the elements of Table 1 are gradually produced as indicated in the following arrangement (the zeros completing elements consisting of less than three non-zero intervals are omitted):

Fig. 10


All diagonals, whether from right to left or left to right, express expansions of a similar order where zeros are generally replaced with 1, 1 with 2, and so on, + upwards and - downwards. The horizontals correspond to the columns of Fig. 7 and Fig. 9, but now it is possible to replace the conflicting notions of division and reduction with expansion working both ways. Any element can be divided into two parts, left and right, and either part into two intervals (0(03) or (00)3 for instance), and a two-way expansion is simply brought about by adding 1 to the left and subtracting 1 from the right or vise versa. Divisions and reductions then result from the same operation applied repeatedly, as illustrated by the completion of the third line of Fig. 10.

If the process is continued, combinations of intervals in opposite directions result, for example in the case of the upward expansion:

The first two-way combinations produced in this manner from the ranges of 0 to 3 correspond to the elements of Table 1 with opposite signs for two intervals (for instance, as above: +2+2-1 and +4-1 in place of +2+2+1 and +4+1).
Thus including zeros in the system simplifies the basic relations since extension, reduction, and division are replaced by expansion, and this leads at the same time to a wider interpretation of the sequence of Table 1. Evidently only a few of the possible alternatives can be used in a single syllabic stanza, i.e., a stanza with one note and occasionally two notes to a syllable. In larger samples, however, they may be assumed to be used equally if the system makes no selection besides its division into narrow (1 primary), medium (1-2 primary), and wide (1-3 primary). Proportionate use of intervals for medium examples could then be expected to be in accordance with Table 1, where frequency of occurrence of the intervals 1, 2, 3, and 4 respectively is 28, 28, 7, 5. The narrow examples, on the other hand, are limited for the most part to the first four elements (1, 2, b2, and b3) consisting of four small and two large seconds; the ratio of 2:1 could therefore be expected.
Table 2 shows interval counts for syllabic examples with contours A-D and with a range of not less than 4. Since the number of ascending and descending intervals of each size tends to be the same, separate counts are not given. Articulated zeros (reiterations), on the other hand, are counted with intervals 1-4 and included in the number of total intervals. The only fourth occurring is counted as 4. Grace notes are omitted, and the interval between the first pitch of the stanza and the seimur is not counted:

Table 2

The intervals of the narrow examples, Nos. 3-6, are counted as before with a detailed count given only for No. 6.

Table 3


The proportion between small and large seconds is 2:1 as expected while zeros are half of the total intervals. Besides, there are only a few thirds, so the proportionate use of zeros and small and large seconds is near 3:2:1. Four of the narrow stanzas, Nos. 3, 4(1-2), and 6b4, approach the average rather closely.
Percentages for syllabic E examples are presented in Table 4 and F in Table 5.

Table 4

Table 5


The only wide example (No. 24) and the narrowest example (No. 25) are included in Table 4. Percentages for the remaining examples of the two tables approach the medium and narrow averages of Table 2 and 3 respectively to varying degrees. Thus Nos. 27 and 34 appear to be on the borderline between narrow and medium, and No. 33 between medium and wide. On the whole, the frequency of small seconds tends to be relatively constant while the frequency of larger intervals increases with widening, mainly at the cost of zeros. Ornamentation, on the other hand, increases the total number of intervals by adding seconds for the most part.
As already suggested, the content of Table 1 can be interpreted as interval combinations in one or two directions resulting from two-way expansion. If two-way expansion is simply considered to be the basic operation of the system, narrowing and widening appear as two aspects of the same process, and the sequences may be seen as the result as well as the material to which the process is applied.
What is meant by the operation 11 (+1-1 or -1+1; underlining indicates the opposite sign) is applied repeatedly to itself in the same was as previously 003.

The sequences are evidently combinations of a,b, and b' elements with opposite signs summing up to zero; at the same time they can be regarded as operations increasing in weight from left to right, weight meaning at this stage the number of applications of 11 of a given length. The difference between adjacent sequences of the same weight is 121 reading the columns from top to bottom (for instance, 123,+121 = 2(0)2), which leads to a second set by applying 11 as well as 11.

The two sets could also result as the relations (the differences) between all progressions in the same direction within the range 4; i.e., no other operations are needed to change any progression into another (for instance 013+023 = 130) whether close to each other or not when all are brought to the same length and arranged as in (3).

On the whole the operations of the two sets, with a few additions to the second (in the bracket above), seem to contain the possible three-interval operations of the system. In the simplest cases only 11 occurs, as in the following example, where the last line of b (in what follows only articulated zeros will be written in such progressions), and the relations, the operations needed to change a into b, are written below the brackets.

It is clear that (-1+1) lowers by 1 two melodic segments which are separated by the "f" on the third syllable, the only pitch both lines have in common besides the "a" at the beginning and the final "f." But the length of the two segments is different, the first two eighth-notes and the second three quarter-notes. In the following, the length of an operation will be determined by the length of the segment after it has been raised or lowered.
It might seem that Kristín uses two relatively fixed formulas at the beginning of her A' couplets, the first ascending on the second syllable and the second on the fourth syllable. Of her nine recorded 1a stanzas with this contour one begins as 10b with a variant of the second ascent 0031, whereas two of the remaining stanzas begin the second couplet with 1030 and 0310 respectively. Thus intermediate forms exist related to both extremes as indicated below (3310 and 22000 besides 11200 are the variants of the first ascent).

If the material were greater perhaps most or all progressions of (4) would appear there as in Thórur's A stanzas, where a similar displacement of two syllables takes place but with the limits one further to the left; i.e., a segment corresponding to the first two syllables of the first line, instead of the second and third syllables, is raised or lowered by approximately 4, the extremes being roughly:

In terms of line contours a' changes into 'a' and occasionally a into a' or vice versa. Kristín's second ascent on the other hand never occurs. Thus displacements of ascents and descents in A contours of range 4 can be seen to result from two-way expansion within a limit of two syllables. It is even possible to connect the B contours with this process, at least in Thórur's performances, where the first couplet of several stanzas is 'A", i.e., A" beginning a third above the seimur but ending below it as B. In a few cases it begins with an ascent changing it into A". Then Bleads to 'A" which leads to A" and finally A" leads to A' expresses the general relation. The first operation would evidently be -1-2+3 for B as it is in No. 22b if the first line of 'A" consists of zeros only:

The relatively simple operations mentioned so far with regard to parts varying in length from one syllable to a whole line could mean that not only intervals and details of melodic contour but also the generalized contours of Fig. 1 are at least partly governed by the same principle. Before examining this matter closer, however, temporal organization must be considered and limits of length for the operations in question found.

Generally, it is possible to divide metric syllables according to length into two classes, long and short, disregarding lengthened syllables at the end of lines for the moment. When all syllables are of the same length, as in Nos. 10 and 16 for example, they will be considered long. Short syllables appear then only with the long as approximately half of their length. If 1 stands for short and 2 (usually transcribed as a quarter-note) for long, their combinations into feet are 21, 12, and 1.5 1.5 (usually transcribed as two dotted eighth-notes). Such feet can be indicated by 3-, and feet of two longs by 4-, and by 5- when one of the syllables is lengthened to 3. These three varieties are combined freely; two shorts (2-) on the other hand occur only as the first foot of a measure of two feet.
Average absolute syllable length tends to be the same from one stanza to another even in long performances and consequently the length of individual stanzas to be stable irrespective of the varieties of feet used. For instance, eight of Thórur's seventeen stanzas from Rímur af Thóri Hreu (see p. 13), including the first and last, are close to the average stanza length of 10.2 seconds (seimur not included) while the remaining vary at most ± 1.3. Tempo appears in fact to be a no less stable factor in kvæaskapur than the pitch of the\seimur . Frequently absolute syllable length is similar or the same in different performances of the same performer. The long in Kristján's Nos. 1 and 28b measured in seconds is 0.4 and also in No. 39d, where the foot 3- (of length 0.6) is used exclusively. The feet of Nos. 7 and 39a-b are 0.6 too but the syllables equal and therefore interpreted as longs of absolute length 0.3 of a second. Only one of Kristján's seven examples, No. 39c, departs from this remarkable norm with feet between two standard lengths 0.8 and 0.6. In Thórur's 1 performances 4- varies in length from about 1 second to 0.7; the ratio is 4:3 as in Kristján's performances, but intermediate values are not unusual. The first measure of his I stanzas contain only 4- and 5- feet, as 3- is introduced earliest at the beginning of the second measure; the same is characteristic of Nos. 17-20. Such an arrangement can cause an impression of momentary change of tempo within a stanza, for instance in No. 11, where the stanzas 3-7 continue with combinations of 2-, 3-, and 4- feet. The second and third lines are nevertheless of approximately the same length as the first line, summing up to about seven longs. Similarly No. 23a, the fifth of the seventeen stanzas discussed above, sums up exactly to 25 longs up to the seimur, and as its absolute length of 10 seconds is almost the average, the whole performance would take about the same time if one long to a syllable were a rule. In such cases lengthened syllables at the end of lines redress the balance, but add 2-3 to the minimal 25 when longs are used for the most part, as in No. 22, or exclusively, as in No. 23c. Besides Thórur's stanzas most of the I examples other than Id fall within the range 25-31 measured in longs (i.e., 2's) with an average close to 28. Of the remaining I examples, Nos. 4, 5, 6, 12, 15, and 29b vary from 20 to 22 longs with 21 the average. No. 24 stands apart from the rest, but the average is 23 for a and c if the dotted eighth-note is the basic unit. Since the average 28 corresponds to fourteen 4- feet in length and 21 to the number of 3- feet, I performances apparently divide into two groups, the first with 2 and the second mainly with 1.5 as basic units of length. Combinations 3-3-, 2-4-, and 2-3- measures characterize the second group except No. 4, where 2-4- measures are used for the most part, with 2 probably as the unit of length (No. 4 stands then to the first group in a similar way as No. 24c to the second). In II performances, on the other hand, uniform measures are the rule. Gunnar and Karl however deviate from this norm (Nos. 35-37 and No. 31b), and Gumundur's II stemma (No. 32) is of a particular order. The first half is always based on strict 3-3-, the most unusual measure in II performances, but the more varied second half is mainly based of 4-4-.
Units of length defined as above can now be used to specify length of operations in the same way as average small seconds define width. The displacement of two syllables in the A' stanzas of Kristín and Thórur discussed above is then estimated to correspond to two longs on the average and the operation 44 therefore to be limited by two units of length. The length of the operation 11, on the other hand, is at most 7-8 units of length in the examples, for instance 7 units (quarter-notes) in the first relation between No. 22b and 22a3, lowering the first line by 1, and about 8 units (dotted eighth-notes) between the first line of No. 29a and the first line of b, whereas No. 30 presents an ambiguous case.
In the first set of sequences above 44 results from 11 applied eight times within the limits of twice its length. If the weight of 11 is 1 when its length is one unit of length, the weight of 44 will be 8 and the values of the remaining six groups from 2 to 7. It is clear too that doubling the length of 11 by applying it twice doubles its weight:

Continuing in this way its weight will be 8 when it has been lengthened eight times, which means that to raise or lower two units of length by 4 and eight units by 1 is the same with regard to weight. The three-interval sequences of both sets may be regarded as combinations of two pairs. But a pair corresponds evidently to a pitch (a p) of expression (1) when measured from a reference level within a stanza (secondary reference levels will be dealt with later) and to the difference between two corresponding segments of one or more pitches when expression a relation or part of a relation between two stanzas. Thus for instance 123 can be written


Weight can now be defined as the quantitative measure of pairs obtained by multiplying interval value by length. If 11 and 33 respectively correspond to two units as in No. 22b, the weight of the first is 2, the second 6, and the total weight of the sequence is 8. Then the change from B to 'A" in Thórdur's performance (see p. 35) is approximately of the same weight as to change 'A" into A", a weight apparently close to the maximal weight for any operation (denoted in the following by M).
In order to illustrate better the two-pair operations between stanzas and the difference between the two sets above, the operations changing the second line and the last line of No.22a2 into the corresponding lines of 22b are compared:

The two (-2+1+1) operations of weight 3 and 2 respectively only lower the two segments framed by "a" and "g," and "g" and "f," whereas (-1+2-1) of weight 2.5 lowers the first half and raises the second half of the segment framed by "f-sharp" and "g."
Generalizing the two sets of sequences leads to the following formulations:


(5) x' x'
x x

(6) x' x'
x x


where 0 <x <4 and 0 < x' <4, and the two variables vary independently of each other, except x' < 2 if 2 < x and x < 2 if 2 < x' in (6). The weight w for the length l (in units of length) of xx and l' for x'x' is then in both cases

(7) xl + x'l' = w

The formulation above evidently reinstates the variability of the intervals as x and x' can take any value within the defined limits. When x' = x, two- interval sequences (i.e., xx) result from (5) and x2xx from (6). The general relation between (5) and (6) is 2x2x, the relation between xx and its inversion. This relation is limited to pairs at or below 2 since its value does not on the whole exceed the higher limit 4 in the examples.
At the lower limit ambiguity sets it, and it is difficult to decide exactly the smallest values both for interval and length which should be taken into account. No. 30, already mentioned, is a case in point. The first line of No. 30b, apparently directly related to the first line of a, ends on a trill between "a-flat" and "g," the first pitch both lines have in common. If the two pitches are distinguished the first operation +1-1 of length 7 at most ends at the beginning of the trill, followed by six +1-1 operations of length 0.2 each and one of length 1 until the end of the line. Otherwise a pitch midway between "a-flat" and "g" must be reckoned with, and only one operation +1-0.5-0.5 of the same weight as the total weight of the former alternative (approximately 9) is needed. Both interpretations fit the formulations above and hardly exceed the defined limits if M (maximal weight) is considered to be near 8.
Interval values as small as 0.5 should apparently be taken into account in No. 4 too. Then each couplet of the second stanza is related to the corresponding part of the first +1-0.5-0.5 (the second couplet exactly but the first only approximately).The first operation ends of the "f" at the end of the couplet and the second on the trill of the seimur. Both stretch over the greater part of the couplet, but the weight of the first is at most 8 and the second 7. As mentioned earlier the range expands further in the fourth stanza. It may be seen simply as the result of applying +0.5-0.5 to the first couplet of the third stanza in such a way that only the first and the last interval are widened to 2.5 and 2 respectively. In the first couplet of the fifth stanza it is partly narrowed and in the sixth it is the same as in the second and the third. Thus the range above the pitch "f" varies from 1 to 2.5 in two definite stages. One stage could be added if the first stanza is raised from zero, but before discussing this matter the limits governing the balance between operation upwards and downwards (+- and -+) must be defined more precisely.
Since xx is the only directly related to its inversion if x < 2, the limit 2 is the limit of the inversion (denoted by 1 in the following). Sequences at or below this value can be replaced immediately by their inversion, for instance +2-2 in the fourth measure of No. 7, 1 is replaced by -2+2 in the in the second stanza. But it is also possible to arrive at the inversion of a sequence by applying the value of its inversion twice; a third application would then result in twice the value of its inversion. The general rule seems to be that two operations in the same direction can be applied successively to the same point only if the value of at least one of them is not greater than 1 (the limit of inversion) and that three successive operations are possible in the same direction only if the value of no one is greater than 1. The limits for combining two pairs as in (5) and (6) may be stated in terms of the same general rule: the value of at least one of the two pairs must not be greater 1 if they are in opposite directions (as in (6); in other respects their value is at most 21. Further combinations of pairs as defined for (5) and (6) are only restricted by the rule that not more than three pairs in the same direction can combine immediately if they are of different values. Such chains are limited by M in the same way as before (w is the sum of the weight of all the pairs as in (7) for (5) and (6)) and consist of less than six pairs for the most part in the examples. However, a sustained pair, whether representing a pitch or a relation, can always be considered to consist of an indefinite number of pairs of the same interval value in the same direction or to include an indefinite number of unarticulated zeros.
To raise a melody from zero means simply to interpret it in terms of operations starting from p0 (defined earlier, p. 26). Below, the first couplet of No. 4 stanza 1 up to the hnykkur at the end is written as a sequence of intervals (a) and then as two operations (b), the second broken down into a chain of six pairs in such a way that pairs combined as in (6) are left intact. The relation of the first couplet of stanza 2 to the first couplet of 1 follows (c) broken down in the same way (the operations are two as the short hnykkur on the third syllable is taken into account), and finally (d) the first couplet of stanza 2 is notated as a sequence of intervals.

The weight for the two operations from zero in b is 3 and 5.5 respectively and 0.5 and 7.5 for the operations relating the couplets if a quarter-note is considered the unit of length. The total weight for either is then close to M.
Obviously it is only possible to raise a few of the examples directly from zero without exceeding the limits. The first couplet of No. 7 illustrates this point clearly. The couplets of both stanzas are written as sequences of intervals with the relations in the brackets below.

Both couplets are easily raised from zero with "f" as p0 until the ascent n the second measure, but from that point on two stages are needed. By applying the operation +2-2 of weight 8 at the point of ascent the following sequence results (the line indicates "g" and a note head one unit of length).

The two couplets are then produced by the following operations (written below the sequence for each stanza):

the four "g's" serving thus as a secondary reference level.
If the first or the second metric syllable is the point of ascent the following sequences (i) result from the operations +1-1, +2-2, and +4-4 of weight 8 each:

The first two (i a-b) correspond to the contour of A" of Fig. 1, the next two (i c-d) to the line-contour of a, and the last two (i e-f) to the measure contours å and å' respectively. By a second application at or immediately after the point of descent, i c-f can be extended in various ways.

The first two are again A" and could be obtained as well from i a-b by applying +1-1 of weight 8 at the point of ascent as from i c-d by a second application of +2-2 of the same weight at the point of descent. Besides the couplets of No. 4 and 'A" couplets of Nos. 5 and 6 are also close to a ia and iia while the A" couplets of the same examples (the second couplets) are close to ib and iib. It is interesting to note in this connection that 'A" contours are for the most part narrow; otherwise they appear as levelled-down B contours. Medium A", on the other hand, apparently derived from a , and a' line-contours appear occasionally in Thórur's performances besides those directly connected with 'A" and B. Nos. 12 and 13 may be regarded as medium A" too. The former could result directly from iib, the first couplet for instance by applying (-1+2-1) (+1-1) (+1+1-2) (+2-2), but intermediate stages would be needed for the latter.
Apart from No. 1 derivable from iic, A and A' couplets of range 4 consist for the most part of a' and a close to the iie and iif. It is clear too that iid is not far from the second line of No. 10 and the falling measure contours of the first line of No. 22a1. In the performances of Sigurur the measures "ffga" and "fga" or "fgg" appear frequently besides a' similar to the first line of No. 9b, and several couplets can even be raised directly from zero, for instance the second couplet of No. 9a by (+2.5+2-4.5) (+2-2) (+2+2-2-2) of weights 8, 4, and 7 respectively.
Interpreted as above, the A contours appear to rise from the constraints of the system, and the result would evidently be the same if the lengths 8, 4, and 2 raised upwards at M produced the ranges 1, 2, and 4. But it is not possible to assume that melodic outlines or contours generally originate in this way, i.e., from zero in several stages. Some of them may even have originated outside the system, and a second level of reference follows from interpreting individual stanzas in the same terms as the relations between them. Such levels are not fixed pitches but fluctuate within narrow limits. The same applies to the pitch of the seimur in some cases (see p. 23) where the primary level has the range 1. It will be denoted by s in the following, whether a range or a single pitch, and any secondary level of reference r.
Only one r with a similar range as s is needed in the examples for the contours A-F if not raised directly from zero, with the exceptions of Nos. 12, 26, and 33. In Thórur's A, B, D, and E stanzas alike its average distance from s is the same. The highest values (transcribed as "b-flat") appear most often in the first half of stanzas, especially in the first line. Inversely s is as a rule low at its first appearance and high at the latest in the seimur . The use of 'a' mainly as the first line of A stanzas and the general use of a last line possible to raise directly from zero reflect the same tendency. No. 23c presents as exceptional case possibly due to particular circumstances (see p.46 below) with "b-flat" as r in the first two lines heightened to "b-natural" at the beginning of the third line. From the end of the third line it is similar to No. 23a, a typical E stanza where the high s is introduced at the beginning of the last line. The complete stanza is produced by the operations (-2+1+1) (+2-2) (-2+1+1) (-2+2) from "b-flat," (+2+2-4) applied to (00)+1 between "b-flat" and "b", (-2+2) (-3+3) (+3-3) from "b," (-2+2) applied to (0)-6 between "b" and "f," and finally (+1+2-2-1) from "f-sharp" of weight 5, 3, 3, 1, 6, 1, 3, 3, 1, and 8 respectively totaling 34. A similar value is obtained for No. 33 if a "g" at the end of the first line and "b" in the second line are both r. Leaving No. 24 aside, the greatest weight for a single stanza interpreted in terms of operations from r and s is about 40 for the IIa stanza No. 35 if only "a" is r. Nos. 6b and 25, the narrowest stanzas on the basis of interval counts, are at the other extreme with a total weight close to 8.
Values for total weight of relations between stanzas also fall within a range approximately between M and 4M for I stanzas as far as the examples go. The highest value calculated is 36 for the relations between No. 31a and b possible to connect by operations not exceeding the limits of the first (+3+2-5) is admitted, but such operations with the interval 5, staying otherwise within the limits, are occasionally found in the examples. The lowest total weight, about 5, between the second and the third stanza of No. 4, is reached only once in the whole performance. At this stage the pitch "g" could be taken as r; the weight of the stanza is then 9, or the same as the weight of the operations changing into the fourth.
In estimating operations within and between ornamented stanzas all transcribed pitches are included, except that grace notes and trills are regarded as wide vibrato around and intermediate pitch. Below, for example, the relations between the third line of the third stanza of No. 14 and the third line of the fourth are written as before.

If either line is raised from s and r two pitches appear as r, "a" and "b-flat" in the first and "a-flat" and "b-flat" in the second, and if the lines of preceding stanzas are considered too, "a-natural" and "b-natural." This particular case of two r levels at most a large second apart but each varying within the range 1 in the usual way could be seen in connections with couplets, as in No. 15 and diatonic couplets discussed later. In Gunnar's F stanzas, on the other hand, only "a" suffices as r in No. 35 and "a-flat" ("g-sharp") in No. 36. The weight of individual stanzas in No. 14 interpreted as above does not exceed 27, and it is almost constant between stanzas (near 20). Margrét's No. 2 is easily raised from a single level of range 1 ("f-sharp" to "f-natural"), but another possibility is to regard "f-sharp" of the first stanza as r heightened to "g" and slightly lowered "g" in the second. The weight of each stanza is then not more than 12 whereas the weight between them is close to 15 anyway.
As pointed out earlier, the difference between pitch inflections and ornaments (hnykkir) is not always clear-cut. Therefore inflections clearly affecting the aural impression, including barely discernible pitches transcribed as grace notes, should probably be taken into account with regard to weight. Only it is not certain that measurements of the physical signal would yield meaningful results if weight is assumed to reflect the degree of perceived change of pitch. Dynamics is another factor not included which might be relevant here. Basically, all syllables seem to be equally stressed in performance whether stressed by verse metre of not and equally capable of receiving a dynamic accent. Such accents are apparently connected with melodic motion in the first place, as they are found most often on or near the high points of the melody.

Because of its constraints the system cannot produce directly more than three successive intervals in the same direction, and operations containing more than two pairs in the same direction are not applied to the range between s and r irrespective of size. Accordingly if four intervals occur in the same direction at least one is divided from the rest by either of the two reference levels. This means with regard to the elements of Table I that c6 is as all other four-interval progressions only produced in this way. Generally the elements of Table I like the contours if Fig. 1 amount to no more than rather crude approximations which could now be dispensed with, and the content of Table I and Figs. 6-8 reduced to the following array (Fig. 11) where the numbers stand for intervals or ranges as limits rather than elements:

1 2 3
2 4 6
3 6 9

Fig. 11

The mid row and mid column represent 1, 21, and their sum 6, the tritone, which together as 264 or 462 appear as the highest value expressions (6) can take. With respect to the former classification the first row could represent primary, the second secondary, and the third wide. Interval use may now be seen to rise from the hierarchical order and the basic ratio of the system, as the first interval of a row or column has twice the value of the second and the second twice the value of the last with regard to frequency of use. Thus if the small second is given the value 4, the large has the value 2, the small and large thirds 1, the tritone 0.5, and the sixth not occurring as undivided interval 0.

4 2 1
2 1 0.5
1 0.5 0

Fig. 12

Counting only the first two intervals of the first column of Fig. 11 according to the values of Fig. 12 and the remaining seven intervals as one zero yields 4, 2, and 7 (31, 15, and 54 percent), whereas the first column and the first two intervals of the second counted in the same way result in 4, 4, 1, 1, and 4 zeros. Intermediate frequencies corresponding to those of Table 4 and 5 are obtained in the same manner: 4, 4, and 6 from the first two intervals of the first column and the first of the second, and the first of the third. The averages of both 4, 4, 1, 0.5, and 4.5 correspond exactly to the average percentages presented in Table 2. It is not possible, on the other hand, to arrive at a frequency ratio for No. 24 in this way unless the frequency value of all intervals except the seconds is doubled. Counting all intervals of Fig. 11 except the last two results then in 4, 4, 4, 2, 1, and 2. For pairs combined as in (5) at least, this change in value seems to be connected with higher limit (x and x' < 6). Without the change the frequencies would be 4, 4, 2, 1, 0.5, and 2 similar to the frequencies for No. 33 (23, 33, 14, 7, 3, 20 percent) and No. 38b (33, 28, 20, 3, 3, 13 percent), representing to all appearances not an intermediate stage but the system normally at it widest.
Generally speaking narrow, medium, and wide examples alike share the same essential characteristics which may be seen to result from the same basic organization of measure distances. However, exceptions must be made for Nos. 9, 15, and 29 besides the different limits for No. 24 already mentioned. No. 9a and b represent the extremes of Sigurur's I stemma.. Between the second couplet of a and both couplets of b there are various intermediate forms, but the narrow first couplet appears only twice in the 123 stanzas recorded, the second line in No. 9a. It is brought into accord with the rest in the third stanza of the same performance where "ffggggfgggfff" is the first couplet. On the whole, narrow characteristics are absent from Sigurur's performances, as operations with values near 1 are limited to downward applications (- +) within stanzas mainly to s and occur only exceptionally between stanzas. Thus the continuity between zero and 2 is interrupted and progressions through the range of 4 restricted to two intervals, but otherwise the limits remain the same. It would therefore be possible in this case to describe the resulting pitches in terms of variable tones approaching more or less the tones of the diatonic scale. In No. 15, also with a lower percentage of small seconds, the limits for operations are not changed but two pitches above s, "b-flat" and "c," fixed.Since only two pairs are admitted between s and r ("b-flat"), it is possible here too to speak of two variable tones ("g" and "a"). The first couplet of the first stanza of 15 seems to be levelled-down B even if it is r in the first line as in the first line of the remaining couplets. Such a fixed frame of a fourth and fifth stems evidently from diatonic kvæaskapur frequently based on progressions through penta- and hexachords with prominent fourths and fifths. Thórur's couplet labeled earlier Cw belongs here too. Below, it is compared with the second couplet of Kristín's III stemma (b), a variant of one of the most usual diatonic kvæa -melodies in Breiafjörur, and with the second couplet of a stanza (c), especially pointed out by Gunnar as söngstemma , a stemma only fit for singing.

Thórur's couplet (starting on "a") ends on "g" in a few stanzas continuing as E, otherwise only the first measure varies to any extent when it appears as the first couplet. The rhythm tends to be fixed too. Long and short syllables alternate regularly as in a I performance (Thórur's II stemma appears also to be a diatonic melody only partially adapted to the kvæa-system).
Interpreted in terms of the system, "a" is r in the examples above besides "b-flat" in the first and "c" in the last two. Similarly "b-flat" and "c" in the second and third line and "g" in the last line of No. 28a must be r. In Kristján's stanza No. 28b, on the other hand, only "d" is needed as r until the end of the third line, where "g" is introduced.This example, possibly derived from the diatonic melody, accords entirely with the system if 9 is considered the highest limit for the range between s and r. No. 26 has also "d" for r until it is replaced by "g" in the last line. In the remaining examples r is never higher than 6 and most often near 4 or 2 except in No. 24 where "c" to "c-sharp" is r besides "b" (In 24b1) and "a" (in 24c). An expansion bringing in 9 but omitting 7 agrees with the array of numbers (Fig. 11) and is directly expressed in the progression -3-2-4 of No. 23c, where 3 is added to 6 divided into 2 and 4, and in -3-4-2 of No. 28b, where 9 may be regarded as divided into 3 and 6 and 6 into 4 and 2 by the operation -3+5-2 applied to (0)-9(0). This progression is followed by -2-2-1 at the end of the line with note values approximately halved and thus narrowed and shortened roughly according to the ratio 1:2.
Melodic motion around 9 as r is considerably restricted, as the system has to all appearances a fixed total range of a seventh (10-11) above s and a large third below s. This means that operations at the higher limit (21) are not applied downwards to a point below s and upwards to a point above the tritone. This is also valid for No. 24, even if 3 is of the same order as 1 and 2, and 363 is admitted as the higher limit for (6) and 6 for (5). It is clear too that this shift appears rather in operations within than between stanzas. Thus b2 is directly related to both a1 and a2 without their second measure by operations not exceeding the normal limits.4 If most of the range above s is used but r is not higher than 4-5, as in Nos. 24c and 29, the discrepancy increases between relations and progressions within stanzas barely contained by the system.The particular character of Karl's I stemma seems to be due to a diatonic original. No. 29a is directly related (total weight about 20) to one of his wife's melodies learned from her father, whereas b is at the opposite extreme. It is directly related to a (total weight about 30) by operations where the small second figures prominently in the usual way even if it is rare within stanzas.






1 Anderson (1964), 278.

2 Nielsen (1972), 66.

3 Helgason (1972), 20-22.

4 One may note too that Jónína's Id stanza No. 17b is directly related to 17a2 without the second measure. The adaption seems therefore to be the usual one masked by operations keeping the general outline intact rather than exceptional as assumed earlier (p.12).