Chapter 23: Rhythm

The subject of rhythm touches on nearly every aspect of music. Musical elements such as pitch, harmony, and dynamics can all be regarded from the point-of-view of temporal patterns of events. A number of complex tasks arise from rhythm-related In this chapter, two rhythm-related tools are introduced: dur and metpos.

The **recip Representation

For many types of processing tasks it is helpful to have a representation that encodes rhythmic information only. The **recip representation is simply a subset of **kernthat excludes all information apart from the nominal note durations and common system barlines. In addition, **recip distinguishes rests from notes by including the r‘ signifier. Without an accompanying r‘ a duration is assumed to pertain to a note.

Generating **recip data from **kern is straightforward using humsed. For a single-spine input, the following command will make the translation:

humsed '/^[^=]/s/[^0-9.r ]//g; s/^$/./' input.krn \ > >> | sed 's/ |**kern|_\ /**recip/'

The first humsed substitution eliminates all data other than the numbers 0 to 9, the period, the lower-case r, and the space (for multiple-stops). Barlines remain untouched in the output. The second humsed substitution changes any empty lines to null data tokens; this might be necessary in the case of grace notes. The ensuing sed command is used simply to change the exclusive interpretation from **kern to **recip.

A simple type of processing might entail creating an inventory of rhythmic patterns. Suppose we wanted to determine the most common rhythmic pattern spanning a measure. Using a monophonic **recip input, we could use context to amalgamate the appropriate data tokens:

context -b ^= -o ^= input.recip | rid -GLId | sort \ > >> | uniq -c | sort -nr

The output for the combined voices of Bach’s two-part Invention No. 5 shows just seven patterns. The most characteristic patterns are the second one: 8r 16 16 8 8 4 4 and the fourth one: 8 16 16 8 8 4 4.

3016 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 > 128r 16 16 8 8 4 4 > 118 8 8 8 16 16 16 16 8 8 > 88 16 16 8 8 4 4 > 18 8 8 8 2 > 18 32 32 32 32 4 2 > 14 4 16r 16 16 16 16 16 16 16

The dur Command

The dur command produces **dur output from either a **kern or **recip input. The **dur representation scheme consists simply of the elapsed duration of notes and rests, expressed in seconds. The following example shows a simple **dur representation (right spine) with a corresponding **kern input:

As in the case of **recip, the **dur representation designates rests via the lower-case r and uses the common system for barlines. Notice that **dur assumes a metronome indication of quarter-note equals 60 beats per minute if no other metronome marking is given.

Suppose that we wanted to estimate the total duration of some monophonic passage (ignoring rubato). We can do this by translating the score to **dur, eliminating everything but notes and rests, and sending the output to the**stats**command:

dur -d inputfile.krn | rid -GLId | grep -v '^=' | stats

The -d option for dur suppresses the outputting of duplicate durations arising from multiple-stops. Note that outputs from dur will adapt to any changes of metronome marking found in the input, so if the work accelerates the durations will be reduced proportionally.

The -M option will over-ride any metronome markings found in the input stream. For example, if we wanted to estimate the duration of a monophonic passage for a metronome marking of 72 quarter-notes per minute we could use the command:

dur -M 72 -d input.krn | rid -GLId | grep -v '^=' | stats

Of course, the duration of a passage is not the same as the length of time a given instrument sounds. Suppose, for example, that we wanted to compare the duration of trumpet activity in the final movements of Beethoven’s symphonies. We need to make a distinction between the duration of notes and the duration of rests. Since the duration values for rests are distinguished by the trailing letter `r’, we can use grep -v to eliminate all rest tokens.

extract -i '*Itromp' inputfile.krn | dur -d | rid -GLId \ > >> | grep -v '^=' | grep -v r | stats

The dur command provides a -e option that allows the user to echo specified signifiers in the output. The -e option is followed by a regular expression indicating what patterns are to be passed to the output. This option allows us to “mark” notes of special interest. For example, suppose we wanted to determine the longest duration note for which Mozart had marked a staccato.

dur -e \' inputfile | rid -GLId | grep \' | sed 's/\'//' \ > >> | stats

The -e option ensures that **kern staccato marks (‘) are passed along to the output. The rid command eliminates everything but Humdrum data records. Then grep is used to isolate only those notes containing a staccato mark. The sed script is used to eliminate the apostrophe, and finally the numbers are passed to the**stats**command. The max value from**stats**will identify the duration (in seconds) of the longest note marked staccato.

This same basic pipeline can be used for a variety of similar problems. Suppose, for example, that we want to determine whether notes at the ends of phrases tend to be longer than notes at the beginnings of phrases – and if so, how much longer? In this case, we want to have dur echo phrase- related signifiers:

dur -e '{' inputfile | rid -GLId | grep '{' | sed 's/{//' \ > >> | stats > > dur -e '}' inputfile | rid -GLId | grep '{' | sed 's/{//' \ > >> | stats

Similarly, do semitone trills tend to be shorter than whole-tone trills?

dur -e 't' inputfile | rid -GLId | grep 't' | sed 's/{//' \ > >> | stats > > dur -e 'T' inputfile | rid -GLId | grep 'T' | sed 's/{//' \ > >> | stats

Of course, we can also use dur in conjunction with yank in order to investigate particular musical segments or passages. How much shorter is the recapitulation compared with the original exposition?

yank -s 'Exposition' -r 1 inputfile | dur | rid -GLId \ > >> | grep -v '=' | stats > > yank -s 'Recapituation' -r 1 inputfile | dur | rid -GLId \ > >> | grep -v '=' | stats

Do initial phrases in Schubert’s vocal works tend to be shorter than final phrases?

yank -m { -r 1 lied | dur | rid -GLId | grep -v ^= | stats
yank -m { -r $ lied | dur | rid -GLId | grep -v ^= | stats

How much longer is a passage if all the repeats are played?

thru inputfile | dur | rid -GLID | stats -o ^=

Recall that the xdelta command can be used to calculate numerical differences between successive values. If the input to xdelta is **dur duration information, then we can determine rates of change of duration. Most music exhibits lengthy passages of similar duration notes – as in a sequence of sixteenth notes. In French overtures, successive notes are often of highly contrasting durations (longer, very-short, long, etc.). Using xdelta we can identify such large changes of duration. For example, the following pipeline can be used to determine the magnitude of the differences between successive notes.

dur inputfile | xdelta -s ^= | rid -GLId | stats -o ^=

A small mean from**stats**will be indicative of works that tend to have smoother or less angular note-to-note rhythms.

Classifying Durations

We can use the recode command to classify durations into a finite set of categories. Suppose, for example, we wish to create a inventory of long/short rhythmic patterns. We might use recode with reassignments such as the following:

>=0.4long > elseshort

For a monophonic input, we can create an inventory of (say) 3-note long/short rhythmic patterns as follows:

dur inputfile | recode -f reassign -i '**dur' -s ^= | \ > >> context -n 3 -o = | rid -GLId | sort | uniq -c | sort -n

A typical output might appears as follows:

230long long long > 3422short short short > 114long long short > 202short short long > 38long short long > 117short long long > 194long short short > 114short long short

Notice that we might do a similar inventory based on durational differences rather than on durations. For example, the xdelta command will allow us to distinguish short*er* note relationships from long*er* relationships. Our reassignment file would be as follows:

==0equal > >0shorter > <0longer

And our processing would be:

dur inputfile | xdelta -s ^= | recode -f reassign \ > >> -i '**Xdur' -s ^= | context -n 2 -o = \ | rid -GLId | sort | uniq -c | sort -n

Using yank with the timebase Command

Recall that the timebase command can be used to reformat an input so that each data record represents an equivalent elapsed duration. For example, in a 4/4 meter, the following command will format the output so that each full measure consists of precisely 16 data records (not including the barline itself):

timebase -t 16 input.krn

Suppose we wanted to isolate all sonorities in a 4/4 work that occur only on the fourth beat of a measure. If we use timebase , we can ensure that the fourth beat always occurs a certain number of data records following the barline. For example, with the following command, the onset of the fourth beat will always occur 4 records follow the barline:

timebase -t 4 input.krn

We can now use yank -m to extract all appropriate sonorities. The “marker” is the barline and the “range” is 4 records following the marker, hence:

timebase -t 4 input.krn | yank -m ^= -r 4

Note that this process will extract only those notes that begin sounding with the onset of the fourth beat. Some notes may have begun prior to the fourth beat and yet are sustained into the beat. If we want to extract the sounded sonority, we can use the ditto command. Begin by expanding the work with a timebase that ensures all notes are present. For a work whose shortest note is a 32nd note, we can use an appropriately small timebase value. Then use the ditto command to propagate all sustained notes forward through the successive sonorities:

timebase -t 32 input.krn | ditto -s ^=

Now we can yank the data records that are of interest. Notice that the -r (range) option for yank -m allows us to select more than one record. This might allow us, say, to extract only those sonorities that occur on off- beats. For example, the following command extracts all notes played by the horns during beats 2 and 4 in a 4/4 meter work:

extract -i '*Icor' input.krn | timebase -t 16 \ > >> | yank -m ^= -r 5-8,13-16

In some cases, we would like to yank materials that do not themselves contain explicit durational information. Suppose, for example, that for a waltz repertory, we want to contrast those chord functions that tend to occur on the first beat with those that happen on the third beat. We will need to have an input that includes both a **harm spine encoding the Roman numeral harmonic analysis, as well as one or more **kern or **recip spines that include the durational information. We can use the timebase command to expand the output accordingly – cuing on the duration information provided by **kern or **recip. Having suitable expanded the input, we can dispense with everything but the **harm spine. For works in 3/4 meter, the following pipeline would provide an inventory of chords occurring on the first beat of each bar:

timebase -t 8 input | extract -i '**harm' \ > >> | yank -m ^= -r 1 | rid -GLId | sort | uniq -c | sort -n

And the following variation would provide an inventory of chords occurring on the third beat of each bar. (There are 6 eighth durations in a bar of 3/4, therefore the beginning of the third beat will coincide with the 5th eighth – hence the range -r 5:

timebase -t 8 input | extract -i '**harm' \ > >> | yank -m ^= -r 5 | rid -GLId | sort | uniq -c | sort -n

The metpos Command

The metpos command generates a **metpos output spine containing numbers that indicate the metric strength of each sonority. By “metric position” we mean the position of importance in the metric hierarchy for a measure.

The highest position in any given metric hierarchy is given by the value `1’. This value is assigned to the first event at the beginning of each measure. In duple and quadruple meters, the second level in the metric hierarchy occurs in the middle of the measure and is assigned the output value `2’. (In triple meters, metpos assumes that the second and third beats in the measure are both assigned to the second level in the metric hierarchy.) All other metric positions in the measure (beats, sub-beats, sub-sub-beats, etc.) are assigned successively increasing numerical values according to their placement in the metric hierarchy. In summary, larger **metpos values signify sonorities of lesser metric significance.

By way of illustration, consider the case of successive eighth notes in a 2/4 meter. The metric hierarchy values for successive eighths are: 1, 3, 2, 3. In the case of successive sixteenth notes in 2/4, the metric hierarchy values are: 1,4,3,4,2,4,3,4. In the case of 6/8 meter, successive sixteenth durations exhibit a metric hierarchy of: 1,4,3,4,3,4,2,4,3,4,3,4.

For correct operation, the metpos command must be supplied with an input that has been formatted using the timebase command. That is, each data record (ignoring barlines) must represent an equivalent duration of time. In addition, metpos must be informed of both the meter signature and the timebase for the given input passage. This information can be specified via the command line, however it is usually available in the input stream via appropriate tandem interpretations.

The following extract from Bartok’s “Two-Part Study” No. 121 from Mikrokosmos demonstrates the effect of the metpos command. The two left-most columns show the original input; all three columns show the corresponding output from metpos :

Notice that metpos adapts to changing meter signatures, and correctly distinguishes between metric accent patterns such as 6/4 (measure 16) and 3/2 (measure 19).

The **metpos values provide additional ways of addressing various rhythmic questions. We might use recode for example, to recode the numerical outputs from metpos into a smaller set of discrete categories. For example, we might classify metric positions using the following reassignment file:

==1strong > >=3secondary > elseweak

The words strong’, `secondary’, and `weak’ can then be sought by `grep or yank -m, allowing us to isolate points of particular metric stress. Since metpos adapts to changing meters, we can confidently process inputs that may contain mixtures of meters.

Changes of Stress

Once again we can make use of xdelta to identify relationships between successive metric position values. Suppose we had a collection of Hungarian melodies and we wanted to determine how each degree is approached in terms of metric strength. That is, we would like to count the number of tonic pitches that are approached by a weak-to-strong context versus the number of tonic pitches approached by a strong-to-weak context. We also want similar measures for supertonic, mediant, subdominant, etc. scale degrees.

This task involves creating an inventory where fourteen different items are possible: (1) tonic strong-to-weak, (2) tonic weak-to-strong, (3) supertonic, strong-to-weak, etc. A suitable inventory will involve creating two spines of information – scale-degree and relative metric strength.

Assuming that our Hungarian melodies encode key information, creating a **deg spine is straightforward. Recall that the -a option for deg avoids distinguishing the direction of approach (from above or below):

deg -a magyar*.krn > magyar.deg

Creating a spine encoding relative metric strength will be more involved. First we need to expand our input according to the shortest note. We use `**census -k**`_ to determine the shortest duration, and then expand our input using timebase .

census -k magyar*.krn
timebase -t 16 magyar*.krn > magyar.tb

Using metpos will allow us to create a spine with the metric position data.

metpos magyar.tb > magyar.mp

Note that metpos automatically echoes the input along with the new **metpos spine. At this point, the result might look as follows:

!!!OTL: Graf Friedrich In Oesterraaich sin di Gassen sou enge
**kern**metpos
ICvox`` Ivox`` M3/4``*M3/4 *k[f#]`` G:`` *tb16``*tb16 {8g``2 .``4 8b``3 .``4 =1``=1 8dd``1 .``4 etc.

We want to be able to say that the relationship between the first eighth-note G and the eighth-note B is “strong-to-weak” and that the relationship between the eighth-note B and the eighth-note D is “weak-to-strong.” In order to procede we need to eliminate all of the data records that contain only a metpos value – that is, there is no pitch present in the **kern_ spine. We can do this using humsed ; we simply delete all lines that begin with a period character:

humsed '/^\./d' magyar.mp

The result is as follows:

!!!OTL: Graf Friedrich In Oesterraaich sin di Gassen sou enge
**kern**metpos
ICvox`` Ivox`` M3/4``*M3/4 *k[f#]`` G:`` *tb16``*tb16 {8g``2 8b``3 =1``=1 8dd``1 etc.

Notice that the successive **metpos values will now allow us to characterize the changes in stress between successive notes: 2 followed by 3 indicates a strong-to-weak change of metric position, 3 followed by 1 indicates a weak-to-strong change of metric position. We can use xdelta to calculate the differences in metric position values: positive differences will indicate weak-to-strong changes and negative differences will indicate strong-to-weak changes. If both values have the same metric position value, then the successive notes hold equal positions in the metric hierarchy. Before using xdelta we need to isolate the **metpos spine using |extract|_ :

humsed '/^\./d' magyar.mp | extract -i '**metpos' \ > >> | xdelta -s ^=

The result is:

!!!OTL: Graf Friedrich In Oesterraaich sin di Gassen sou enge
**Xmetpos
*
*
*M3/4
*
*
*tb16
.
1
=1
-2
etc.

Now we can use recode to classify the changes of metric position according. Our reassignment file (named reassign):

>0strong-to-weak > <0weak-to-strong > ==0equal

Appending the appropriate command:

humsed '/^\./d' magyar.mp | extract -i '**metpos' \ > >> | xdelta -s ^= | recode -f reassign -i '**Xmetpos' -s ^= > magyar.xmp

Now we can assemble the resulting metric change spine with our original **deg spine. Each data record will contain the scale degree in the first spine and the change of metric position data in the second spine. The final task is to create an inventory using rid,**sort**and uniq:

assemble magyar.deg magyar.xmp | rid -GLId | grep -v ^= \ > >> | sort | uniq -c

The final result will appear as below. The first output line indicates that there were three instances of a tonic pitch approached by a note of equivalent position in the metric hierarchy. The second line indicates that there were twenty-five instances of a tonic pitch approached by a note having a stronger metric position:

31equal > 251strong-to-weak > 301weak-to-strong > 32equal > 142strong-to-weak > 132weak-to-strong > 13equal > 393strong-to-weak > 343weak-to-strong > 34equal > 264strong-to-weak > 174weak-to-strong > 135equal > 495strong-to-weak > 425weak-to-strong > 16equal > 136strong-to-weak > 146weak-to-strong > 37strong-to-weak > 67weak-to-strong > 17-weak-to-strong > 3requal > 10rstrong-to-weak

Instead of scale degree, any other Humdrum spine might be used. For example, if the input contained functional harmony data (**harm) then the output inventory would identify how particular chord functions tend to be approached. For example, we could establish whether the submediant chord is more likely to be approached in a strong-to-weak or weak-to-strong rhythmic context. Similarly, this same technique can be used to determine whether particular melodic or harmonic intervals tend to be approached using particular stress relationships.

In addition, our input spine might also be transformed via the context command. Given a **harm spine, for example, context could be used to generate two-chord harmonic progressions. This would permit us to determine, for example, whether a specific progression such as ii-V tends to fall in strong-to-weak or weak-to-strong contexts.


Reprise

There are a vast number of issues raised in rhythm-related processing. In this chapter we have touched on a few of the more basic tasks. These include identifying the durations of various passages using dur ; classifying and contextualizing durations using recode and context; isolating particular rhythmic moments using timebase and yank-m; determining relative metric positions using metpos; and characterizing metric syncopation using synco.

Processing data that does not explicitly contain duration-related information (such as **harm or **deg) often requires some preparation. It is often useful to maintain a coordinated file where the spines of interest are linked with duration-related spines that assist in processing.

One further topic related to rhythm remains to be discussed. The**accent**command allows the user to distinguish notes according to their estimated perceptual importance. We will consider**accent**in Chapter 31.