In this chapter we introduce three pre-defined representations pertaining to musical dynamics. One representation (dyn) represents dynamic markings as they appear visually in a printed score. A second representation (dynam) represents a rationalized interpretation of the notated dynamic markings in a score. A third scheme (dB) represents continuous dynamic levels in decibels.

The **dynam and **dyn Representations

Musical scores commonly contain dynamic markings that include both written text (such as “subito forte” and “dimin.”) and graphic representations (such as hairpin or wedge-shaped crescendo markings). Unfortunately, traditional dynamic markings are often confusing or ambiguous. Consider, for example, the following sequence of dynamic markings from a Beethoven piano sonata:

pp cresc.  cresc.  p cresc.  pp

What are we to make of these markings? Does the music gradually crescendo from pianissimo to piano? Does this initial crescendo occur in two distinct phases or does the repetition of the term “cresc.” merely indicate a continuation of a single crescendo? Does this crescendo move to a dynamic level above piano and abruptly reduce to piano? Does the final crescendo begin at a piano level and get louder — followed by a relatively abrupt reduction to pianissimo? Or does the final crescendo begin below piano and gradually reach pianissimo? Such ambiguities are rampant in printed musical scores. We can examine the accompanying musical context to help us resolve questions of interpretation, but computers are unable to bring such sophistication to the task.

Humdrum provides two pre-defined representations for score-related dynamic markings. One representation (dyn) attempts to represent the dynamic markings as they appear in a visual rendering of a score. That is, dyn represents the visual or orthographic information. A second representation (dynam) provides a “rationalized” or canonical means for interpreting score-related dynamic indications. Users will want to choose one or the other representation depending on the analytic task being pursued.

In the first instance, dynam uses standardized data tokens to represent particular dynamic levels. Table 28.1 shows the standard representations for dynam. For example, the token pp represents the concept of pianissimo, even if the visual rendering may be pp or pianissimo or pianiss., etc.

Table 28.1.

| value | meaning | | ===== | ======= | | p | piano | | pp | pianissimo | | ppp | triple piano | | pppp | quadruple piano, [etc] | | f | forte | | ff | fortissimo | | fff | triple forte | | ffff | quadruple forte, [etc] | | mp | mezzo-piano | | mf | mezzo-forte | | s | subito (suddenly), e.g. spp (*subito pianissimo*), sf (*subito forte*) | | z | sforzando = fp (forte-piano) | | < | begin crescendo | | > | begin diminuendo | | ( | continuing crescendo | | ) | continuing diminuendo | | [ | end crescendo | | ] | end diminuendo | | X | explicit interpretation (not indicated in the score) | | x | published interpretation (indicated in the score, often in parentheses) | | r | rest (silence) | | v | notated accent or stress |

In the case of crescendo and diminuendo markings, dynam requires an explicit interpretation of where the dynamic marking begins and ends. The beginning of a crescendo is indicated by the less-than sign (<). The end of the crescendo is marked by the open square bracket (]). Between the beginning and end points, continuation signifiers are encoded. For crescendos, continuations are indicated using the open parenthesis; for diminuendos, continuations are indicated using the closed parenthesis.

In the dynam representation, no distinction is made for various ways a composer might indicate a crescendo or a diminuendo. For example, it doesn’t matter whether a diminuendo is notated as dim., dimin., diminuendo, decres., decresc., decresendo, calando, morendo, se perdant, cédéz, gradually quieter, or via a hairpin or wedge graphic diminuendo. All are represented by > … ) … ].

The dynam representation also requires explicit resolution of possibly ambiguous dynamic markings. In many cases, the user will be required to add dynamic markings that are only implicit in the original score. Interpreted dynamics are followed by the upper-case letter X, so an interpreted diminuendo will be represented by >X … )X … ]X.

Often published editions will include dynamic markings that have been introduced by the editor. In scholarly publications these editorialisms are indicated in parentheses or square brackets. Such interpreted dynamics are followed by the lower-case letter x.

The use of the dynam representation is illustrated in Example 28.1.

Example 28.1

This example might be encoded as follows:

**kern	**dynam
*staff1	*staff1
=	=
2c 2e 2g 2cc	p
.	<
2G 2d 2g 2b	(
=	=
2A 2c 2e 2a	(
.	[
2E 2B 2e 2g	pp
=	=
*-	*-

The dynam encoding is interpreted as follows: the level begins piano with a crescendo beginning prior to the second chord; the crescendo continues until after the third chord and then the level abruptly drops to pianissimo with the onset of the fourth chord. Notice that dynamic markings are “read from left-to-right”; that is, we presume that the crescendo begins piano and that the pianissimo is an abrupt reduction in level, rather than presuming that the crescendo builds to the pianissimo level and so there is an abrupt reduction in level after the initial piano just before the cresendo begins. In short, a crescendo (or diminuendo) marking is always assumed to increase (or decrease) the dynamic level from the preceding indication.

If appropriate, a user can render implicit dynamic shading explicitly. For example, a user might choose to re-code Example 28.1 as either

**kern		**dynam
*staff1		*staff1
=		=
2c 2e 2g 2cc	p
2G 2d 2g 2b		<
	=	=
2A 2c 2e 2a		[
.		>X
.		]X
2E 2B 2e 2g		pp
=		=
*-		*-

or

**kern		**dynam
*staff1		*staff1
=		=
2c 2e 2g 2cc	p
.		pppX
2G 2d 2g 2b		<
=		=
2A 2c 2e 2a		(
.		[
2E 2B 2e 2g		pp
=		=
*-	*	-

Notice that null data records may be inserted as necessary to clarify the moment of dynamic change. The dynam representation makes use of the common system for representing barlines.

The **dyn Representation

The dyn representation provides a method for representing the orthographic appearance of notated dynamic markings. Unlike dynam, the dyn representation distinguishes between different ways of identifying a dynamic marking. For example, dim., dimin., diminuendo, decres., decresc., decresendo, are all regarded as different from each other. Composers often have idiosyncratic ways of writing dynamic markings. As a result, the specific terms used may have repercussions, for example, in resolving cases of disputed composership. In some circumstances, it is thought that individual composers distinguish the terms in their own minds. For example, a composer might use decrescendo as a general term to indicate a temporary descending dynamic shape, whereas diminuendo might have a more specific meaning of a ‘dying away’ or ‘fade-out’ gesture.

In the dyn representation the horizontal position of dynamic markings is indicated in quarter-durations with respect to the previous barline. This number appears prior to the dynamic signifier, hence 4.1f means a forte (e.g., f) marking just after the horizontal position of the fourth quarter in the measure. The vertical position of dynamic markings is indicated with respect to the middle line of a corresponding staff; this number appears in curly braces.

By way of illustration, consider Example 28.2.

Example 28.2: Arnold Schoenberg, Three Piano Pieces, Op. 11, No. 2, excerpt.

Using the dyn representation, Example 28.2 might be encoded as follows:

**kern	**kern	**dyn
*staff2	*staff1	*staff1/2
*clefF4	*clefG2	*
*M12/8	*M12/8	*
=	=	=
*	*^	*
(8F# 8enL	2.ffn	8r	0.8f{-4}
8A#)	.	8r	.
(8Dn 8cnJ	.	([8cc#	.
8F#)L	.	8cc#]L	.
(8AAn 8Gn	.	8ccn	.
8C#)J	.	8bnJ	.
(8FFn 8E-L	[4.aan	8b-L	.
8AAn)	.	8ddn	.
(8DD- 8C-J	.	8b-J	.
8FFn)L	8aa]	4.dd)	.
(8BBB- 8AA-	4ff#	.	.
8DDn)J	.	.	.
*	*v	*v	*
=	=	=
8r	8r	.
(8d-L	(>8gn 8ccn 8ffnL>	1.4fp{-4.5}
.	.	1.6>{-4.3}/1.5/
8fnJ	8bn 8eenJ	)
.	.	2.4]{-4.3}
24A-L	4.cn 4.fn 4.b)	2.5pp{-5}
24d-	.	.
24Dn	.	2.8>/1.4/
8.GG	.	)
16BB-)J	.	4.2]
8r	8r	.
(<8AnL	(>8e- 8a- 8dd-L	4.4fp{-4.5}
.	.	4.6>/1.5/
8d-J	8gn 8ccnJ	)
.	.	5.2]
24F#L	4.B- 4.en 4.a)	5.4pp{-4.5}
24Bn	.	5.7>/1.4/
24BB-	.	)
8.DD	.	)
16GG)J	.	6.8]
=	=	=
*-	*-	*-
!!!RDF**kern: > = above
!!!RDF**kern: < = below

The *staff1/2 tandem interpretation indicates that the dynamic markings pertain to both staffs 1 and 2, however all vertical dyn distance measures are encoded with respect to staff 1. (Reversing the numerical order — *staff2/1 — would cause all distances to be measured with respect to staff 2.) The token 0.8f{-4} means that the signifier f is located 0.8 quarter-duration spaces from the beginning of the bar and 4 staff-line steps below the center line of staff 1. The token 1.6>{-4.3}/1.5/ means that a wedge diminuendo marking begins 1.6 quarter-durations from the beginning of the bar; the size of the opening of the wedge is 1.5 staff-line steps wide and the center of the opening is located 4.3 staff-line steps below the center line for staff 1. The token 2.4]{-4.3} means that a wedge diminuendo marking ends 2.4 quarter-durations from the beginning of the bar; the tip of the wedge converges at a point 4.3 staff-line steps below the center line for staff 1. Changing this value allows tilted wedges to be represented.

The **dB Representation

The dB representation provides a way to represent intensity in decibels. Decibels can be expressed in relative or absolute terms. Absolute values are represented according to sound pressure level (SPL). An absolute representation is indicated by the presence of the *SPL tandem interpretation. Zero decibels (SPL) corresponds roughly to the quietest sound detectable under ideal circumstances. A quiet room is roughly 40 dB in intensity; a conversation produces roughly 70 dB, a vacuum cleaner produces roughly 80 dB, a noisy factory produces roughly 90 dB, and a passing loud motorcycle generates roughly 100 dB (SPL).

The dB representation provides a convenient way to represent sound intensity in a numerical form. A numerical representation allows us to carry out a variety of calculations and comparisons.

The db Command

The db command translates dynamic markings to dynamic level expressed in decibels; specifically, db translates from the dynam representation to dB representation. By default, db uses the following mapping:

Notice the presence of the accent signifier v; the assigned value +5 means that any encoded accents will receive a decibel level 5 dB higher than the basic sound pressure level at that point in the score. For example, an explicitly accented note occurring in a fortissimo passage will be assigned a value of 85 dB SPL.

Users can define other mappings by using the f option for db. With f the user provides a filename that contains the non-default mapping values. The format for this file is the same as that shown in the above table. Each table entry specifies a dynamic marking, followed by a tab, followed by a numerical value.

In the case of crescendo and diminuendo markings, the db command attempts to interpolate a series of values between any preceding and subsequent dynamic markings. The following example shows a pianissimo marking at the beginning of measure 5; a crescendo marking spans all of measure 6, and a mezzo-forte marking appears in measure 7. The right-most spine shows the corresponding output generated by the db command. It shows an interpolation between the two dynamic levels.

**dynam	**dB
*SPL	*
=5	=5
pp	55
.	.
.	.
.	.
=6	=6
<	58
(	61
.	.
(	64
]	67
=7	=7
.	.
mf	70
.	.

The interpolation begins with the crescendo indicator and increments for each continuation signifier (i.e., the open parentheses). Interpolations are linear and continue up to the crescendo termination signifier. The size of the increment value depends on starting and ending dynamic levels as well as the number of crescendo-continuation signifiers. In the above case four pertinent crescendo signifiers separate the pianissimo and mezzo-forte markings; each of these records has been incremented by 3 decibels. Where necessary, decimal values are output. Notice that null tokens (such as those in the middle of measure 6) are ignored in the calculation.

Processing Dynamic Information

The dB representation can be used to assist a number of tasks related to musical dynamics. Suppose, for example, that we want to compare the average overall dynamic levels for two arabesques:

extract -i '**dynam' arabesque1 | db | rid -GLId | stats
extract -i '**dynam' arabesque2 | db | rid -GLId | stats

Similarly, we might compare the overall dynamic levels between two sections of a single work. Perhaps we wish to know whether the exposition is on average louder than the development section:

yank -s Exposition -r 1 symphony3 | extract -i '**dynam' \
     | db | rid -GLId | stats
yank -s Development -r 1 symphony3 | extract -i '**dynam' \
     | db | rid -GLId | stats

Does a work tend to begin quietly and end loudly, or vice versa? Here we might compare the first 10 measures with the final 10 measures. Notice the use of ditto to increase the number of values participating in the calculation of the average dynamic level:

yank -n = -r 1-10 janacek | extract -i '**dynam' \
     | ditto -s = | db | rid -GLId | stats
yank -n = -r '$-10-$' janacek | extract -i '**dynam' \
     | ditto -s = | db | rid -GLId | stats

Suppose we want to determine whether there is an association between dynamic levels and pitch height for Klezmer music. That is, does the music tend to be quieter for lower pitches and louder for higher pitches? A straightforward way to determine this is to compare dynamic level with pitch height — represented in semitones (semits). The correl command can then be used to measure Pearson’s coefficient of correlation. If there is a relationship between pitch height and dynamic level then the correlation should be positive.

semits klezmer | correl -s ^= -m

This command assumes an input consisting of two spines — one pitch-related and a dB spine. The s option for correl is used to skip barlines so bar numbers aren’t included in the calculation. The m option for correl disables the “matched pairs” criterion. Normally, if a number is found in one spine but not the other then correl will complain and terminate. With the m option, each encoded pitch need not have a corresponding dynamic level indication and vice versa.

Similarly, we could use this same approach to determine whether there is a relationship between duration and dynamic level. Are longer notes more likely to be louder in Klezmer music?

dur klezmer | correl -s ^= -m

A variation on this procedure might be to restrict the comparison over a specified pitch range. For example, one might think that higher pitches tend to be louder but that lower pitches are neither softer nor louder than usual. In order to test this view we can use the recode command to reassign “low” pitches to a single value. By way of illustration, the reassignment might presume that below G4 (semits=7) there is no relationship between pitch height and dynamic level. We might recode all values lower than 7 to a unique string (such as XXX) and then use grep -v to eliminate these notes from a subsequent correlation:

extract -i '**kern' klezmer | semits recode > temp1  extract -i '**dB' klezmer > temp2  assemble temp1 temp2 | grep -v 'XXX' | correl -s ^= -m

Terraced Dynamics

Suppose we want to identify whether various works exhibit “terraced” or “graduated” dynamics. In the case of terraced dynamics, we would expect to see many relatively abrupt dynamic contrasts, such as alternations between forte and piano. There are several ways of approaching this question. One approach might translate dynam data to dB data and then calculate the average (or maximum) changes in dynamic level. If a work contains many crescendos and diminuendos markings, then most of the changes in dB values will be small. Conversely, alternations between contrasting dynamic levels will cause the average decibel differences to be larger. The xdelta command can be used to calculate the changes in dynamic level. Notice that it is important to avoid using the ditto command since repeated dynamic level values will cause the average dynamic difference to approach zero.

extract -i '**dynam' haendel | db | xdelta -a -s = | rid -d | stats

Another approach to this problem might be to count the number of dynamic contrasts, avoiding the use of the db command. In the following pipeline, we use context to generate pairs of dynamic markings, and then use grep to count the number of alternations between f and p.

extract -i '**dynam' haendel | grep -v '[][()=rX]' | rid -d \
     | context -n 2 | grep -c 'f p'
extract -i '**dynam' haendel | grep -v '[][()=rX]' | rid -d \
     | context -n 2 | grep -c 'p f'

Dynamic Swells

Conceptually, crescendos and diminuendos can be paired to form one of two dynamic gestures. A “swell” gesture consists of a crescendo followed by a diminuendo. Conversely, a “dip” gesture would consist of a diminuendo followed by a crescendo. Musical intuition would suggest that swell gestures are more common than dip gestures. We could test this view as follows:

extract -i '**dynam' grieg | grep -v '[][()=rX]' | rid -d \
| context -n 2 | grep -c '< >'
extract -i '**dynam' grieg | grep -v '[][()=rX]' | rid -d \
| context -n 2 | grep -c '> <'

MIDI Dynamics

Dynamic level data is not always easily available. One possible source is to translate MIDI key-velocity data to an estimated decibel value. Actual sound pressure levels will depend on the timbre of the MIDI sounds, the specific pitch played, and the volume on the output amplifier. Nevertheless, a rough estimate of sound pressure level may be useful for various analytic tasks. Recall that in the MIDI representation, key-velocity data is encoded as the final number in three-number tokens where numbers are separated by slashes. The first value in the triplet is elapsed clock ticks and the second value is the MIDI key number (positive for key-on events, negative for key-off events). By way of reminder, the following example shows three kern notes with a corresponding MIDI representation.

**kern	**MIDI
*	*Ch1
4c	72/60/64
4d	72/-60/64 72/62/64
4e	72/-62/64 72/64/64
.	72/-64/64
*-	*-

In order to translate to a dB representation, we must first isolate the key velocity values for key-on events. The following humsed command simply eliminates all data up to (and including) the last slash character:

extract -i '**MIDI' mono_input | humsed 's/.*\///'

This will leave us with just the key-down velocity data. Let’s suppose that the following rough decibel equivalents are established:

key	velocity approximate dB SPL
127	85
100	80
90	77
80	74
70	70
60	65
50	60
40	53
30	44
20	32
10	21
1	10
0	0

An appropriate reassignment file for recode would begin as follows:

=127	85
=100	80
=90	77
=80	74
=70	70

etc.

The completed translation would be accomplished by the following pipeline:

extract -i '**MIDI' mono_input | humsed 's/.*\///' \
     | recode -f reassign | sed 's/**MIDI/**dB/'

Notice the use of the sed command to replace the MIDI interpretation by a dB interpretation.

Reprise

In this chapter we have introduced three representations related to musical dynamics. The dyn representation allows us to encode dynamic markings as they appear visually in a printed score. Unfortunately, traditional notated dynamic markings are often confusing or ambiguous. In order to facilitate some types of analytic processing it is useful to generate a more rationalized interpretation of the dynamics of a work. The dynam representation provides a canonical scheme for representing basic notated dynamic markings where ambiguities are resolved by explicitly interpreting the meaning of dynamic markings. A third scheme (dB) provides a scheme for representing continuous dynamic levels in decibels. We have seen that the db command (which translates from dynam to dB) allows us to pose and answer a variety of questions related to the dynamic organization of music.