Chapter4

In this chapter we introduce some basic musical translations. We will limit this initial discussion to translating between various representations related to pitch. These will include French, German, and international pitch designations, frequency, semitones, cents, solfège, scale-degree and MIDI pitch representations. In later chapters we will describe additional types of translations.

The best-known system for representing equally-tempered pitches is the International Standards Organization (ISO) format consisting of a letter (A-G) followed by an optional sharp or flat, followed by an octave number. According to the ISO representation middle C is designated C4. Octave numbers change between B and C so that the B a semitone below C4 is B3. Humdrum provides a predefined ISO-like representation called **pitch illustrated below. Here we see an ascending chromatic scale in the left spine, with a concurrent descending chromatic scale in the right spine:

**pitch	**pitch
C4	C5
C#4	B4
D4	Bb4
D#4	A4
E4	Ab4
F4	G4
F#4	Gb4
G4	F4
G#4	E4
A4	Eb4
A#4	D4
B4	Db4
C5	C4
*-	*-

A4-19

Other pitch representations (such as **kern) can be translated to the ISO-inspired **pitch representation by invoking the pitch command. For example, consider the following **kern input:

**kern

=1

4g

4g#

4a

4cc

=2

*-

**pitch

=1

G4

G#4

A4

C5

=2

*-

pitch filename

Notice that the **pitch representation uses the same system for representing barlines as **kern. In fact, all of the pitch-related representations described in this chapter make use of the so-called `common system' for representing barlines.

The German system of pitch designations (Tonhöhe) is available in the Humdrum **Tonh representation. The German system is similar to the ISO system with the following exceptions. The pitch letter names include A,B,C,D,E,F,G,H and S. Sharps and flats are indicated by suffixes: `is' for sharps and `es' for flats, hence `Cis' for C-sharp and `Ges' for G-flat. Suffixes are repeated for double and triple sharps and flats. Special exceptions include the following: `B' for B-flat, `H' for B-natural, `Heses' for B double-flat (rather than `Bes'), and `As' and `Es' rather than `Aes' or `Ees'. `S' is a short-hand for `Es' (E-flat). As in the ISO pitch system, octaves are indicated by integer numbers with middle C represented as C4.

Although modern German practice has gravitated toward the ISO system, the traditional German system for representing pitches remains important in historically related studies, such as searching for `B-A-C-H' and the pitch signature used by Dmitri Shostakovich (`D-S-C-H').

Data in the **pitch or **kern representations can be translated to **Tonh via the tonh command:

tonh filename

The common French system for pitch naming uses a so-called "fixed-do" method of diatonic pitch designations: do, ré, mi, fa, sol, la and si (rather than ti), where do corresponds to the English/German `C'. In the Humdrum **solfg representation, solfège pitch names are used. Flats (bémol) and sharps (dièse) are abbreviated b and d respectively. When accidentals are encoded, the tilde character (~) is encoded following the solfège syllable and before the accidental. Double and triple sharps and flats are encoded via repetition. Hence, `do dièse' (do~d) for C-sharp, `la bémol' (la~b) for A-flat, `sol double-dièse' (sol~dd) for G double-sharp, `si double-bémol' (si~bb) for B double-flat, and so on. As with the German and ISO pitch representations, octave is designated by integers with do4 representing middle C.

For acoustic-related applications it may be helpful to translate to frequency. The Humdrum **freq representation can be used to represent frequencies for either pure or complex tones. Frequencies are encoded in hertz (abbreviated Hz) where 440 Hz means 440 cycles per second. In the **freq representation frequencies may be specified as integer or real values (with a decimal point).

The **cents representation provides a means for representing pitches in absolute units with respect to middle C (= 0 cents). In the **cents representation, all pitches are represented with respect to this reference. Thus C#4 is represented by the number 100, A4 is represented by 900, and A3 is represented by -300. As in the case of **freq, cents may be specified as integer numbers or as real values (with a decimal point).

A related pitch representation is **semits. In this case, all pitches are represented in numerical semitones with respect to middle C (= 0 semits). An ascending chromatic scale beginning on C4 would be represented by the ascending integers from 0 to 12. Pitches below middle C are represented by negative values. Fractional values can be represented using decimal points.

Another way of representing pitch is provided by the Humdrum **MIDI representation. This representation closely mimics the commercial MIDI specification. The **MIDI representation allows MIDI inputs and outputs to be exported or imported by various Humdrum tools. A complete description of **MIDI will be given in Chapter 7.

Two different Humdrum representations are provided to describe scale-degree related information: **deg and **solfa. Both of these representations emphasize slightly different aspects of scale-degree information. Both representations assume some established or pre-defined tonal center or tonic pitch.

The **solfa representation represents pitch according to tonic solfa syllables. Pitches are designated by the syllables do, re, mi, fa, so, la and ti or their chromatic alterations as indicated in the following table:

basic	raised	lowered
do (doe)	di (dee)	de (day)
re (ray)	ri (ree)	ra (raw)
mi (me)	my (my)	me (may)
fa (fah)	fi (fee)	fe (fay)
so (so)	si (see)	se (say)
la (la)	li (lee)	le (lay)
ti (tee)	ty (tie)	te (tay)

Summary of solfa Signifiers

The **solfa representation differs from **deg in that pitches are represented without regard to major or minor mode. For example, in the key of C major, **deg will characterize A-flat as a lowered sixth scale degree (6-), whereas the same pitch will be a normal (unaltered) sixth scale degree in the key of C minor (6). In the case of **solfa, the A-flat will be represented as le -- whether or not the key is C major or C minor. Like **deg, the amount of chromatic alteration is not represented in **solfa. Once a pitch is raised, raising it further will not change the representation. For example, if the tonic is B-flat, then both B-natural and B-sharp will be represented by di in the **solfa representation.

In the case of the minor mode, **deg characterizes scale degrees with respect to the harmonic minor scale only.

Another difference between **solfa and **deg is that the **deg representation provides a way for encoding melodic approach. The caret (^) denotes an ascending melodic approach to the current note, whereas the lower-case letter v denotes a descending melodic approach. Repeated pitches carry no melodic approach signifier.

Some of the differences between the **solfa and **deg representations are illustrated in Example 4.1. (The corresponding **kern representation is given in the first spine.) Notice that **solfa does not encode any octave information. The **deg representation does not encode the octave of the starting pitch, but it does indicate contour information using the caret (^) for ascending and the lower-case v for descending pitches. Notice also the different ways of characterizing accidentals.

Example 4.1

!! Comparison	of pitch-related	representations.
**kern	**solfa	**deg
*M2/4	*M2/4	*M2/4
*c:	*c:	*c:
8.cc	do	1
16dd	re	^2
=1	=1	=1
8.ee-	me	^3
16dd	re	v2
4een	mi	^3+
=2	=2	=2
8r	r	r
8b-	te	v7-
8an	la	v6+
8cc	do	^1
=3	=3	=3
2bn	ti	v7
==	==	==
*-	*-	*-

Pitch Translations

Humdrum provides a number of commands for translating between the various pitch-related representations described above. Typically, the command name is the same as the name of the output representation. For example, translating to the **solfg representation can be accomplished with:

solfg inputfile > outputfile

Translating to the German **Tonh representation:

tonh inputfile > outputfile

Translating to ISO **pitch:

pitch inputfile > outputfile

Similarly, the freq command translates pitch-related inputs to the **freq representation, the cents command translates appropriate inputs to the **cents representation, and so on.

In a few cases, the command names are slightly modified. All Humdrum command names employ lower-case letters only, so **MIDI output is generated by the midi command (rather than the MIDI command), and **Tonh output is generated by the tonh command.

Examples 4.2 and 4.3 compare several parallel representations of the same pitch-related information. In both examples, the pitch information has been derived from the **kern data shown in the left-most spine. The duration information in the **kern data is not available in the other representations. However, the `common system' for barlines is used throughout.

Example 4.2 shows four pitch naming systems: ISO pitch, German Tonhöhe, French solfège, as well as **kern. Notice the different ways of treating accidentals such as the D-sharp and B-flat. Also note the German use of H for B-natural.

Example 4.2

!! Comparison	of pitch-related	representations.
**kern	**pitch	**Tonh	**solfg
*M2/4	*M2/4	*M2/4	*M2/4
*C:	*C:	*C:	*C:
8.cc	C5	C5	do5
16dd	D5	D5	re5
=1	=1	=1	=1
8.ee	E5	E5	mi5
16dd#	D#5	Dis5	re~d5
4ee	E5	E5	mi5
=2	=2	=2	=2
8r	r	r	r
8b-	Bb4	B4	si~b4
8a	A4	A4	la4
8c	C4	C4	do4
=3	=3	=3	=3
2bn	B4	H4	si4
==	==	==	==
*-	*-	*-	*-

!! Comparison	of pitch-related	representations	(continued).
**kern	**semits	**cents	**MIDI	**freq
*M2/4	*M2/4	*M2/4	*Ch1	*M2/4
*C:	*C:	*C:	*M2/4	*C:
*	*	*	*C:	*
8.cc	12	1200	/72/	523.25
16dd	14	1400	/-72/ /74/	587.33
=1	=1	=1	=1	=1
8.ee	16	1600	/-74/ /76/	659.26
16dd#	15	1500	/-76/ /75/	622.25
4ee	16	1600	/-75/ /76/	659.26
=2	=2	=2	=2	=2
8r	r	r	/-76/	r
8b-	10	1000	/70/	466.16
8a	9	900	/-70/ /69/	440.00
8c	0	0	/-69/ /60/	261.63
=3	=3	=3	=3	=3
2bn	11	1100	/-60/ /71/	493.88
==	==	==	==	==
.	.	.	/-71/	.
*-	*-	*-	*-	*-

	**cents	**deg	**freq	**kern	**MIDI	**pitch	**semits	**solfa	**solfg	**Tonh
cents			¤	¤		¤	¤		¤	¤
cocho			¤
deg				¤		¤			¤	¤
freq	¤			¤	¤	¤	¤		¤	¤
kern	¤		¤		¤	¤	¤	¤	¤	¤
midi				¤
pitch	¤		¤	¤	¤		¤	¤	¤	¤
semits	¤		¤	¤	¤	¤			¤	¤
solfa				¤		¤			¤	¤
solfg	¤		¤	¤	¤	¤	¤		¤
tonh	¤		¤	¤	¤	¤	¤	¤	¤

A common pitch-related manipulation is transposition. The trans command has the user specify a diatonic offset and a chromatic offset. The diatonic offset affects the pitch-letter name used to spell a note. The chromatic offset affects the number of semitones shifted from the original pitch height. The two types of offset are completely independent of each other. For common transpositions, both the diatonic and chromatic offsets will need to be specified. For example, in transposing up a minor third (e.g. C to E-flat), the diatonic offset is `up two pitch-letter names,' and the chromatic offset is `up three semitones.' The appropriate command invocation is:

trans -d +2 -c +3 input > output

The diatonic offset can be a little confusing because traditional terminology labels perfect unisons by the number 1 (e.g. P1) rather than zero. So transposing up a perfect fifth involves a diatonic offset of +4 letter names, and a chromatic offset of +7 semitones:

trans -d +4 -c +7 input > output

For some changes of mode (such as melodic to harmonic minor), you may need to use the humsed command described in Chapter 14 to modify accidentals for specific scale degrees.

Notice the addition of a "tandem interpretation" to the above example (*Trd1). Whenever trans is invoked, it adds a record indicating that the encoding is no longer at the original pitch. Transposition tandem interpretations are similar in syntax to the trans command itself. In the above example, *Trd1 indicates a diatonic shift up one letter name. The tandem interpretation *Trd-1c-2 would indicate that a score has been transposed down a major second. The trans command also provides a -k option that allows the user to specify a replacement key signature for the output.

The trans command can be used in conjunction with any of the appropriate pitch-related representations, such as **pitch, **kern, **Tonh, and **solfg.

The proportion of subdominant pitches can be calculated by simply comparing the resulting pattern count with the number of notes identified by census.

Recall that one of the differences between the **solfa and **deg representations is that the **deg output contains an indication of the direction of melodic approach. The caret (^) indicates approach from below, whereas the lower-case v indicates approach from above. Suppose we wanted to determine whether the dominant pitch is more commonly approached from above or from below. Assuming a monophonic input, we can once again use grep to answer this question. First let's count how many dominant pitches (`5') are approached from above (`v'):

deg repertory.krn | grep -c v5

The caret has a special meaning for grep which will be discussed in Chapter 9. We can escape the special meaning by preceding the caret by a backslash. In order to count the number of dominant pitches approached from below we can use the following:

deg repertory.krn | grep -c \^5

Recall that some scale tones are spelled differently depending on whether the mode is major or minor. For example, in A major the mediant pitch is C sharp; but in A minor the mediant pitch is C natural. The deg and solfa commands produce subtly contrasting outputs that make one or the other command better suited depending on the user's goal. The deg command would represent C sharp in A major, and C natural in A minor by the same scale degree -- 3. In the key of A major, C natural would be characterized as a lowered mediant (3-) and in A minor, C sharp would be characterized as a raised mediant (3+). By contrast, the solfa command characterizes pitches with respect to the tonic alone and ignores the mode. Hence, solfa would designate C sharp as `mi' whether the key was A major or A minor. Similarly, C natural would be designated `me' in both A major and A minor. The differences between deg and solfa allow users to distinguish chromatically altered scale tones in a manner appropriate to the task.

Occasionally it is useful to process a given representation to the same representation. The kern command translates various pitch-related representations to the **kern format. The -x option eliminates any input data that do not pertain to pitch. When applied to a **kern input, this option allows us to filter out durations, articulation marks, phrasing, and other non-pitch data. Suppose, for example, that we wanted to determine the proportion of successively repeated notes in a vocal melody: how often is a pitch followed immediately by the same pitch? We might begin by first determining the total number of notes in the melody using census with the -k option.

census -k melody.krn

We can use the uniq command to eliminate successive repeated pitches -- but only if the note tokens are identical. First we can use kern -x to translate "from **kern to **kern" while eliminating non-pitch-related data. Then we need to remove barlines so they don't interfere with pitches that are repeated across the measure. Using uniq will then eliminate all of the successively duplicated records, so a sequence of six G's will be reduced to a single G. Finally, we pipe the output to census -k to count the total number of notes.

kern -x melody.krn | uniq | census -k

semits melody.krn | kern | uniq | census -k

Incidentally, given **semits input, the kern command will spell pitches according to any key or key signatures it encounters. For example, if the key signature contains sharps, then G-sharp will be output; if the key or key signature contains flats, then A-flat will be output.

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  © Copyright 1999 David Huron