Chapter 4: Basic Pitch Translations¶
Many musical processes entail some sort of data translation in which one form of representation is transformed into another form of representation. There are innumerable examples of such musical “translations.” For example, we might rewrite guitar tablatures as notated pitches. Pitches might be translated to tonic-solfa syllables. Scale-degrees might be analyzed as Roman-numeral harmonies. Figured bass notations might be “realized” as pitches. Successive pitches might be characterized as melodic intervals. Intervals might be rewritten as semitone distances. Pitch-class sets might be transformed to interval-class vectors.
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, solfege, scale-degree and MIDI pitch representations. In later chapters we will describe additional types of translations.
ISO Pitch Representation¶
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 = = C1 C1 *- *-
Notice that only upper-case letters are used for pitch-name and that the flat is represented by the lower-case letter b‘. The small letter x‘ can be used to indicate double-sharps and the double-flat is represented by two small successive letters bb‘. If a given pitch deviates from equal temperament, cents deviation can be indicated by trailing integers preceded by either a plus sign (tuned sharp) or a minus sign (tuned flat). The unit of the cent is one one-hundredth of a semitone, so the distance between C4 and C#4 is 100 cents. For example, the pitch corresponding to a tuning of A-435 Hz is 19 cents flat compared with the standard A-440, hence it is represented in **pitch as:
**kern =1 4g 4g# 4a 4cc =2 *-
It can be translated to this **pitch output:
**pitch =1 G4 G#4 A4 C5 =2 *-
using the following command:
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.
German Tonhohe –
The German system of pitch designations (Tonhohe) 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’).
French Solfege –
The common French system for pitch naming uses a so-called “fixed-do” method of diatonic pitch designations: do, re, mi, fa, sol, la and si (rather than ti), where do corresponds to the English/German C’. In the Humdrum |**solfg|_ representation, solfege pitch names are used. Flats (*bemol*) and sharps (*diese*) are abbreviated *b* and *d* respectively. When accidentals are encoded, the tilde character (``~`) is encoded following the solfege syllable and before the accidental. Double and triple sharps and flats are encoded via repetition. Hence, do diese’ (``do~d`) for C-sharp, la bemol’ (``la~b`) for A-flat, sol double-diese’ (``sol~dd`) for G double-sharp, si double-bemol’ (``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.
Scale Degree – **solfa and **deg¶
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 **deg representation identifies scale-degrees by the numbers 1 (tonic) to 7 (leading-tone). These values may be chromatically altered by raising (+) or lowering (-). The amount of chromatic alteration is not indicated; for example, both a raised supertonic and a doubly-raised supertonic are represented as 2+. A lowered dominant is represented as 5-.
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.
!! 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 == == == *- *- *-
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
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 Tonhohe, French solfege, 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.
!! 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 8 r r r 8b- Bb4 B4 si4~b4 8a A4 A4 la4 8c C4 C4 do4 =3 =3 =3 =3 2bn B4 H4 si4 == == == == *- *- *- *-
In Example 4.3 four of the more technical representations are illustrated, including frequency and cents. Notice that the **MIDI representation uses key-numbers to represent pitch: key-on events are indicated by positive integers (between two slashes) and key-off events are indicated by negative integers. More detail concerning **MIDI is given in Chapter 7.
!! 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 8 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/ . *- *- *- *- *-
Not all of the above pitch-related representations can be translated directly from one to another. Table 4.1 shows the possible translations supported by Humdrum Release 2.0 commands. The input representations are listed from right to left. Under each column, those commands that will translate from the given format are identified. For example, the **cents representation can be translated to **freq, **kern, **pitch, **semits, **solfg, and |**tonh|_. Notice that **deg data cannot be translated to any other format since **deg representations do not encode absolute pitch height. Note also that when translating to the **kern representation, only pitch-related information is translated: duration, articulation marks, and other **kern signifiers are not magically generated.
Transposition Using the trans Command¶
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
We can transpose without changing the diatonic pitch names. For example, the following command will transpose down an augmented unison (e.g. C# to C):
trans -d 0 -c -1 input > output
Conversely, we can respell the diatonic pitches without changing the overall pitch height. For example, the following transposition will transpose “up” a diminished second (e.g. from F-sharp to G-flat):
trans -d +1 -c 0 input > output
Modal transpositions are also possible by omitting the chromatic offset option. Consider, for example, the following C major scale:
**kern d e f g a b cc *-
We can transform this using the following diatonic transposition:
trans -d +1
The resulting output is the Dorian mode:
**kern *Trd1 d e f g a b cc dd *-
When using the -d option alone, trans eliminates all accidentals in the input. This can be potentially confusing, but it is often useful. Suppose you have a passage in the key of E major which you would like to translate to E Dorian. First transpose so the tonic is D using only the -d option; then transpose exactly so the tonic is E again:
trans -d -1 Emajor | trans -d +1 -c +2 > Edorian
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.
In order for the |solfa|_ or |deg|_ commands to translate from other pitch representations, the encoded music must contain an explicit key indication. Keys are explicitly represented by a single asterisk, followed by an upper- or lower-case letter, followed by an optional accidental, followed by a colon. The octothorpe (#) indicates a sharp and the hyphen (-) indicates a flat.
Upper-case letters indicate major keys; lower-case letters indicate minor keys. By way of illustration, the following key interpretations indicate the keys of C major, C minor, B-flat major, and F-sharp minor:
Key interpretations usually appear near the beginning of a representation, and key interpretations can be redefined at any place in a score.
Apart from transposition, translating from one representation to another provides opportunities for different sorts of processing. Suppose, for example, we wanted to know whether the subdominant pitch occurs more frequently in one vocal repertory than in another repertory. We can use |solfa|_ in conjunction with grep‘s -c option to count the number of occurrences. (For the following examples, we will assume that the inputs consist of only a single spine, that barlines are absent, and that appropriate interpretations are provided indicating the key of each work.) First we need to count the total number of notes in each repertory.
census -k repertory1.krn
census -k repertory2.krn
solfa repertory1.krn | grep -c fa
solfa repertory2.krn | grep -c fa
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.
Uses for Pitch Translations¶
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
A variation on this approach would entail translating to a representation that does not distinguish enharmonic pitches. For example, translating our melody to **semits and then back to **kernwill standarize all of the enharmonic spellings. If our melody contains a G-sharp that undergoes an enharmonic shift to A-flat, then the pitches will be deemed identical. The following command carries out the same task as above, but ignores possible enharmonic spellings:
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.
In this chapter we have introduced a number of pre-defined pitch-related representations. Simple commands can be used to translate from one representation to another. Which representation is most appropriate depends on the user’s goal.
There is a wealth of other representation formats related to pitch distances, tablatures, timing, and other types of musical information. These representations will be explored in later chapters. In addition, we’ll describe how to design your own representations – representations that may be better tailored to a specific application. However, before we continue discussing further representations, this is an appropriate point to present a more formal description of the general Humdrum representation syntax.