Ear Training on Music Theory Print

Ear training

Mon, 01 Jan 0001 00:00:00 +0000

Functional pitch recognition#

Functional pitch recognition involves identifying the function or role of a single pitch in the context of an established tonic. Once a tonic has been established, each subsequent pitch may be classified without direct reference to accompanying pitches. For example, once the tonic G has been established, listeners may recognize that the pitch D plays the role of the dominant in the key of G. No reference to any other pitch is required to establish this fact. Many musicians use functional pitch recognition in order to identify, understand, and appreciate the roles and meanings of pitches within a key. To this end, scale-degree numbers or movable-do solmization (do, re, mi, etc.) can be quite helpful. Using such systems, pitches with identical functions (the key note or tonic, for example) are associated with identical labels (1 or do, for example). Functional pitch recognition is not the same as fixed-do solfège, e.g. do, re, mi, etc. Functional pitch recognition emphasizes the role of a pitch with respect to the tonic, while fixed-do solfège symbols are labels for absolute pitch values (do=C, re=D, etc., in any key). In the fixed-do system (used in the conservatories of the Romance language nations, e.g. Paris, Madrid, Rome, as well as the Juilliard School and the Curtis Institute in the USA), solfège symbols do not describe the role of pitches relative to a tonic, but rather actual pitches. In the movable-do system, there happens to be a correspondence between the solfège symbol and a pitch’s role. However, there is no requirement that musicians associate the solfège symbols with the scale degrees. In fact, musicians may utilize the movable-do system to label pitches while mentally tracking intervals to determine the sequence of solfège symbols. Functional pitch recognition has several strengths. Since a large body of music is tonal, the technique is widely applicable. Since reference pitches are not required, music may be broken up by complex and difficult to analyze pitch clusters, for example, a percussion sequence, and pitch analysis may resume immediately once an easier to identify pitch is played, for example, by a trumpet—no need to keep track of the last note of the previous line or solo nor any need to keep track of a series of intervals going back all the way to the start of a piece. Since the function of pitch classes is a key element, the problem of compound intervals with interval recognition is not an issue—whether the notes in a melody are played within a single octave or over many octaves is irrelevant. Functional pitch recognition has some weaknesses. Music with no tonic or ambiguous tonality does not provide the frame of reference necessary for this type of analysis. When dealing with key changes, a student must know how to account for pitch function recognition after the key changes: retain the original tonic or change the frame of reference to the new tonic. This last aspect in particular, requires an ongoing real-time (even anticipatory) analysis of the music that is complicated by modulations and is the chief detriment to the movable-do system.

Equal temperament

Mon, 01 Jan 0001 00:00:00 +0000

Comparison with just intonation#

The intervals of 12 closely approximate some intervals in just intonation. The fifths and fourths are almost indistinguishably close to just intervals, while thirds and sixths are further away. In the following table, the sizes of various just intervals are compared to their equal-tempered counterparts, given as a ratio as well as cents. :

History#

The two figures frequently credited with the achievement of exact calculation of twelve-tone equal temperament are Zhu Zaiyu (also romanized as Chu-Tsaiyu, ) in 1584 and Simon Stevin in 1585. According to Fritz A. Kuttner, a critic of the theory, it is known that “Chu-Tsaiyu presented a highly precise, simple and ingenious method for arithmetic calculation of equal temperament mono-chords in 1584” and that “Simon Stevin offered a mathematical definition of equal temperament plus a somewhat less precise computation of the corresponding numerical values in 1585 or later.” The developments occurred independently. Kenneth Robinson attributes the invention of equal temperament to Zhu Zaiyu and provides textual quotations as evidence. Zhu Zaiyu is quoted as saying that, in a text dating from 1584, “I have founded a new system. I establish one foot as the number from which the others are to be extracted, and using proportions I extract them. Altogether one has to find the exact figures for the pitch-pipers in twelve operations.” Kuttner disagrees and remarks that his claim “cannot be considered correct without major qualifications.” Kuttner proposes that neither Zhu Zaiyu or Simon Stevin achieved equal temperament and that neither of the two should be treated as inventors.

Just intonation

Mon, 01 Jan 0001 00:00:00 +0000

Definition#

Any time an interval is sounded without acoustical beats it is in just intonation. The sound is also described as pure. The frequency of each note in a pure interval will correspond to the whole number ratios in the harmonic series. In the harmonic series on C, the 1st and 2nd notes form an octave in a 2:1 ratio. The fifth between the G and C is in a 3:2 ratio. The fourth is a 4:3 ratio. When its frequency is doubled, A 440 Hertz sounds an octave higher at 880 Hz. The pitch sounds an octave lower when the frequency is halved to 220 Hz. Just intonation also describes a tuning system that contains five or more pure intervals in an octave. There have been many attempts to construct scales composed completely of justly tuned intervals.

Pythagorean tuning

Mon, 01 Jan 0001 00:00:00 +0000

Method#

12-tone Pythagorean temperament is based on a sequence of perfect fifths, each tuned in the ratio 3:2, the next simplest ratio after 2:1 (the octave). Starting from D for example (D-based tuning), six other notes are produced by moving six times a ratio 3:2 up, and the remaining ones by moving the same ratio down: :E♭–B♭–F–C–G–D–A–E–B–F♯–C♯–G♯ This succession of eleven 3:2 intervals spans across a wide range of frequency (on a piano keyboard, it encompasses 77 keys). Since notes differing in frequency by a factor of 2 are perceived as similar and given the same name (octave equivalence), it is customary to divide or multiply the frequencies of some of these notes by 2 or by a power of 2. The purpose of this adjustment is to move the 12 notes within a smaller range of frequency, namely within the interval between the base note D and the D above it (a note with twice its frequency). This interval is typically called the basic octave (on a piano keyboard, an octave has only 12 keys). This dates to antiquity: in Ancient Mesopotamia, rather than stacking fifths, tuning was based on alternating ascending fifths and descending fourths (equal to an ascending fifth followed by a descending octave), resulting in the notes of a pentatonic or heptatonic scale falling within an octave. : In the formulas, the ratios 3:2 or 2:3 represent an ascending or descending perfect fifth (i.e. an increase or decrease in frequency by a perfect fifth, while 2:1 or 1:2 represent a rising or lowering octave). The formulas can also be expressed in terms of powers of the third and the second harmonics. The major scale based on C, obtained from this tuning is: : In equal temperament, pairs of enharmonic notes such as A and G are thought of as being exactly the same note—however, as the above table indicates, in Pythagorean tuning they have different ratios with respect to D, which means they are at a different frequency. This discrepancy, of about 23.46 cents, or nearly one quarter of a semitone, is known as a Pythagorean comma. To get around this problem, Pythagorean tuning constructs only twelve notes as above, with eleven fifths between them. For example, one may use only the 12 notes from E to G. This, as shown above, implies that only eleven just fifths are used to build the entire chromatic scale. The remaining interval (the diminished sixth from G to E) is left badly out-of-tune, meaning that any music which combines those two notes is unplayable in this tuning. A very out-of-tune interval such as this one is known as a wolf interval. In the case of Pythagorean tuning, all the fifths are 701.96 cents wide, in the exact ratio 3:2, except the wolf fifth, which is only 678.49 cents wide, nearly a quarter of a semitone flatter. If the notes G and E need to be sounded together, the position of the wolf fifth can be changed. For example, a C-based Pythagorean tuning would produce a stack of fifths running from D to F, making F-D the wolf interval. However, there will always be one wolf fifth in Pythagorean tuning, making it impossible to play in all keys in tune.

Solfège

Mon, 01 Jan 0001 00:00:00 +0000

Etymology#

The words solfège and solfeggio both derive from the names of two of the syllables used: sol and fa. The generic term “solmization”, referring to any system of denoting pitches of a musical scale by syllables, including those used in India and Japan as well as solfège, comes from French , from the Latin syllables sol and mi. The verb “to sol-fa” means to sing the solfège syllables of a passage (as opposed to singing the lyrics, humming, etc).