Modulation (music)

In music, modulation is the change from one key (tonic, or tonal center) to another. This may or may not be accompanied by a change in key signature. Modulations articulate or create the structure or form of many pieces, as well as add interest. Treatment of a chord as the tonic for less than a phrase is considered tonicization.

Modulation is the essential part of the art. Without it there is little music, for a piece derives its true beauty not from the large number of fixed modes which it embraces but rather from the subtle fabric of its modulation.

— Charles-Henri Blainville (1767)[2]
Modulation vocal music example duple labelled
Example of modulation from the tonic to the dominant.[1] Play 
Key change example
Key signature change example: C major to C minor.

Requirements

The quasi-tonic is the tonic of the new key established by the modulation. The modulating dominant is the dominant of the quasi-tonic. The pivot chord is a predominant to the modulating dominant and a chord common to both the keys of the tonic and the quasi-tonic. For example, in a modulation to the dominant, ii/V–V/V–V could be a pivot chord, modulating dominant, and quasi-tonic.

Types

Common-chord modulation

Chopin - Prelude in C minor opening modulation
Common-chord modulation in the opening of Chopin's Prelude in C minor, Op. 28, No. 20.[4] Play 
Common-chord modulation in Tchaikovsky, Mazurka Op. 39, No. 10
Common-chord modulation in Tchaikovsky's Album pour enfants (1887), Op. 39, No. 10, Mazurka[5] Play 
Common chord modulation in Mozart, Sonata in D Major, K. 284, III, m. 1-8
Common-chord modulation in the opening of Mozart's, Sonata in D Major, K. 284, III[6] Play 

Common-chord modulation (also known as diatonic-pivot-chord modulation) moves from the original key to the destination key (usually a closely related key) by way of a chord both keys share: "Most modulations are made smoother by using one or more chords that are common to both keys."[7] For example, G major and D major have four chords in common: G major, B minor, D major and E minor. This can be easily determined by a chart similar to the one below, which compares triad qualities. The I chord in G major—a G major chord—is also the IV chord in D major, so I in G major and IV in D major are aligned on the chart.

G major I
G
ii
Am
iii
Bm
IV
C
V
D
vi
Em
viio
Fo
D major IV
G
V
A
vi
Bm
viio
Co
I
D
ii
Em
iii
Fm

Any chord with the same root note and chord quality (major, minor, diminished) can be used as the pivot chord. However, chords that are not generally found in the style of the piece (for example, major VII chords in a J. S. Bach-style chorale) are also not likely to be chosen as the pivot chord. The most common pivot chords are the predominant chords (ii and IV) in the new key. In analysis of a piece that uses this style of modulation, the common chord is labeled with its function in both the original and the destination keys, as it can be heard either way.

Where an altered chord is used as a pivot chord in either the old or new key (or both), this would be referred to as altered common chord modulation, in order to distinguish the chromaticism that would be introduced from the otherwise, diatonic method.

Enharmonic modulation

Schubert - op.9 D.365, mm.17-24 German sixth modulation
Modulation from D major to D major in Schubert's Op. 9, No. 14, D. 365, mm. 17–24, using the German sixth, in the new key, that is enharmonic to the dominant seventh in the old key.[8] Play 
Schubert - op. 29, D.804, I, mm.144-49 enharmonic modulation
Modulation from A minor to E minor in Schubert's Op.29, D. 804, I, mm.144-49, using viio7: Go7 ≡ Do7 (≡ Bo7 ≡ Fo7)[9] Play 

An enharmonic modulation takes place when one treats a chord as if it were spelled enharmonically as a functional chord in the destination key, and then proceeds in the destination key. There are two main types of enharmonic modulations: dominant seventh/augmented sixth, and (fully) diminished seventh. Any dominant seventh or German sixth can be reinterpreted as the other by respelling the m7 or A6 chord tone (respectively) in order to modulate to a key a half-step away (descending or ascending); if the fifth from root chord tone of a German sixth is omitted, the result is an Italian sixth. A diminished seventh chord meanwhile, can be respelled in multiple other ways to form a diminished seventh chord in a key a minor third (m3 as root), tritone (d5 as root) or major sixth (d7 as root) away.[10] Where the dominant seventh is found in all diatonic scales, the diminished seventh is found only in the harmonic scale naturally; an augmented sixth is itself an altered chord, relying on the raised fourth scale degree.

By combining the diminished seventh with a dominant seventh and/or augmented sixth, altering only one pivot note (by a half tone), it is possible to modulate quite smoothly from any key to any other in at most three chords, no matter how distant the starting and ending keys (be aware that, only when modulating between key signatures featuring double-sharps/flats, may the need to respell natural notes enharmonically arise); however, this may or may not require the use of altered chords (operating in the harmonic minor without augmented sixth would not) where the effect can be less subtle than other modulations. The following are examples used to describe this in chord progressions starting from the key of D minor (these chords may instead be used in other keys as borrowed chords, such as the parallel major, or other forms of the minor):

  • C–E–G–B (dim. 7th), C–E–G–B (lowering the root a semitone to a modulating dom. 7th), F–A–C (quasi-tonic) takes us to F major—a relative major modulation (though not enharmonic); but exactly the same progression enharmonically C–E–G–B, C–E–G–A (Ger. aug. 6th), E–G–B–E (quasi-tonic) takes us somewhat unexpectedly to E natural/harmonic minor—a half-step modulation (ascending).
  • C–E–G–B (dim. 7th), A–C–E–G (lowering the 7th a semitone and respelling as a modulating dom. 7th), D–F–A (quasi-tonic) takes us to the key of D major—a parallel modulation (though not enharmonic). Enharmonically: C–E–G–B, A–C–E–Fdouble sharp (Ger. aug. 6th), C–E–G (quasi-tonic) modulates to C minor—a major seventh modulation/half-step descending.
  • C–E–G–B (dim. 7th), C–E–G–B ≡ E–G–B–D (lowering the major third a half tone and respelling as a modulating dom. 7th), A–C–E (quasi-tonic) leads to A major—a minor third and relative modulation (or tritone modulation if starting in D Major).

Note that in standard voice leading practice, any type of augmented sixth chord favours a resolution to the dominant chord (see: augmented sixth chord), with the exception of the German sixth, where it is difficult to avoid incurring parallel fifths; to prevent this, a cadential six four is commonly introduced before the dominant chord (which would then typically resolve to the tonic to establish tonality in the new key), or an Italian/French sixth is used instead.

In short, lowering any note of a diminished seventh chord a half tone leads to a dominant seventh chord (or German sixth enharmonically), the lowered note being the root of the new chord. Raising any note of a diminished seventh chord a half tone leads to a half-diminished seventh chord, the root of which is a whole step above the raised note. This means that any diminished chord can be modulated to eight different chords by simply lowering or raising any of its notes. If also employing enharmonic respelling of the diminished seventh chord, such as that beginning the modulation in the above examples (allowing for three other possible diminished seventh chords in other keys), it quickly becomes apparent the versatility of this combination technique and the wide range of available options in key modulation.

This type of modulation is particularly common in Romantic music, in which chromaticism rose to prominence.

Other types of enharmonic modulation include the augmented triad (III+) and French sixth (Fr+6). Augmented triad modulation occurs in the same fashion as the diminished seventh, that is, to modulate to another augmented triad in a key: a major third (M3 as root) or minor sixth (A5 as root) away. French augmented sixth (Fr+6) modulation is achieved similarly but by respelling both notes of either the top or bottom major third (i.e. root and major third or diminished fifth and augmented sixth) enharmonically and inverting with the other major third (i.e. diminished fifth and augmented sixth becomes root and major third of the new Fr+6); either choice results in the same chord and key modulation (a tritone away), as the diminished fifth always becomes the new root.

Common-tone modulation

Schubert - Op.163 (D.956), i common-tone modulation
Modulation between relative keys, C minor and E major, using a common tone, G, in Schubert's Op. 163 (D. 956).[11] Play 
Common tone modulation between chromatic mediants in Mozart K 475
Common-tone modulation between chromatic mediants in Mozart's K.475[12] Play 

Common-tone modulation uses a sustained or repeated pitch from the old key as a bridge between it and the new key (common tone). Usually, this pitch will be held alone before the music continues in the new key. For example, a held F from a section in B major could be used to transition to F major. This is used, for example, in Schubert's Unfinished Symphony. "If all of the notes in the chord are common to both scales (major or minor), then we call it a common chord modulation. If only one or two of the notes are common, then we call it common tone modulation."[13]

Starting from a major chord, for example G major (G–B–D), there are twelve potential goals using a common-tone modulation: G minor, G minor, B major, B major, B minor, C major, C minor, D minor, D major, E major, E major, E minor.[14] Thus common-tone modulations are convenient for modulation by diatonic or chromatic third.

Chromatic modulation

Chromatic modulation in Bach BWV 300, m. 5-6
Chromatic modulation in Bach's Du grosser Schmerzensmann, BWV 300, mm. 5–6[15] (Play  with half cadence, Play  with PAC) transitions from F major to D minor through the inflection of C to C between the second and third chords. Note that there is no common chord.

A chromatic modulation is so named because it occurs at the point of a chromatic progression, one which involves the chromatic inflection of one or more notes whose letter name, thus, remains the same though altered through an accidental.[15] Chromatic modulations are often between keys which are not closely related.[15] A secondary dominant or other chromatically altered chord may be used to lead one voice chromatically up or down on the way to the new key. (In standard four-part chorale-style writing, this chromatic line will most often be in one voice.) For example, a chromatic modulation from C major to D minor:

C major IV
F
V/ii
A
ii
Dm
D minor i
Dm
(...)

In this case, the IV chord in C major (F major) would be spelled F–A–C, the V/ii chord in C major (A major) spelled A–C–E, and the ii chord in C major (D minor), D–F–A. Thus the chromaticism, C–C–D, along the three chords; this could easily be part-written so those notes all occurred in one voice. Despite the common chord (ii in C major or i in D minor), this modulation is chromatic due to this inflection.

In the example pictured, a chromatic modulation from F major to D minor:

F major I
F
V
C
D minor V
A
i
Dm
iv
Gm
V
A

In this case, the V chord in F major (C major) would be spelled C–E–G, the V in D minor (A major) would be spelled A–C–E. Thus the chromaticism, C–C–D, which is here split between voices but may often easily be part-written so that all three notes occur in one voice.

The combination of chromatic modulation with enharmonic modulation in late Romantic music led to extremely complex progressions in the music of such composers as César Franck, in which two or three key shifts may occur in the space of a single bar, each phrase ends in a key harmonically remote from its beginning, and great dramatic tension is built while all sense of underlying tonality is temporarily in abeyance. Good examples are to be found in the opening of his Symphony in D minor, of which he himself said (see Wikiquote) "I dared much, but the next time, you will see, I will dare even more..."; and his Trois Chorals for organ, especially the first and third of these, indeed fulfill that promise.

Phrase modulation

Mozart - K.331, III, mm.6-10 phrase modulation
Phrase modulation in Mozart's Sonata in A major, K.331, III (Alla turca), mm. 6–10.[16] Play 

Phrase (also called direct, static, or abrupt) modulation is a modulation in which one phrase ends with a cadence in the original key, and begins the next phrase in the destination key without any transition material linking the two keys. This type of modulation is frequently done to a closely related key—particularly the dominant or the relative major/minor key.

An unprepared modulation is a modulation "without any harmonic bridge", characteristic of impressionism.[17]

For example:

A E A F B F
A major I V I
F major I IV I

Sequential modulation

Sequential modulation in Beethoven, Sonata Op. 53, movement I
Sequential modulation in Beethoven's Sonata Op. 53, movement I[18] Play 
Sequential modulation in Schubert, Sonata in E Major, movement III
Sequential modulation in Schubert's Piano Sonata in E Major, D. 459, movement III[18] Play 

"A passage in a given key ending in a cadence might be followed by the same passage transposed (up or down) to another key," this being known as sequential modulation.[19] Although a sequence does not have to modulate, it is also possible to modulate by way of a sequence. A sequential modulation is also called rosalia. The sequential passage will begin in the home key, and may move either diatonically or chromatically. Harmonic function is generally disregarded in a sequence, or, at least, it is far less important than the sequential motion. For this reason, a sequence may end at a point that suggests a different tonality than the home key, and the composition may continue naturally in that key.

Chain modulation

Sequential modulation through the circle of fifths in Haydn, Quartet Op. 3, No. 3, IV
Sequential modulation through the circle of fifths in Quartet Op. 3, No. 3, IV, Hob. III:15,[20] formerly attributed to Haydn (ca. 1840) Play 

Distant keys may be reached sequentially through closely related keys by chain modulation, for example C to G to D or C to C minor to E major.[21] A common technique is the addition of the minor seventh after each tonic is reached, thus turning it into a dominant seventh chord:

D D7 G G7 C C7 F
I V7 I V7 I V7 I

Parallel key modulation

A parallel key modulation is a change of mode, but maintains the same tonal center. For example, one section of a composition may be in the key of E major and then modulate to E minor. This can be done directly or facilitated by the various modulation techniques described above. Depending on the length of the modulation and whether or not it returns to the original key, it may or may not be designated by a change of key signature.

Common modulations

Pitch class space star
The circle of fifths drawn within the chromatic circle as a dodecagram[22]
Truck driver's gear change Because the Night
Modulation up a whole step at the end of "Because the Night" Play 

The most common modulations are to closely related keys (I, V, IV, vi, iii, ii).[23] V (dominant) is the most frequent goal and, in minor, III (relative key) is also a common goal.[24] Modulation to the dominant or the subdominant is relatively simple as they are adjacent steps on the circle of fifths. Modulations to the relative major or minor are also simple, as these keys share all pitches in common. Modulation to distantly related keys is often done smoothly through using chords in successive related keys, such as through the circle of fifths, the entirety of which may be used in either direction:

D – A – E – B/C – F/G – C/D – G/A – D/E – A/B – F – C – G – D

If a given key were G major, the following chart could be used:

C G D

From G (which is the given key), a musician would go P5 (a perfect fifth) above G (which is D) and also P5 below G (which is C).

From this, the musician would go to G major's relative minor which is E minor, and potentially to C major and D major's related minor as well (a musician who does not know the related minor for C and D major may also go P5 below or above E minor).

C G D
| | |
Am Em Bm

By using the relative minor keys one can find the specific key that the key can modulate into.

Many musicians use the circle of fifths to find these keys and make similar charts to help with the modulation.

Significance

In certain classical music forms, a modulation can have structural significance. In sonata form, for example, a modulation separates the first subject from the second subject. Frequent changes of key characterize the development section of sonatas. Moving to the subdominant is a standard practice in the trio section of a march in a major key, while a minor march will typically move to the relative major.

Changes of key may also represent changes in mood. In many genres of music, moving from a lower key to a higher often indicates an increase in energy.

Change of key is not possible in the full chromatic or the twelve tone technique, as the modulatory space is completely filled; i.e., if every pitch is equal and ubiquitous there is nowhere else to go. Thus other differentiating methods are used, most importantly ordering and permutation. However, certain pitch formations may be used as a "tonic" or home area.

Other types

Though modulation generally refers to changes of key, any parameter may be modulated, particularly in music of the 20th and 21st century. Metric modulation (known also as tempo modulation) is the most common, while timbral modulation (gradual changes in tone color), and spatial modulation (changing the location from which sound occurs) are also used.

Modulation may also occur from a single tonality to a polytonality, often by beginning with a duplicated tonic chord and modulating the chords in contrary motion until the desired polytonality is reached.

See also

Further reading

  • Vincent Persichetti, Twentieth-Century Harmony. W.W. Norton and Company, 1961. ISBN 0-393-09539-8.

References

  1. ^ Boston Academy of Music, Lowell Mason (1836). The Boston Academy's Collection of Church Music, pp. 16–18. Fourth edition. J. H. Wilkins and R. B. Carter.
  2. ^ Forte, Allen (1979). Tonal Harmony in Concept & Practice, p. 265. ISBN 0-03-020756-8.
  3. ^ a b c Forte (1979), p. 267.
  4. ^ Benward and Saker (2009). Music in Theory and Practice, Vol. II, p. 214. ISBN 978-0-07-310188-0.
  5. ^ Forte (1979), p. 307.
  6. ^ Benward and Saker (2009), p. 244.
  7. ^ Forte (1979), p. 305.
  8. ^ Benward & Saker (2009), pp. 214–15.
  9. ^ Benward & Saker (2009), p.220.
  10. ^ "Enharmonic Reinterpretation" (PDF). Feezell, M. Retrieved 2016-04-05.
  11. ^ Meyer, Leonard B. (1989). Style and Music: Theory, History, and Ideology, p. 299. ISBN 9780226521527.
  12. ^ Kostka, Stefan and Payne, Dorothy (1995). Tonal Harmony, p. 321. McGraw-Hill. ISBN 0-07-035874-5.
  13. ^ Briggs, Kendall Durelle (2014). The Language and Materials of Music, p. 198. Lulu.com. ISBN 9781257996148.
  14. ^ Kopp, David (2006). Chromatic Transformations in Nineteenth-Century Music, p. 50. Cambridge University Press. ISBN 9780521028493. After Marx, Adolph Bernard. Theory and Practice (1837). Trans. Saroni.
  15. ^ a b c Benward and Saker (2003). Music: In Theory and Practice, Vol. I, p. 245. Seventh Edition. ISBN 978-0-07-294262-0.
  16. ^ Benward and Saker (2003), Vol. I, p. 244.
  17. ^ Reti, Rudolph (1978). Tonality, Atonality, Pantonality. Westport, CT: Greenwood Press. ISBN 0-313-20478-0.
  18. ^ a b Forte (1979), p.319.
  19. ^ Heussenstamm, George (2011). Hal Leonard Harmony & Theory – Part 2: Chromatic. ISBN 9781476841212.
  20. ^ Forte (1979), p.320.
  21. ^ Jones, George T. (1994). HarperCollins College Outline Music Theory, p. 217. ISBN 0-06-467168-2.
  22. ^ "Prelude to Musical Geometry", p. 364, Brian J. McCartin, The College Mathematics Journal, Vol. 29, No. 5 (Nov., 1998), pp. 354–70. (abstract) (JSTOR).
  23. ^ Benward & Saker (2003). Music: In Theory and Practice, Vol. I, p. 243. 7th edition. McGraw-Hill. ISBN 978-0-07-294262-0. "Most modulations occur between closely related keys, which are those keys that differ by no more than one accidental in the key signature."
  24. ^ Forte (1979), p. 269.

External links

Index of music articles

This page is an alphabetized index of articles about music.

Interdominant

In music, an interdominant is a temporary dominant, the dominant of a key other than the tonic. Since a composition generally begins and ends with the tonic, the dominant of notes other than the tonic would be found in the middle.

José Asunción Silva

José Asunción Silva (27 November 1865 in Bogotá – 23 May 1896 in Bogotá) was a Colombian poet. He is considered one of the founders of Spanish–American Modernism.

List of general music articles in Rees's Cyclopaedia

The music articles in the Rees's Cyclopaedia were written by Charles Burney (1726–1814), with additional material by John Farey Sr (1766–1826), and John Farey Jr (1791–1851).The Cyclopædia was illustrated using 53 plates as well as a numerous examples of music typset within the articles.

The general musical articles list all those that are not biographical, which form a separate list. They were written mostly by Charles Burney. Others on the scientific basis of music were by John Farey Sr, and technical descriptions of some musical instruments were given by his son.

It had been Burney's intention to write a Dictionary of Music, but for various reasons he never did so. In 1801 when he was aged 75 he was offered the chance of writing music articles in Rees's Cyclopaedia, and this occupied him to around 1805 or '06. His fee was £1000. Burney's brief was to 'include definitions in all the languages of Europe where Music has been much cultivated, with its history, biography, Criticism and discussions'The articles contain an enormous amount of musical information, much of which being augmented versions of what he had already published in his earlier writings In addition, he took the opportunity to add new topics covering the years of the last quarter of the eighteenth century and much of the first decade of the nineteenth, in particular the London musical scene. In all he wrote 996 general articles. While the majority of the articles have some length, a good proportion are brief (3 or 4 lines or fewer), dictionary definitions, or cross references. Many of the former are terms derived from French and Italian. The encyclopaedic-length articles (as distinct from the dictionary-length articles) are usually longer than Burney's earlier published writings on the same topic.

Dr Percy Scholes cites a statement by Dr Rees from Preface to the Cyclopaedia that he had 'constantly interpolated his own additions to the articles of his specialists'. and quotes a passage from the article about Dance (Vol 11) describing 'in our own memory', Welsh church-goers being played out of church by a fiddle.John Farey, sr was by profession a geologist, but was greatly interested music. He was involved with the Choral Fund, the Cecilian Society and the Surrey Chapel Society, and also sang in oratorios in Drury Lane Theatre. He was particularly interested in the mathematics of music and temperament, and wrote all the 215 scientific articles on these topics for Rees. Farey's investigations of temperament involved discussing (and criticising) the various schemes in use then. He described them in a series of letters to the Philosophical Magazine, and the Monthly Magazine as well as the American Journal of Science. He also contributed 25 articles to the Edinburgh Encyclopædia on this topic .John Farey, jr was a prolific contributor to Rees's Cyclopaedia, not only as a writer but also as an artist of many of the plates. For the four articles he wrote, he drew plates for two illustrating John Isaac Hawkins's Finger-keyed Viol, and two different patterns of pipe organ. The plates were keyed to the texts of the articles, and for this reason it was necessary for Farey to have produced both, since writing the texts describing the plates would have required technical knowledge which Burney would not have possessed.

Alphabetisation of articles:

The work followed the common practice of conflating the letters I and J and U and V into single sequences. The topics included in this list therefore follow the sequence they appear in the original volumes.

Annotations:

The articles are annotated to Mercer's edition of Burney's History (1935) and Scholes's edition of Burney's travels, Travels (1959). Where a page reference is given the text can be found there. Where a book is cited, but with no page, index entries were found, and Burney is presumed to have written his article using the information there. Where there is no annotation, the article must be unique to the Cyclopaedia.

Scale (music)

In music theory, a scale is any set of musical notes ordered by fundamental frequency or pitch. A scale ordered by increasing pitch is an ascending scale, and a scale ordered by decreasing pitch is a descending scale. Some scales contain different pitches when ascending than when descending, for example, the melodic minor scale.

Often, especially in the context of the common practice period, most or all of the melody and harmony of a musical work is built using the notes of a single scale, which can be conveniently represented on a staff with a standard key signature.Due to the principle of octave equivalence, scales are generally considered to span a single octave, with higher or lower octaves simply repeating the pattern. A musical scale represents a division of the octave space into a certain number of scale steps, a scale step being the recognizable distance (or interval) between two successive notes of the scale. However, there is no need for scale steps to be equal within any scale and, particularly as demonstrated by microtonal music, there is no limit to how many notes can be injected within any given musical interval.

A measure of the width of each scale step provides a method to classify scales. For instance, in a chromatic scale each scale step represents a semitone interval, while a major scale is defined by the interval pattern T–T–S–T–T–T–S, where T stands for whole tone (an interval spanning two semitones), and S stands for semitone. Based on their interval patterns, scales are put into categories including diatonic, chromatic, major, minor, and others.

A specific scale is defined by its characteristic interval pattern and by a special note, known as its first degree (or tonic). The tonic of a scale is the note selected as the beginning of the octave, and therefore as the beginning of the adopted interval pattern. Typically, the name of the scale specifies both its tonic and its interval pattern. For example, C major indicates a major scale with a C tonic.

Thematic transformation

Thematic transformation (also known as thematic metamorphosis or thematic development) is a musical technique in which a leitmotif, or theme, is developed by changing the theme by using permutation (transposition or modulation, inversion, and retrograde), augmentation, diminution, and fragmentation. It was primarily developed by Franz Liszt and Hector Berlioz. The technique is essentially one of variation. A basic theme is reprised throughout a musical work, but it undergoes constant transformations and disguises and is made to appear in several contrasting roles. However, the transformations of this theme will always serve the purpose of "unity within variety" that was the architectural role of sonata form in the classical symphony. The difference here is that thematic transformation can accommodate the dramatically charged phrases, highly coloured melodies and atmospheric harmonies favored by the Romantic composers, whereas sonata form was geared more toward the more objective characteristics of absolute music. Also, while thematic transformation is similar to variation, the effect is usually different since the transformed theme has a life of its own and is no longer a sibling to the original theme.

Transposition (music)

In music transposition refers to the process, or operation, of moving a collection of notes (pitches or pitch classes) up or down in pitch by a constant interval.

The shifting of a melody, a harmonic progression or an entire musical piece to another key, while maintaining the same tone structure, i.e. the same succession of whole tones and semitones and remaining melodic intervals.

For example, one might transpose an entire piece of music into another key. Similarly, one might transpose a tone row or an unordered collection of pitches such as a chord so that it begins on another pitch.

The transposition of a set A by n semitones is designated by Tn(A), representing the addition (mod 12) of an integer n to each of the pitch class integers of the set A. Thus the set (A) consisting of 0–1–2 transposed by 5 semitones is 5–6–7 (T5(A)) since 0 + 5 = 5, 1 + 5 = 6, and 2 + 5 = 7.

Yamaha TX81Z

The Yamaha TX81Z is a rack version of Yamaha DX11 and rack-mounted (keyboard-less) frequency modulation music synthesizer, which was released in 1987. Unlike previous FM synthesizers of the era, the TX81Z was the first to offer a range of oscillator waveforms other than just sine waves, conferring the new timbres of some of its patches when compared to older, sine-only FM synths. The TX81Z has developed a famous reputation, largely based on some of its preset bass sounds. A keyboard version with more onboard editing abilities was released the following year as the Yamaha DX11.

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