music theory online : small intervalslesson 7
Dr. Brian Blood





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I occasionally play works by contemporary composers for two reasons. First, to discourage the composer from writing any more and secondly to remind myself how much I appreciate Beethoven.
Jascha Heifetz (1901-87) Russian-American violinist

The Tone and Semitone :: The Octave :: The Chromatic Scale :: Microtones


Important: To see and hear our 'live' music examples you will need to install the free Scorch plug-in for PC and MAC systems.


The Tone and Semitone :: top

Key words:
keyboard
tone
semitone
whole step
half step
1

The Tone and Semitone

Returning to the keyboard we introduced in the previous lesson, we now consider the difference in pitch between one key and its neighbour.

C
D
E
F
G
A
B
*
C
D
E
F
G
A
B
C
D
E
F
G
A
B

If we count the number of keys between the key C with the asterix * (we call this middle C) and the key marked with a B that lies just below the C above (i.e. to the right of) middle C we find that there are twelve keys (five black and seven white) that match the twelve different notes on the stave below.

The difference in pitch (the pitch 'interval' or just interval), between a key and its immediate neighbour is called a semitone, meaning 'half' a tone. Two semitones are equivalent to a 'whole' tone. Where there is no black key between them (for example, between B and C) neighbouring white keys are a semitone interval apart. If there is a black key between them (for example, between F and G) neighbouring white keys are a tone interval apart. In this case the black key (F sharp / G flat) is the white key's immediate neighbour and the interval between the white key F and the black key, F sharp/ G flat, is a semitone.

This is shown clearly in the diagram below.

The Octave :: top

Key word:
octave
natural
double flat
flat
sharp
double sharp
1

The Octave

Sharpening or flattening the pitch of a note changes the pitch by a semitone, in the former case sharpening, increasing or raising the pitch by a semitone and in the latter case flattening, reducing or lowering the pitch by a semitone. Raising the pitch of a note by twelve semitones raises the pitch by one 'octave'. If the original note was C, the new note one octave higher will also be called C. In a similar way, lowering the pitch by twelve semitones lowers the pitch by an 'octave'. You might wonder why the word 'octave' which seems to have something to do with the Greek word for 'eight' (e.g. octagon - a shape with eight sides) is used in this situation. If you count the number of white keys that lie across an octave - as, for example, C, D, E, F, G, A, B and C again - you will see that there are eight, hence the word 'octave'.

Notice that the interval between each successive pair of notes on the two staves above is a semitone.

The pitch of a note can be raised by two semitones, in which case a double sharp sign is employed, while the equivalent for lowering the pitch by two semitones is the double flat sign. The double sharp and double flat signs are illustrated in the table below.

notation English French German Italian Spanish Catalan
double sharp double dièse Doppelkreuz doppio diesis doble sostenido,
elevación de dos semitonos
doble diesi,
elevació de dos semitons
sharp dièse Kreuz diesis sostenido,
elevación de un semitono
diesi,
sostingut
natural,
natural sign
bécarre,
naturel
Auflösungszeichen, Quadrat bequadro becuadro becaire
flat bémol B, Be bemolle bemol,
bajada de un semitono
bemoll,
disminució d’un semitò
double flat double bémol Doppel-B, Doppel-Be doppio bemolle doble bemol,
bajada de dos semitonos
doble bemoll,
disminució de dos semitons

The Chromatic Scale :: top

Key word:
chromatic scale
1

The Chromatic Scale

The succession of all twelve semitones in ascending or descending order is the chromatic scale.

The standard 'convention', which we have mentioned before, is that sharp signs are using for rising chromatic scales and flat signs are used for falling chromatic scales.

We will return to this matter again in lesson 11.

Microtones :: top

Key words:
microtone
microtonal
quarter tone
eighth tone
comma
cent
1

Microtones

The Boston Microtonal Society Website discusses the problem of what a microtone is.

Interpretations of the term microtonal vary widely, and for our purposes it is useful to note the following two basic types of usage:

1) The most literal and narrow definition of the word microtone has as its reference point the Western tone (or whole tone). If a semitone is half of a tone (in terms of cents*), then anything smaller is classified as a microtone, or microinterval, according to this definition. (More specific names are "quarter-tone," "fifth-tone," "eighth-tone," "sixth-tone," etc.)

* Alexander J. Ellis' system for measurement of musical intervals, in which the equal-tempered semitone equals 100 cents, the whole tone 200 cents, the octave 1200 cents, and so on.

2) The most general, inclusive—and most common—usage of the term microtonal is its application to any music made using intervals other than the traditional intervals of 12-note equal temperament (with its multiples of 100-cent semitones and 200-cent whole tones), which has been the standard tuning for Western music since the mid-nineteenth century.

If we consider this second, more general application, we can easily see that there are a variety of artistic, theoretical and philosophical channels through which musicians may be drawn to those 'other intervals'. As a result there are a few different disciplines, only loosely inter-related, all of which may fall into the category 'microtonality'. These include:

    • the practice of simply adding pitches to 12-note equal temperament (most often through microtonal equal temperaments such as 24-note (quarter-tones), 36-note (sixth-tones), 48-note (eighth-tones), 72-note (twelfth-tones), 96-note (sixteenth-tones), etc.)
    • contemporary pure tuning methods such as the various modern forms of just intonation, Pythagorean and mean-tone tunings
    • historically accurate tunings of Medieval, Renaissance, Baroque and Classical pieces
    • the study of non-Western tuning systems

The quarter-tone is defined as that pitch which exists midway (48-50 cents or 2 Pythagorean commas) between any two semitones (half-tones), with neither semitone predominating. A semitone is equal to 100 cents. The quarter-tone may be considered a universal interval, like the tone and semitone, as it exists in numerous Eastern and Western musical cultures. Irish folk tunes, for example, sometimes feature the inclusion of 'half-sharp' notes, quarter tones mid-way between natural and sharp. [ref: Peter Cooper, Mel Bay's Complete Irish Fiddle Player, Mel Bay Publications, 1995]

Other divisions of the tone have their place in Eastern and Western musical cultures. The eighth-tone is measured at 24-25 cents (or, for example in Turkish music, a Pythagorean comma).

Quartertone Accidental Signs (there are other sign conventions)
3/4 tone flat
sesquiflat
trois-demi de bémol (French)
double bémol barré (French)
bemol y medio (Spanish)
1/4 tone flat
semiflat
semi-bemolle (Italian)
bémol barré (French)
demi bémol(French)
bémol inversé (French)
semibemol (Spanish)
1/4 tone sharp
semisharp
semi-diesis (Italian)
dièse barré (French)
demi dièse avec une seule barre verticale (French)
semisostenido (Spanish)
3/4 tone sharp
sesquisharp
trois-demi de dièse (French)
dièse avec trois barres verticales (French)
sostenido y medio (Spanish)

We include below a quarter-tone/semitone or half-step interval chart.

no. of quarter tones from unison no. of semitones or half steps from unisoninterval description abreviated name
00unisonu
1 quarter-sharp unison+u
quarter-flat minor second-m2
21minor secondm2
3 quarter-sharp minor second+m2
quarter-flat major second-M2
42major secondM2
5 quarter-sharp major second+M2
quarter-flat minor third-m3
63minor thirdm3
7 quarter-sharp minor third+m3
quarter-flat major third-M3
84major thirdM3
9 quarter-sharp major third+M3
quarter-flat perfect fourth-P4
105perfect fourthP4
11 quarter-sharp perfect fourth+P4
quarter-flat augmented fourth-A4
126augmented fourth = diminish fifthA4
13 quarter-sharp augmented fourth+A4
quarter-flat perfect fifth-P5
147perfect fifthP5
15 quarter-sharp perfect fifth+P5
quarter-flat minor sixth-m6
168minor sixthm6
17 quarter-sharp minor sixth+m6
quarter-flat major sixth-M6
189major sixthM6
19 quarter-sharp major sixth+M6
quarter-flat minor seventh-m7
2010minor seventhm7
21 quarter-sharp minor seventh+m7
quarter-flat major seventh-M7
2211major seventhmM
23 quarter-sharp major seventh+M7
quarter-flat octave-8ve
2412octave8ve

While experimenting with his violin in 1895, Julian Carrillo discovered sixteenths of a tone, i.e., sixteen clearly different sounds between the pitches of G and A emitted by the fourth violin string. Because there are six whole tones in conventional tuning to the next octave, a musical scale made with sixteenths of each tone has 96 different notes or pitches. In contrast to this, the scale made with half-tones has only 12 different pitches.

References: (many taken from the Boston Microtonal Society Links Page)

  • What is Microtonality?
  • Microtonal Music
  • Micromegas
  • Microtonal.org
  • Centre for Microtonal Music
  • John Starrett's Microtonal Music Page
  • Graham's Microtonal Website
  • Huygens-Fokker Foundation
  • NewMusicBox.org - Issue on Microtonality
  • Where Are the Microtones? by Julia Werntz
  • Legacy of Microtonal Music in the 20th Century by Anton Rovner
  • World Scale Depository
  • Graphic Representation of Some Gamelan Tunings by Bill Alves
  • Mande Jeli Balafon Tuning
  • Xian Shi Yue Tuningby Mercedes DuJunco
  • A Gentle Introduction to South Indian Classical (Karnatic) Music by Mahadevan Ramesh
  • Hear Various Bagpipe Chanter Tunings by Ewan Macpherson

    This whole field is considered in greater depth in Pitch, Temperament and Timbre.

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