Intra‐octave tuning, as the name suggests, regulates pitches inter‐
nally within the octave which it organises into a number of constitu‐
ent pitches and intervals. The main functions of intra‐octave tuning are: [1] to enable any particular pitch included in a perform‐
ance or recording session to be sounded in unison by all ensemble members designated to play that pitch; [2] to regulate intervals be‐
tween the octave’s constituent pitches so that they are sounded in a reasonably consistent fashion. This brief description of intra‐oc‐
tave tuning begs questions about the term INTERVAL. Intervals
In everyday speech an interval usually means the ‘horizontal’ dis‐
tance in time between one event from another. In music theory, however, an interval is the ‘vertical’ distance in pitch between one tone and another. If temporal intervals are quantified in units like milliseconds or millennia, intervals of pitch are quantified in terms of octaves, tones, semitones and cents (hundredths of a semitone, sometimes abbreviated ‘¢’). Intervals are produced and under‐
stood in two ways: [1] melodically, as the pitch gap between two notes sounded one immediately or very soon after the other; [2]
harmonically, as the pitch gap between two simultaneously sound‐
ing notes. As already implied, one such pitch distance, the OCTAVE, is central to the understanding of all other intervals in music.
Octave
Two tones at the same pitch —in unison— are in a pitch frequency ratio of 1:1. Two tones an octave apart are separated by a frequency factor of 2. For example, the first note in each of the pairs aÌ (220 hz) and aÒ (440 hz), or cÒ (261.63 hz) and cÙ (523.25), or e$Ì (155.56) and e$Ò (311.13), is each one octave below the second (Figure 9 ).
With its simple frequency ratio of 2:1, the octave is also the interval between a note’s fundamental pitch and that of its first harmonic, which, in its turn, is an intrinsic part of the timbre of every singing voice and of most acoustic tonal instruments. This interval is called
‘octave’ because it’s the eighth note you reach in the heptatonic (seven‐note) scale if you ascend or descend one step at a time, for example a b c d e f g [a] (Â Ê Î Ô Û â ê [î], rising) or a g f e d c b [a]
(î ê â Û Ô Î Ê [Â], descending).
All known music traditions tend to treat two pitches an octave apart as the same note in another register. Men are understood to be singing the same tune as women and children if both parties fol‐
low the same pitch contour at the same time in parallel octaves.
The octave’s property of unison in another register is also illus‐
trated by the fact that: [1] a common chord consisting of the tonic, third, fifth and octave (i.e. as, say, cÒ eÒ gÒ cÙ) is treated as a triad, not a tetrad, because it contains only three, not four, differ‐
ently named notes (e.g. just cÒ eÒ gÒ as tonic, third, fifth, i.e.
and no î); [2] any single note sounded on instruments like the twelve‐string guitar, or using common types of organ registration, device; [5] the octave is associated with the concept of REGISTER. Music’s range of audible fundamental pitches is often divided into octaves so that REGISTER can be referred to without having to men‐
tion cycles per second (Hz). A standard piano keyboard spans just over eight octaves from a0 (27.5 hz) to c8 (4186 hz; see Figure 9).
The average human singing voice usually spans about two octaves.
According to this system of labelling octaves, the first note of the Rolling Stones’ Satisfaction riff (1965a) is bp, concert pitch is ar and the first sung note of Abba’s Dancing Queen (1975c) is c#Ù.
Fig. 9. The piano keyboard’s 88 notes: a0 (27.5 Hz) to c8 (4186 Hz)
Figure 9 shows a piano keyboard divided into seven octaves plus three extra notes at the bottom and one at the top . Octave numbers appear to the left of the keyboard and the identity of the 88 individ‐
ual notes, each with its fundamental frequency in cycles per sec‐
ond (Hz), to its right. Figure 10 (p. 74) also shows the familiar pattern of seven white and five black notes (twelve in all) that re‐
curs in each octave. The eleven intervals inside the Western equal‐
tempered octave are set out in Table 4 (p. 74).
Intervals and intra‐octave tuning
Table 4. Western intra‐octave intervals (ascending from cn to cn+1)
Table 4 presents all twelve tones in the West‐
ern chromatic scale. Column 1 gives the note names of those twelve pitches in an ascending scale with c@ as its tonic (see also Fig. 10 ).
Column 2 in Table 4 presents the number of semitones separating each note from the lower tonic (c), and column 3 the heptatonic scale‐
degree shorthand for each of the twelve notes
1. Note name(doh = c) 2. Semitonesabove doh 3. Scale degreeshorthand 4. Frequencyratio to tonic 5. × > frequency of tonic (just temperament) 6. × > frequency of tonic (equal temperament)
7. Interval name
POPULAR)
c 0 1 1:1 1 1 prime (unison) tonic: ONE
c# 1 # 25:24 1.042 1.060 [raised prime] ‐
d$ 1 25:24 1.042 1.060 minor 2nd
or semitone
flat supertonic FLAT TWO
d 2 9:8 1.125 1.123 major 2nd or
whole tone
supertonic:
TWO
d# 3 # 6:5 1.2 1.189 augmented 2nd SHARP TWO
e$ 3 6:5 1.2 1.189 minor 3rd FLAT THREE
e 4 5:4 1.25 1.260 major 3rd mediant: THREE or
MAJOR THREE f 5 4:3 1.333 1.335 perfect 4th subdominant: FOUR f# 6 # 45:32 1.406 1.414 augmented 4th
or tritone or
[raised subdominant]
SHARP FOUR g$ 6 45:32 1.406 1.414 diminished 5th FLAT FIVE
g 7 3:2 1.5 1.498 perfect 5th dominant: FIVE
g# 8 8:5 1.6 1.587 augmented 5th SHARP FIVE
a$ 8 $6 8:5 1.6 1.587 minor 6th flat submediant:
FLAT SIX a 9 6 5:3 1.667 1.682 major 6th submediant: SIX or
MAJOR SIX
[a#] 10 #6 9:5 1.8 1.782 augmented 6th ‐
b$ 10 $7 9:5 1.8 1.782 minor 7th subtonic: FLAT SEVEN b 11 7 15:8 1.875 1.888 major 7th leading note:
SHARP SEVEN
c 12 8 2:1 2 2 (perfect) octave tonic: EIGHT
Fig. 10. One octave
( = ‘flat two’, = ‘sharp four’, etc.). Column 4 shows the pitch frequency ratio in just temperament (p. 78,ff.) between each note and the lower tonic, while columns 5 and 6 show the same pitch differences as multiples of the tonic’s fundamental frequency, us‐
ing just and equal temperament respectively.5 Column 7 presents the most widely used interval names in Western music theory. Fi‐
nally, column 8 lists two types of scale degree designation: [1] in italics, those used in theories of euroclassical harmony; and [2], in small capitals, the popular practice used by anglophone musicians when pronouncing the scale‐degree symbols in column 2.6 The dif‐
ference between the labels in columns 7 and those in italics in col‐
umn 8 can be explained as follows.
Although the interval names in column 7 of Table 4 are all given in relation to the lower tonic (c@), they can in fact be applied in rela‐
tion to any note. For example, f@ is located, as shown in Table 4, a perfect fourth (5 semitones or guitar frets) above c, but it is also a perfect fourth below b$ and a perfect fifth (7 semitones) below c, as well as a semitone or minor second (or a single guitar fret) above e;
f is also a major third (4 semitones) above d$, a major sixth (9 semi‐
tones) below d, and a major second or whole tone below g, as well as a minor seventh (10 semitones) above g.
The terms in italics in column 8 of Table 4, on the other hand, are used almost exclusively about music in the euroclassical tradition and can only be applied in relation to the relevant keynote or tonic of music in that tradition.7 For example, although six different rising perfect fifths exist within the tonal vocabulary of a C major scale (f<c, c<g, g<d, d<a, a<e, e<b),8 only g, the note situated a perfect fifth above (or a perfect fourth below) the tonic ( ), and tertial
5. i.e. how much higher than the tonic (e.g. c), in terms of how many times faster each pitch frequency is in relation to that lower tonic.
6. This popular practice varies considerably. , for example, can be called ‘flat five’, ‘flat fifth’, ‘flatted fifth’; (e.g. in C) can be ‘six’ or ‘major six’, etc.
7. See also ‘Classical harmony’, pp.255‐274, esp. p. 259, ff. and p. 266, ff.
8. Of course, those ascending perfect fifths can be inverted at the octave into descending perfect fourths (f>c, c>g, g>d, d>a, a>e, e>b in C).
chords based on that same scale degree (G, G7, etc. in the key of C), can be called dominant. By the same token, the note f and tertial chords based on f (F, F7, Fm, etc.) can be called dominant only in the key of B$, mediant only in the key of D$, submediant only in A$, su‐
pertonic only in E$, leading note only in G$, and subdominant only in C. Although useful in the analysis of musics following the tonal habits of euroclassical music and most types of jazz, terms like dominant and subdominant are of little or no relevance to music based on other tonal principles.9 For example, the common three‐
chord mixolydian loop heard throughout Sweet Home Alabama ({D-C-G} in D) and repeated at the end of Hey Jude ({G-F-C} in G) is referred to as I-$VII-IV (’one, flat seven, four’), not ‘tonic, subtonic, subdominant’.10 And that’s not because the first designa‐
tion of the same sequence is more concise: it’s because the chord on IV (the G in D, the C in G) just doesn’t work like a euroclassical sub‐
dominant and because the sequence includes no dominant (V) to which a chord on the fourth degree (IV) can reasonably be ‘sub’.11 Another ethnocentric problem with column 8 in Table 5 (p. 78) con‐
cerns the scale’s seventh degree: the ‘leading note’. It’s a problem best explained by example.
Ex. 1. Subtonic or leading note? (a) Handel: hymn tune Antioch (‘Joy To The World’); (b) The Foggy Dew (Irish trad.).
Example 1 includes seven sevenths of which only one is strictly speaking a leading note. Example 1a contains two sevenths, both major or ‘sharp sevens’ ( ), the first one descending from the tonic, the other [nº 2] rising back up to the tonic. The five sevenths
9. See Chapter 9, p. 277, ff.
10. Lynyrd Skynyrd (1974), Beatles (1968b). For more examples of that mixoly‐
dian chord loop, see Table.35, p. 435.
11. For non‐ionian harmony, see p.277,ff. For roman‐numeral chords, see p. 224.
in example 1b are all minor or ‘flat sevens’ ( ), two of them [4, 5]
descending from the tonic, two [3, 6] ascending to the tonic and one [7] going in both directions. So which of the seven sevenths is definitely a leading note? Well, the seventh degree in the euroclas‐
sical major, ascending minor and harmonic minor scales (see p. 95,ff.) is called leading note because in those modes it’s the major seventh ( , ‘sharp seven’) which is supposed to lead to the tonic ( ) a semitone above, (e.g. b@?c in C, f#?g in G). That means the only unequivocal leading note in example 1 is number 2.
LEADING NOTE can also designate any tone that leads by a single semi‐
tone step, ascending or descending, to a subsequent note heard as conso‐
nant, as with an f@, either in a G7 chord descending one semitone to e@ in a C major tonic triad ( > , see p. 256, ff.), or like the second scale degree in E phrygian descending to its tonic ( > , see pp. 126 and 443 ).12 Now, in conventional music theory leading note tends to mean the note situated one semitone below the tonic and which is assumed to lead up to that keynote ( < = ), even if it can also de‐
scend from it. One obvious problem with this terminology is that, as example 1b suggests, widely disseminated types of popular mu‐
sic often use the minor seventh ( , the subtonic, ‘flat seven’), which is located not a semitone but a whole tone below the tonic and just as likely to descend to the sixth or fifth as ascend to the tonic, or ar‐
rive from or depart to other scale degrees. And, as the first seventh in example 1a shows, not even a major seventh necesarily leads to the tonic. In short, the term leading note is misleading if it designates the sort of minor sevenths shown in example 1b because none of them have to lead to any other place in particular. It is for these rea‐
sons advisable, when referring in relative terms to the seventh scale degree, to use the term subtonic for flat sevens and to restrict the meaning of leading note to a scale degree that literally leads by a semitone (or less) up to its tonic.
12. See also The Other Leading Note (Moore, 2013).
Equal‐tone tuning
The most widely accepted intra‐octave tuning system for music in the urban West is equal temperament or equal‐tone tuning. It divides the octave into twelve equal intervals (semitones) and has been in use since the late eighteenth century. It was developed to solve problems caused by discrepancies between certain intervals as constituent parts of the octave and the same intervals in their
‘pure’ form.13
Table 5. Intra‐octave intervals in just and equal tuning, with scale degrees 1‐8 and note names in C
As shown in Figure 11 ( ), the top note of three stacked pure major thirds, each at the frequency ratio 5:4 above the previous one, is out of tune at the octave with the bottom note. That means the at the top of the pile of the three major thirds , , is, in just intonation, one fifth of a tone (40¢) lower than the octave above the ini‐
tial a$. Similarly, the top in the four stacked
natural minor thirds14 g#-b-d-f-a$ is more than a quarter‐tone (>50¢) lower than the octave above the initial g#. These natural acoustic discrepancies posed particular problems for keyboard players needing to produce, say, both (as in an E major triad) and (as in an F minor triad) in the same piece: one or the other
13. ‘Pure’ means in this context the acoustically unadjusted simple frequency ratios of intervals used in just intonation (see Table 5).
? Interval
Tuning type ñ
Prime/Tonic Minor 2nd Major 2nd Minor 3rd Major 3rd Perfect 4th Augm. 4th/ Dimin. 5th Perfect 5th Minor 6th Major 6th Minor 7th Major 7th Octave/Tonic
Just 1:1 1 Equal 1 1.060 1.123 1.189 1.260 1.335 1.414 1.498 1.587 1.682 1.782 1.888 2 Degree
in C
14. ‘Pure’ minor thirds are intervals separated by a frequency ratio of 6:5 (= ×1.2).
Fig. 11. g#≠a$
would be seriously out of tune.15 Equal temperament tackled the problem by slightly detuning eleven of the octave’s constituent semitones so that the interval between each of them became iden‐
tical. As Table 5 shows, the equal‐temperament perfect fourths (e.g. ) and fifths ( ) have almost the same values as their just‐
tone equivalents. Thirds, sixths and sevenths, on the other hand,
sic requiring no enharmonic alignment (between d# and e$, g# and a$ etc.) for purposes of modulation or harmonic colour. Moreover, equal temperament is either unnecessary or inappropriate in, for example, most types of blues, bluegrass, blues‐based rock, folk rock, not to mention the traditional musics of Africa, the Arab world, the Balkans, the British Isles, the Indian subcontinent, Scan‐
dinavia etc., i.e. in any music whose tonality is non‐euroclassical and/or drone‐based.16 One reason for the relative incompatibility of such music with equal‐tone tuning may be the use of drone notes to produce an overall sound that is rich in natural overtones and thereby inconsistent with equal‐temperament intervals. An‐
other reason might be the centrality of each interval’s expressive character in relation to a permanent tonic, as in the rāga traditions of India whose aesthetics also often require microtonal pitch dis‐
tinctions. Artificially adjusting intervals by as much as a quarter‐
tone, as in equal‐tone tuning, is incompatible with the principles of such music.
Another important consideration is, as shown in Table 6 (p. 80), the pitch location of scale degrees incompatible with the Western assumption that semitones are the smallest possible intervals.
15. If you’re in C major and need to make first a perfect cadence in the relative minor (E7-Am) and later an altered plagal cadence in C (Fm-C), you won’t want your g# and a$ to be out of tune by a quarter tone. The {G-B-C-Cm} loop in Creep (Radiohead, 1992) would also suffer if played in just tuning (d# and e$).
16. See p. 211, ff. and ‘Open tuning and drones’ (p. 344, ff.).
Table 6. Intra‐octave interval pitches for five heptatonic modes
Columns 1 and 9 in Table 6 show, in ascending order, the scale de‐
grees (including accidentals, where appropriate) of a heptatonic octave.17 Column 2 lists the twelve semitones in an octave ascend‐
ing in equal‐tone tuning from an to an+1, specifying a pitch differ‐
ence of 100 cents between each semitone step. Column 8 provides
17. See pp. 37‐39, ff. for explanation of scale‐degree shorthand (§ = ¼‐tone flat).
an incremental listing in cents of each semitone step from the ini‐
tial an (‘0’=no interval) to an+1, located 1200¢, twelve semitones or one octave higher. Please note that columns 1 and 2 are in complete horizontal alignment with columns 8 and 9.
Columns 3‐7 show, in cents, the pitch difference between each of the seven scale degrees in five different modes. The pitch location of scale degrees in the ionian and aeolian modes (columns 3 and 5) align entirely with the Western equal‐tone semitone pitches given in columns 2 (100¢) and 8 (multiples of 100¢), as do those of Rast (column 4), except for the latter’s two 150¢ (¾‐tone) steps § and § . In a similar way, Bayati (col. 6) resembles the aeolian mode (col. 5), except for the four ¾‐tone steps (150¢) ‐§ , §
§â and § .18 The Javanese Pelog scale (col. 7) diverges even more radically from Western equal‐tone tuning: neither its nor align with those of the other modes in the table.19 The point is that in many types of tonality scale degree pitches do not fit into the sim‐
ple twelve‐semitone grid of Western intra‐octave tuning systems.
Moreover, as highlighted by the thicker horizontal lines above the start and end of each scale degree in Table 6 and by the varying number of cents given for the interval between scale degrees, pitch placement of an octave’s constituent tones can vary radically from one mode to another.
Within the general framework of just intonation discussed earlier, a wide variety of intra‐octave tunings are used in different music traditions. Despite a few exceptions, such as the Pelog and Slendro systems of Java, many intra‐octave tunings include, as suggested by the thick horizontal line above and in Table 6, the natural fourth (4:3), and most include the natural fifth (3:2).20 At the same time, Arab and Indian music theories divide the octave into 16 and 22 unequal steps respectively, reflecting intra‐octave tuning con‐
ventions that differ markedly from those of the urban West.21
18. Bayati §â and §ê are sometimes given as $â and $ê (cf. Fig. 19, p. 119).
19. For ionian and aeolian, see pp. 91‐96, 99, 103‐116; for Rast and Bayati, see pp.
119‐121; for Pelog and Slendro, see Malm (1977:45‐47).
20. See also Table 4, p. 74.
The Western adjustment of natural intervals into the twelve equal intervals shown in Tables 4, 5 and 6 (pp. 74, 78, 80) has only been in operation for a couple of centuries in urban Europe and America, but it has during that short period managed to replace many ear‐
lier vernacular tuning patterns in the Western world, patterns that can be heard today in archival recordings from what were rela‐
tively isolated areas like the Outer Hebrides or the Appalachian backwoods.22 It’s impossible to predict if the global spread of An‐
glo‐North‐American music during the latter half of the twentieth century, together with the equal‐tone tuning of piano, organ, ac‐
cordion and synthesiser keyboards —plus the inclusion of general MIDI in personal computers, plus the overwhelming use of equal‐
tone tuning in globally disseminated film and games music—, will eventually bring about the demise of other tuning systems. Even if that were to happen, tonal diversity does not, thankfully, depend solely on a variety of intra‐octave tuning systems to survive and flourish. The vast variety of modes used on a daily basis in differ‐
ent parts of the world is one healthy symptom of tonal diversity;23 another is tuning in the second sense of the word presented at the start of this chapter.
21. Neutral is often used in the West to qualify pitches between ‘major’ and
‘minor’ thirds, sixths and sevenths. It is a eurocentric term implying that those pitches are heard according to that same intervallic grid at all times in all cultures. The historical phenomenon of musica ficta suggests that not even Europeans have always perceived thirds, sixths and sevenths in the same way.
Another ethnocentric notion is that other peoples sing or play ‘in the cracks between the notes’ (of a modern Western piano keyboard, of course). For much more on modes and scales, see Chapters 3 and 4. For maqamat Rast and Bayati, see pp. 119‐121.
22. See, for example, ‘Waulking Song’ on Musique Celtique des Îles Hébrides (1970) and ‘The Lost Soul’ on The Doc Watson Family (Watson 1963/1990)
23. See, for example, the nineteen modes with which Greek popular musicians should ideally be familiar (Λαϊκοι Δρόμοι, p. 119).