![5-limit Tonnetz](https://upload.wikimedia.org/wikipedia/commons/d/d4/5-limit_Tonnetz.svg)
Five-limit tuning, 5-limit tuning, or 5-prime-limit tuning (not to be confused with
5-odd-limit tuning), is any system for
tuning
Tuning can refer to:
Common uses
* Tuning, the process of tuning a tuned amplifier or other electronic component
* Musical tuning, musical systems of tuning, and the act of tuning an instrument or voice
** Guitar tunings
** Piano tuning, adjusti ...
a
musical instrument
A musical instrument is a device created or adapted to make musical sounds. In principle, any object that produces sound can be considered a musical instrument—it is through purpose that the object becomes a musical instrument. A person who pl ...
that obtains the frequency of each note by multiplying the frequency of a given reference note (the base note) by products of
integer powers of 2, 3, or 5 (
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
s limited to 5 or lower), such as .
Powers of 2 represent intervallic movements by octaves. Powers of 3 represent movements by intervals of perfect fifths (plus one octave, which can be removed by multiplying by 1/2, i.e., 2
−1). Powers of 5 represent intervals of major thirds (plus two octaves, removable by multiplying by 1/4, i.e., 2
−2). Thus, 5-limit tunings are constructed entirely from stacking of three basic purely-tuned intervals (octaves, thirds and fifths). Since the perception of consonance seems related to low numbers in the harmonic series, and 5-limit tuning relies on the three lowest primes, 5-limit tuning should be capable of producing very consonant harmonies. Hence, 5-limit tuning is considered a method for obtaining
just intonation
In music, just intonation or pure intonation is the tuning of musical intervals
Interval may refer to:
Mathematics and physics
* Interval (mathematics), a range of numbers
** Partially ordered set#Intervals, its generalization from numbers to ...
.
The number of potential intervals, pitch classes, pitches, key centers, chords, and modulations available to 5-limit tunings is unlimited, because no (nonzero integer) power of any prime equals any power of any other prime, so the available intervals can be imagined to extend indefinitely in a 3-dimensional
lattice
Lattice may refer to:
Arts and design
* Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material
* Lattice (music), an organized grid model of pitch ratios
* Lattice (pastry), an orna ...
(one dimension, or one direction, for each prime). If octaves are ignored, it can be seen as a 2-dimensional lattice of
pitch class
In music, a pitch class (p.c. or pc) is a set of all pitches that are a whole number of octaves apart; for example, the pitch class C consists of the Cs in all octaves. "The pitch class C stands for all possible Cs, in whatever octave positio ...
es (note names) extending indefinitely in two directions.
However, most tuning systems designed for acoustic instruments restrict the total number of pitches for practical reasons. It is also typical (but not always done) to have the same number of pitches in each octave, representing octave transpositions of a fixed set of pitch classes. In that case, the tuning system can also be thought of as an octave-repeating scale of a certain number of pitches per octave.
The frequency of any pitch in a particular 5-limit tuning system can be obtained by multiplying the frequency of a fixed reference pitch chosen for the tuning system (such as
A440 A440 or A-440 may refer to:
* A440 (pitch standard)
* A440 highway (Australia), a road in Victoria, Australia
* Quebec Autoroute 440 (Laval)
* Quebec Autoroute 440 (Quebec City)
See also
* Apollo 440
Apollo 440 (also known as Apollo Four Fo ...
, A442, A432, C256, etc.) by some combination of the powers of 3 and 5 to determine the pitch class and some power of 2 to determine the octave.
For example, if we have a 5-limit tuning system where the base note is C256 (meaning it has 256 cycles per second and we decide to call it C), then ''f
C'' = 256 Hz, or "frequency of C equals 256 Hz." There are several ways to define E above this C. Using thirds, one may go up one factor 5 and down two factors 2, reaching a
frequency ratio
In music, an interval ratio is a ratio of the frequencies of the pitches in a musical interval. For example, a just perfect fifth (for example C to G) is 3:2 (), 1.5, and may be approximated by an equal tempered perfect fifth () which is 27 ...
of 5/4, or using fifths one may go up four factors of 3 and down six factors of 2, reaching 81/64. The frequencies become:
:
or
:
Diatonic scale
Assuming we restrict ourselves to seven pitch classes (seven notes per octave), it is possible to tune the familiar
diatonic scale
In music theory, a diatonic scale is any heptatonic scale that includes five whole steps (whole tones) and two half steps (semitones) in each octave, in which the two half steps are separated from each other by either two or three whole steps, ...
using 5-limit tuning in a number of ways, all of which make most of the triads ideally tuned and as consonant and stable as possible, but leave some triads in less-stable intervalic configurations.
The prominent notes of a given scale are tuned so that their frequencies form ratios of relatively small integers. For example, in the key of
G major
G major (or the key of G) is a major scale based on G, with the pitches G, A, B, C, D, E, and F. Its key signature has one sharp. Its relative minor is E minor and its parallel minor is G minor.
The G major scale is:
Notable compositi ...
, the ratio of the frequencies of the notes G to D (a
perfect fifth
In music theory, a perfect fifth is the Interval (music), musical interval corresponding to a pair of pitch (music), pitches with a frequency ratio of 3:2, or very nearly so.
In classical music from Western culture, a fifth is the interval fro ...
) is 3/2, while that of G to C is 2/3 (a descending perfect fifth) or 4/3 (a
perfect fourth
A fourth is a musical interval encompassing four staff positions in the music notation of Western culture, and a perfect fourth () is the fourth spanning five semitones (half steps, or half tones). For example, the ascending interval from C to ...
) going up, and the major third G to B is 5/4.
![Primary triads in C](https://upload.wikimedia.org/wikipedia/commons/e/ef/Primary_triads_in_C.png)
A just diatonic scale may be derived as follows. Imagining the key of C major, suppose we insist that the subdominant root F and dominant root G be a fifth (3:2) away from the tonic root C on either side, and that the chords FAC, CEG, and GBD be just
major triads (with frequency ratios 4:5:6):
This is known as
Ptolemy's intense diatonic scale
Ptolemy's intense diatonic scale, also known as the Ptolemaic sequence,
justly tuned major scale,
Ptolemy's tense diatonic scale, or the syntonous (or syntonic) diatonic scale, is a tuning for the diatonic scale proposed by Ptolemy, and correspo ...
. Here the row headed "Natural" expresses all these ratios using a common list of
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''Cardinal n ...
s (by multiplying the row above by the
lcm of its denominators). In other words, the lowest occurrence of this one-octave scale shape within the harmonic series is as a subset of 7 of the 24 harmonics found in the octave from harmonics 24 to 48.
The three major thirds are correct (5:4), and three of the minor thirds are as expected (6:5), but D to F is a
semiditone
In music theory, a minor third is a musical interval that encompasses three half steps, or semitones. Staff notation represents the minor third as encompassing three staff positions (see: interval number). The minor third is one of two comm ...
or Pythagorean minor third (equal to three descending just perfect fifths, octave adjusted), a
syntonic comma
In music theory, the syntonic comma, also known as the chromatic diesis, the Didymean comma, the Ptolemaic comma, or the diatonic comma is a small comma type interval between two musical notes, equal to the frequency ratio 81:80 (= 1.0125) ( ...
narrower than a justly tuned (6:5) minor third.
As a consequence, we obtain a scale in which EGB and ACE are just
minor triads (10:12:15), but the DFA triad doesn't have the minor shape or sound we might expect, being (27:32:40). Furthermore, the BDF triad is not the (25:30:36)
diminished triad
In music theory, a diminished triad (also known as the minor flatted fifth) is a triad consisting of two minor thirds above the root. It is a minor triad with a lowered ( flattened) fifth. When using chord symbols, it may be indicated by the s ...
that we would get by stacking two 6:5 minor thirds, being (45:54:64) instead:
It can be seen that basic step-wise scale intervals appear:
* s = 16:15 (
semitone
A semitone, also called a half step or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically.
It is defined as the interval between two adjacent no ...
)
* t = 10:9 (
minor tone
In Western music theory, a major second (sometimes also called whole tone or a whole step) is a second spanning two semitones (). A second is a musical interval encompassing two adjacent staff positions (see Interval number for more deta ...
)
* T = 9:8 (
major tone
In Western music theory, a major second (sometimes also called whole tone or a whole step) is a second spanning two semitones (). A second is a musical interval encompassing two adjacent staff positions (see Interval number for more deta ...
)
Which may be combined to form larger intervals (among others):
* Ts = 6:5 (minor third)
* Tt = 5:4 (major third)
* Tts = 4:3 (perfect fourth)
* TTts = 3:2 (perfect fifth)
* TTTttss 2:1 (octave)
Another way to do it is as follows. Thinking in the relative minor key of A minor and using D, A, and E as our spine of fifths, we can insist that the chords DFA, ACE, and EGB be just
minor triads (10:12:15):
If we contrast that against the earlier scale, we see that for five pairs of successive notes the ratios of the steps remain the same, but one note, D, the steps C-D and D-E have switched their ratios.
The three major thirds are still 5:4, and three of the minor thirds are still 6:5 with the fourth being 32:27, except that now it's BD instead of DF that is 32:27. FAC and CEG still form just
major triads (4:5:6), but GBD is now (108:135:160), and BDF is now (135:160:192).
There are other possibilities such as raising A instead of lowering D, but each adjustment breaks something else.
It is evidently not possible to get all seven diatonic triads in the configuration (4:5:6) for major, (10:12:15) for minor, and (25:30:36) for diminished at the same time if we limit ourselves to seven pitches.
That demonstrates the need for increasing the numbers of pitches to execute the desired harmonies in tune.
Twelve-tone scale
To build a twelve-tone scale in 5-limit tuning, we start by constructing a table containing fifteen justly intonated pitches:
The factors listed in the ''first row'' and ''first column'' are powers of 3 and 5 respectively (e.g., = 3
−2). Colors indicate couples of
enharmonic
In modern musical notation and tuning, an enharmonic equivalent is a note, interval, or key signature that is equivalent to some other note, interval, or key signature but "spelled", or named differently. The enharmonic spelling of a written n ...
notes with almost identical pitch. The ratios are all expressed relative to C in the centre of this diagram (the base note for this scale). They are computed in two steps:
# For each cell of the table, a ''base ratio'' is obtained by multiplying the corresponding factors. For instance, the base ratio for the lower-left cell is 1/9 · 1/5 = 1/45.
# The base ratio is then multiplied by a negative or positive power of 2, as large as needed to bring it within the range of the octave starting from C (from 1/1 to 2/1). For instance, the base ratio for the lower left cell (1/45) is multiplied by 2
6, and the resulting ratio is 64/45, which is a number between 1/1 and 2/1.
Note that the powers of 2 used in the second step may be interpreted as ascending or descending
octave
In music, an octave ( la, octavus: eighth) or perfect octave (sometimes called the diapason) is the interval between one musical pitch and another with double its frequency. The octave relationship is a natural phenomenon that has been refer ...
s. For instance, multiplying the frequency of a note by 2
6 means increasing it by 6 octaves. Moreover, each row of the table may be considered a sequence of
fifths (ascending to the right), and each column a sequence of
major third
In classical music, a third is a musical interval encompassing three staff positions (see Interval number for more details), and the major third () is a third spanning four semitones. Forte, Allen (1979). ''Tonal Harmony in Concept and P ...
s (ascending upward). For instance, in the first row of the table, there is an ascending fifth from D and A, and another one (followed by a descending octave) from A to E. This suggests an alternative but equivalent method for computing the same ratios. For instance, you can obtain A (5/3 ratio), starting from C, by moving one cell to the left and one upward in the table, which means descending by one fifth (2/3) and ascending by one major third (5/4):
:
Since this is below C, you need to move up by an octave to end up within the desired range of ratios (from 1/1 to 2/1):
:
A 12-tone scale is obtained by removing one note for each couple of enharmonic notes. This can be done in at least three ways, which have in common the removal of G, according to a convention valid even for C-based Pythagorean and 1/4-comma meantone scales. Note that it is a
diminished fifth
Diminished may refer to:
*Diminution
In Western music and music theory, diminution (from Medieval Latin ''diminutio'', alteration of Latin ''deminutio'', decrease) has four distinct meanings. Diminution may be a form of embellishment in which ...
, close to half an octave, above the tonic C, which is a disharmonic interval; also its ratio has the largest values in its numerator and denominator of all tones in the scale, which make it least harmonious: all reasons to avoid it.
The first strategy, which we operationally denote here as ''symmetric scale 1'', consists of selecting for removal the tones in the upper left and lower right corners of the table. The second one, denoted as ''symmetric scale 2'', consists of discarding the notes in the first and last cell of the second row (labeled "1"). The third one, denoted as ''asymmetric scale'', consists of discarding the first column (labeled "1/9"). The resulting 12-tone scales are shown below:
In the first and second scale, B and D are exactly the inversion of each other. This is not true for the third one. This is the reason why these two scales are regarded as symmetric (although the removal of G makes all 12 tone scales, including those produced with any other tuning system, slightly asymmetric).
The asymmetric system has the advantage of having the "justest" ratios (those containing smaller numbers), nine pure fifths (factor 3/2), eight pure major thirds (factor 5/4) by design, but also six pure
minor third
In music theory, a minor third is a musical interval that encompasses three half steps, or semitones. Staff notation represents the minor third as encompassing three staff positions (see: interval number). The minor third is one of two com ...
s (factor 6/5). However, it also contains two impure fifths (e.g., D to A is 40/27 rather than 3/2) and three impure minor thirds (e.g., D to F is 32/27 rather than 6/5), which practically limits
modulation
In electronics and telecommunications, modulation is the process of varying one or more properties of a periodic waveform, called the ''carrier signal'', with a separate signal called the ''modulation signal'' that typically contains informatio ...
to a narrow range of keys. The chords of the tonic C, dominant G and subdominant F are pure, as well as D, A, E and the minor chords Fm, Cm, Gm, Am, Bm and Em, but not the Dm.
A drawback of the asymmetric system is that it produces 14 wolf intervals, rather than 12 as for the symmetric ones.
The B in the first symmetric scale differs from the B in the other scales by the
syntonic comma
In music theory, the syntonic comma, also known as the chromatic diesis, the Didymean comma, the Ptolemaic comma, or the diatonic comma is a small comma type interval between two musical notes, equal to the frequency ratio 81:80 (= 1.0125) ( ...
, being over 21 cents. In equally tempered scales, the difference is eliminated by making all steps the same frequency ratio.
![Five-limit tuning ratios Cuisenaire diagram](https://upload.wikimedia.org/wikipedia/commons/d/d7/Five-limit_tuning_ratios_Cuisenaire_diagram.svg)
The construction of the asymmetric scale is graphically shown in the picture. Each block has the height in cents of the constructive frequency ratios 2/1, 3/2 and 5/4. Recurring patterns can be recognised. For example, many times the next note is created by replacing a 5/4-block and a 3/2-block by a 2/1-block, representing a ratio of 16/15.
For a similar image, built using frequency factors 2, 3, and 5, rather than 2/1, 3/2, and 5/4, see
here
Here is an adverb that means "in, on, or at this place". It may also refer to:
Software
* Here Technologies, a mapping company
* Here WeGo (formerly Here Maps), a mobile app and map website by Here
Television
* Here TV (formerly "here!"), a TV ...
.
The just ratios
The just ratios used to build these scales can be used as a reference to evaluate the
consonance
In music, consonance and dissonance are categorizations of simultaneous or successive sounds. Within the Western tradition, some listeners associate consonance with sweetness, pleasantness, and acceptability, and dissonance with harshness, unple ...
of intervals in other scales (for instance, see
this comparison table). However, 5-limit tuning is not the only method to obtain
just intonation
In music, just intonation or pure intonation is the tuning of musical intervals
Interval may refer to:
Mathematics and physics
* Interval (mathematics), a range of numbers
** Partially ordered set#Intervals, its generalization from numbers to ...
. It is possible to construct just intervals with even "juster" ratios, or alternately, with values closer to the equal-tempered equivalents. For instance, a
7-limit
7-limit or septimal tunings and intervals are musical instrument tunings that have a limit of seven: the largest prime factor contained in the interval ratios between pitches is seven. Thus, for example, 50:49 is a 7-limit interval, but 14:11 ...
tuning is sometimes used to obtain a slightly juster and consequently more consonant interval for the minor seventh (7/4) and its inversion, the major second (8/7). A list of these reference ratios, which may be referred to as pure or strictly just intervals or ratios, is provided below:
Cells highlighted in yellow indicate intervals that are juster than those in the non-coloured cells in the same row. Those highlighted in cyan indicate even juster ratios.
Notice that the ratios 45/32 and 64/45 for the
tritone
In music theory, the tritone is defined as a musical interval composed of three adjacent whole tones (six semitones). For instance, the interval from F up to the B above it (in short, F–B) is a tritone as it can be decomposed into the three a ...
s (augmented fourth and diminished fifth) are not in all contexts regarded as strictly just, but they are the justest possible in the above-mentioned 5-limit tuning scales. An extended asymmetric 5-limit scale (see below) provides slightly juster ratios for both the tritones (25/18 and 36/25), the purity of which is also controversial. 7-limit tuning allows for the justest possible ratios, namely 7/5 (about 582.512 cents, also known as
septimal tritone
A septimal tritone is a tritone (about one half of an octave) that involves the factor seven. There are two that are inverses. The lesser septimal tritone (also Huygens' tritone) is the musical interval with ratio 7:5 (582.51 cents). The greater ...
) and 10/7 (about 617.488 cents). These ratios are more consonant than 17/12 (about 603.000 cents) and 24/17 (about 597.000 cents), which can be obtained in 17-limit tuning, yet the latter are also fairly common, as they are closer to the equal-tempered value of 600.000 cents.
The above-mentioned 7/4 interval (about 968.826 cents), also known as the
septimal minor seventh
The harmonic seventh interval, also known as the septimal minor seventh, or subminor seventh, is one with an exact 7:4 ratio (about 969 cents). This is somewhat narrower than and is, "particularly sweet", "sweeter in quality" than an "ordinar ...
, or harmonic seventh, has been a contentious issue throughout the history of music theory; it is 31 cents flatter than an equal-tempered minor seventh.
Size of intervals
The tables above show only the frequency ratios of each tone with respect to the base note C. However, intervals can be formed by starting from each of the twelve notes. Thus, twelve intervals can be defined for each interval type (twelve unisons, twelve
semitone
A semitone, also called a half step or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically.
It is defined as the interval between two adjacent no ...
s, twelve intervals composed of 2 semitones, twelve intervals composed of 3 semitones, etc.).
![Interval ratios in C-based asymmetric 5-limit tuning](https://upload.wikimedia.org/wikipedia/commons/9/99/Interval_ratios_in_C-based_asymmetric_5-limit_tuning.PNG)
In 5-limit tuning, each of the interval types, except for the unisons and the octaves, has three or even four different sizes. This is the price paid for seeking just intonation. The tables on the right and below show their frequency ratios and their approximate sizes in cents, for the "asymmetric scale". Similar tables, for the "symmetric scale 1", are publishe
herean
here Interval names are given in their standard shortened form. For instance, the size of the interval from C to G, which is a perfect fifth (P5), can be found in the seventh column of the row labeled C. Pure intervals, as defined above, are shown in
bold
In typography, emphasis is the strengthening of words in a text with a font in a different style from the rest of the text, to highlight them. It is the equivalent of prosody stress in speech.
Methods and use
The most common methods in W ...
font (notice that, as explained above, the justly intonated ratio 45/32 ≈ 590 cents, for A4, is not considered pure).
A color code distinguishes intervals that deviate from the reference sizes in the construction table, and show the amount of their deviation. Wolf intervals are marked in black.
[Wolf intervals are operationally defined herein as intervals composed of 3, 4, 5, 7, 8, or 9 semitones (i.e., major and minor thirds or sixths, perfect fourths or fifths, and their ]enharmonic equivalent
In modern musical notation and tuning, an enharmonic equivalent is a note, interval, or key signature that is equivalent to some other note, interval, or key signature but "spelled", or named differently. The enharmonic spelling of a written no ...
s) the size of which deviates by more than one syntonic comma
In music theory, the syntonic comma, also known as the chromatic diesis, the Didymean comma, the Ptolemaic comma, or the diatonic comma is a small comma type interval between two musical notes, equal to the frequency ratio 81:80 (= 1.0125) ( ...
(about 21.5 cents) from the corresponding justly intonated interval. Intervals made up of 1, 2, 6, 10, or 11 semitones (e.g., major and minor seconds or sevenths, tritones, and their enharmonic equivalents) are considered dissonant
In music, consonance and dissonance are categorizations of simultaneous or successive Sound, sounds. Within the Western tradition, some listeners associate consonance with sweetness, pleasantness, and acceptability, and dissonance with harshness ...
even when they are justly tuned, thus they are not marked as wolf intervals even when they deviate from just intonation by more than one syntonic comma.
The reason why the interval sizes vary throughout the scale is that the pitches forming the scale are unevenly spaced. Namely, the frequencies defined by construction for the twelve notes determine four different
semitone
A semitone, also called a half step or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically.
It is defined as the interval between two adjacent no ...
s (i.e., intervals between adjacent notes). For instance:
*
("Just"
augmented unison
In modern Western tonal music theory an augmented unison or augmented prime is the interval between two notes on the same staff position, or denoted by the same note letter, whose alterations cause them, in ordinary equal temperament, to be one ...
between E and E)
*
(Augmented unison between D and D)
*
("Just"
minor second
A semitone, also called a half step or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically.
It is defined as the interval between two adjacent no ...
between C and D)
*
(Minor second between A and B)
Conversely, in an
equally tempered
An equal temperament is a musical temperament or tuning system, which approximates just intervals by dividing an octave (or other interval) into equal steps. This means the ratio of the frequencies of any adjacent pair of notes is the same, w ...
chromatic scale, by definition the twelve pitches are equally spaced, all semitones having a size of exactly
*
As a consequence all intervals of any given type have the same size (e.g., all major thirds have the same size, all fifths have the same size, etc.). The price paid, in this case, is that none of them is justly tuned and perfectly consonant, except, of course, for the unison and the octave.
Note that 5-limit tuning was designed to maximize the number of pure intervals, but even in this system several intervals are markedly impure (for instance, as shown in the figures, 60 out of 144 intervals deviate by at least 19.6 cents from the justly intonated reference sizes shown in the construction table). Also, 5-limit tuning yields a much larger number of wolf intervals with respect to
Pythagorean tuning
Pythagorean tuning is a system of musical tuning in which the frequency ratios of all intervals are based on the ratio 3:2.Bruce Benward and Marilyn Nadine Saker (2003). ''Music: In Theory and Practice'', seventh edition, 2 vols. (Boston: Mc ...
, which can be considered a 3-limit just intonation tuning. Namely, while Pythagorean tuning determines only 2 wolf intervals (a fifth and a fourth), the 5-limit symmetric scales produce 12 of them, and the asymmetric scale 14. It is also important to note that the two fifths, three minor thirds, and three major sixths marked in orange in the tables (ratio 40/27, 32/27, and 27/16 (or G−, E−, and A+
), even though they do not completely meet the conditions
to be wolf intervals, deviate from the corresponding pure ratio by an amount (1
syntonic comma
In music theory, the syntonic comma, also known as the chromatic diesis, the Didymean comma, the Ptolemaic comma, or the diatonic comma is a small comma type interval between two musical notes, equal to the frequency ratio 81:80 (= 1.0125) ( ...
, i.e., 81/80, or about 21.5 cents) large enough to be clearly perceived as
dissonant
In music, consonance and dissonance are categorizations of simultaneous or successive Sound, sounds. Within the Western tradition, some listeners associate consonance with sweetness, pleasantness, and acceptability, and dissonance with harshness ...
.
Clearly, the more we try to increase the number of pure and
consonant
In articulatory phonetics, a consonant is a speech sound that is articulated with complete or partial closure of the vocal tract. Examples are and pronounced with the lips; and pronounced with the front of the tongue; and pronounced wit ...
intervals, the more the remaining ones become impure and dissonant, by compensation. Some of the major seconds (M2) and minor sevenths (m7) represent the only exception to this rule. As you can see in the tables, those marked in orange are pure (10/9 and 16/9), even if their size is 81/80 narrower than the corresponding reference size (9/8 and 9/5).
For a comparison with other tuning systems, see also
this table.
Commas
In other tuning systems, a
comma
The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline ...
may be defined as a minute interval, equal to the difference between two kinds of
semitone
A semitone, also called a half step or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically.
It is defined as the interval between two adjacent no ...
s (diatonic and chromatic, also known as minor second, m2, or augmented unison, A1). In this case, however, 4 different kinds of semitones are produced (two A1, S
1 and S
2, and two m2, S
3 and S
4), and 12 different commas can be defined as the differences between their sizes in cents, or equivalently as the ratios between their ratios. Among these, we select the six ascending ones (those with ratio larger than 1/1, and positive size in cents):
The other six ratios are discarded because they are just the opposite of these, and hence they have exactly the same length, but an opposite direction (i.e., a descending direction, a ratio smaller than 1/1, and a negative size in cents). We obtain commas of four different sizes: the diaschisma, the lesser diesis, the syntonic comma, and the greater diesis. Since S
1 (the ''just A1'') and S
3 (the ''just m2'') are the most often occurring semitones in this 12-tone scale (see tables above), the lesser diesis, being defined as the ratio between them, is the most often observed comma.
The syntonic comma is also defined, in 5-limit tuning, as the ratio between the major
tone (M2 with size 9/8) and the minor tone (M2 with size 10/9). Notice that it cannot be defined, in other tuning systems, as the ratio between diatonic and cromatic semitones (m2/A1), but it is an important reference value used to tune the
perfect fifth
In music theory, a perfect fifth is the Interval (music), musical interval corresponding to a pair of pitch (music), pitches with a frequency ratio of 3:2, or very nearly so.
In classical music from Western culture, a fifth is the interval fro ...
in any tuning system in the
syntonic temperament
A regular diatonic tuning is any musical scale consisting of " tones" (T) and "semitones" (S) arranged in any rotation of the sequence TTSTTTS which adds up to the octave with all the T's being the same size and all the S's the being the same s ...
continuum (including also meantone temperaments).
Diminished seconds
Three of the above-mentioned commas, namely the diaschisma, the diesis and the greater diesis, meet the definition of the
diminished second
In modern Western tonal music theory, a diminished second is the interval produced by narrowing a minor second by one chromatic semitone.Bruce Benward and Marilyn Saker (2003). ''Music: In Theory and Practice, Vol. I'', p. 54. . Specific example ...
, being the difference between the sizes in cents of a diatonic and a chromatic semitone (or equivalently the ratio between their frequency ratios).
On the contrary, the syntonic comma is defined either as the difference in cents between two chromatic semitones (S
2 and S
1), or between two diatonic semitones (S
4 and S
3), and cannot be considered a diminished second.
Extension of the twelve-tone scale
The table above uses only low powers of 3 and 5 to build the base ratios. However, it can be easily extended by using higher positive and negative powers of the same numbers, such as 5
2 = 25, 5
−2 = 1/25, 3
3 = 27, or 3
−3 = 1/27. A scale with 25, 35, or even more pitches can be obtained by combining these base ratios.
For instance, one can obtain 35 pitches by adding rows in each direction like this:
The left column (1/9) is sometimes removed (as in the asymmetric scale shown above), thus creating an asymmetric table with a smaller number of pitches. Notice that a juster ratio is produced for the diminished fifth (C-G = 36/25), with respect to the restricted 5-limit tuning described above (where C to G
- = 64/45).
History
In Pythagorean tuning, perhaps the first tuning system theorized in the West,
[The oldest known description of the Pythagorean tuning system appears in Babylonian artifacts. See: West, M. L. "The Babylonian Musical Notation and the Hurrian Melodic Texts", '']Music & Letters
''Music & Letters'' is an academic journal published quarterly by Oxford University Press with a focus on musicology. The journal sponsors the Music & Letters Trust, twice-yearly cash awards of variable amounts to support research in the music fie ...
'', vol. 75, no. 2 (May 1994). pp. 161–179. the only highly consonant intervals were the
perfect fifth
In music theory, a perfect fifth is the Interval (music), musical interval corresponding to a pair of pitch (music), pitches with a frequency ratio of 3:2, or very nearly so.
In classical music from Western culture, a fifth is the interval fro ...
and its inversion, the
perfect fourth
A fourth is a musical interval encompassing four staff positions in the music notation of Western culture, and a perfect fourth () is the fourth spanning five semitones (half steps, or half tones). For example, the ascending interval from C to ...
. The Pythagorean
major third
In classical music, a third is a musical interval encompassing three staff positions (see Interval number for more details), and the major third () is a third spanning four semitones. Forte, Allen (1979). ''Tonal Harmony in Concept and P ...
(81:64) and
minor third
In music theory, a minor third is a musical interval that encompasses three half steps, or semitones. Staff notation represents the minor third as encompassing three staff positions (see: interval number). The minor third is one of two com ...
(32:27) were
dissonant
In music, consonance and dissonance are categorizations of simultaneous or successive Sound, sounds. Within the Western tradition, some listeners associate consonance with sweetness, pleasantness, and acceptability, and dissonance with harshness ...
, and this prevented musicians from using
triads and
chords, forcing them for centuries to write music with relatively simple
texture
Texture may refer to:
Science and technology
* Surface texture, the texture means smoothness, roughness, or bumpiness of the surface of an object
* Texture (roads), road surface characteristics with waves shorter than road roughness
* Texture (c ...
. In late
Middle Ages
In the history of Europe, the Middle Ages or medieval period lasted approximately from the late 5th to the late 15th centuries, similar to the post-classical period of global history. It began with the fall of the Western Roman Empire a ...
, musicians realized that by slightly tempering the pitch of some notes, the Pythagorean thirds could be made
consonant
In articulatory phonetics, a consonant is a speech sound that is articulated with complete or partial closure of the vocal tract. Examples are and pronounced with the lips; and pronounced with the front of the tongue; and pronounced wit ...
. For instance, if you decrease by a
syntonic comma
In music theory, the syntonic comma, also known as the chromatic diesis, the Didymean comma, the Ptolemaic comma, or the diatonic comma is a small comma type interval between two musical notes, equal to the frequency ratio 81:80 (= 1.0125) ( ...
(81:80) the frequency of E, C-E (a major third), and E-G (a minor third) become just. Namely, C-E is narrowed to a justly intonated ratio of
:
and at the same time E-G is widened to the just ratio of
:
The drawback is that the fifths A-E and E-B, by flattening E, become almost as dissonant as the Pythagorean
wolf fifth
In music theory, the wolf fifth (sometimes also called Procrustean fifth,
or imperfect fifth)
Paul, Oscar (1885). A manual of harmony for use in music-schools and seminaries and for self-instruction', p.165. Theodore Baker, trans. G. Schirmer. ...
. But the fifth C-G stays consonant, since only E has been flattened (C-E * E-G = 5/4 * 6/5 = 3/2), and can be used together with C-E to produce a C-
major
Major (commandant in certain jurisdictions) is a military rank of commissioned officer status, with corresponding ranks existing in many military forces throughout the world. When used unhyphenated and in conjunction with no other indicators ...
triad (C-E-G).
By generalizing this simple rationale,
Gioseffo Zarlino
Gioseffo Zarlino (31 January or 22 March 1517 – 4 February 1590) was an Italian music theorist and composer of the Renaissance. He made a large contribution to the theory of counterpoint as well as to musical tuning.
Life and career
Zarlino w ...
, in the late sixteenth century, created the first justly intonated 7-tone (
diatonic
Diatonic and chromatic are terms in music theory that are most often used to characterize Scale (music), scales, and are also applied to musical instruments, Interval (music), intervals, Chord (music), chords, Musical note, notes, musical sty ...
) scale, which contained pure perfect fifths (3:2), pure major thirds, and pure minor thirds:
F → A → C → E → G → B → D
This is a sequence of just major thirds (M3, ratio 5:4) and just minor thirds (m3, ratio 6:5), starting from F:
F + M3 + m3 + M3 + m3 + M3 + m3
Since M3 + m3 = P5 (perfect fifth), i.e., 5/4 * 6/5 = 3/2, this is exactly equivalent to the diatonic scale obtained in 5-limit just intonation, and hence can be viewed as a subset of the construction table used for the 12-tone (
chromatic
Diatonic and chromatic are terms in music theory that are most often used to characterize scales, and are also applied to musical instruments, intervals, chords, notes, musical styles, and kinds of harmony. They are very often used as a pair, ...
) scale:
where both rows are sequences of just fifths, and F-A, C-E, G-B are just major thirds:
See also
*
Mathematics of musical scales
Music theory analyzes the pitch, timing, and structure of music. It uses mathematics to study elements of music such as tempo, chord progression, form, and meter. The attempt to structure and communicate new ways of composing and hearing music ...
*
Microtonal music
Microtonal music or microtonality is the use in music of microtones—intervals smaller than a semitone, also called "microintervals". It may also be extended to include any music using intervals not found in the customary Western tuning of tw ...
*
Microtuner
A microtuner or microtonal tuner is an electronic device or software program designed to modify and test the tuning of musical instruments (in particular synthesizers) with microtonal precision, allowing for the design and construction of microtona ...
*
Pythagorean interval
In musical tuning theory, a Pythagorean interval is a musical interval with frequency ratio equal to a power of two divided by a power of three, or vice versa.Benson, Donald C. (2003). ''A Smoother Pebble: Mathematical Explorations'', p.56. . " ...
*
Semitone
A semitone, also called a half step or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically.
It is defined as the interval between two adjacent no ...
*
List of intervals in 5-limit just intonation
*
List of meantone intervals
The following is a list of intervals of extended meantone temperament. These intervals constitute the standard vocabulary of intervals for the Western common practice era. Here 12-EDO refers to the size of the interval in 12 equal divisions of th ...
*
List of musical intervals
*
List of pitch intervals
Below is a list of intervals expressible in terms of a prime limit (see Terminology), completed by a choice of intervals in various equal subdivisions of the octave or of other intervals.
For commonly encountered harmonic or melodic intervals ...
*
Whole-tone scale
In music, a whole-tone scale is a scale in which each note is separated from its neighbors by the interval of a whole tone. In twelve-tone equal temperament, there are only two complementary whole-tone scales, both six-note or ''hexatonic'' s ...
*
Regular number
Regular numbers are numbers that evenly divide powers of 60 (or, equivalently, powers of 30). Equivalently, they are the numbers whose only prime divisors are 2, 3, and 5. As an example, 602 = 3600 = 48 ×&nb ...
*
Hexany
In musical tuning systems, the hexany, invented by Erv Wilson, represents one of the simplest structures found in his combination product sets.
It is referred to as an uncentered structure, meaning that it implies no tonic. It achieves this b ...
*
Electronic tuner
In music, an electronic tuner is a device that detects and displays the pitch of musical notes played on a musical instrument. "Pitch" is the perceived fundamental frequency of a musical note, which is typically measured in Hertz. Simple tune ...
*
Consonance and dissonance
In music, consonance and dissonance are categorizations of simultaneous or successive Sound, sounds. Within the Western tradition, some listeners associate consonance with sweetness, pleasantness, and acceptability, and dissonance with harshness ...
Notes
External links
Art of the States: microtonal/just intonationworks using just intonation by American composers
Just Intonationb
Mark Nowitzkyby
Kyle Gann
Kyle Eugene Gann (born November 21, 1955, in Dallas, Texas) is an American professor of music, critic, analyst, and composer who has worked primarily in the New York City area. As a music critic for ''The Village Voice'' (from 1986 to 2005) and ...
A selection of Just Intonation works edited by the Just Intonation Networkweb published on the
Tellus Audio Cassette Magazine
Launched from the Lower East Side, Manhattan in 1983 as a subscription only bimonthly publication, the ''Tellus Audio Cassette Magazine'' utilized the audio cassette medium to distribute no wave downtown music and audio art and was in activity ...
project archive at
UbuWeb
UbuWeb is a web-based educational resource for avant-garde material available on the internet, founded in 1996 by poet Kenneth Goldsmith. It offers visual, concrete and sound poetry, expanding to include film and sound art mp3 archives.
Philosop ...
Medieval Music and Arts Foundation* Barbieri, Patrizio
(2008) Latina, Il Levante
22 Note Just Intonation Keyboard Software with 12 Indian Instrument SoundsLibreria Editrice
Plainsound Music Edition– Just intonation (JI) music and research, information about the
Helmholtz
Hermann Ludwig Ferdinand von Helmholtz (31 August 1821 – 8 September 1894) was a German physicist and physician who made significant contributions in several scientific fields, particularly hydrodynamic stability. The Helmholtz Association, ...
-
Ellis
Ellis is a surname of Welsh and English origin. Retrieved 21 January 2014 An independent French origin of the surname is said to derive from the phrase fleur-de-lis.
Surname
A
* Abe Ellis (Stargate), a fictional character in the TV series ' ...
JI
pitch notation
{{Instrument tunings
5-limit tuning and intervals