Five-limit tuning


Five-limit tuning, 5-limit tuning, or 5-prime-limit tuning, is any system for tuning a musical instrument that obtains the frequency of each note by multiplying the frequency of a given reference note by products of integer powers of 2, 3, or 5, such as.
Powers of 2 represent intervallic movements by octaves. Powers of 3 represent movements by intervals of perfect fifths. Powers of 5 represent intervals of major thirds. Thus, 5-limit tunings are constructed entirely from stacking of three basic purely-tuned intervals. 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.
The number of potential intervals, pitch classes, pitches, key centers, chords, and modulations available to 5-limit tunings is unlimited, because no 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. If octaves are ignored, it can be seen as a 2-dimensional lattice of pitch classes 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 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 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, then fC = 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 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, it is possible to tune the familiar diatonic scale 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, the ratio of the frequencies of the notes G to D is 3/2, while that of G to C is 2/3 or 4/3 going up, and the major third G to B is 5/4.
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 away from the tonic root C on either side, and that the chords FAC, CEG, and GBD be just major triads :
This is known as Ptolemy's intense diatonic scale. Here the row headed "Natural" expresses all these ratios using a common list of natural numbers. 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, and three of the minor thirds are as expected, but D to F is a semiditone or Pythagorean minor third, a syntonic comma narrower than a justly tuned minor third.
As a consequence, we obtain a scale in which EGB and ACE are just minor triads, but the DFA triad doesn't have the minor shape or sound we might expect, being. Furthermore, the BDF triad is not the diminished triad that we would get by stacking two 6:5 minor thirds, being instead:
It can be seen that basic step-wise scale intervals appear:
  • s = 16:15
  • t = 10:9
  • T = 9:8
which may be combined to form larger intervals :
  • Ts = 6:5
  • Tt = 5:4
  • Tts = 4:3
  • TTts = 3:2
  • TTTttss 2:1
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 :
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, but GBD is now, and BDF is now.
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 for major, for minor, and for diminished at the same time if we limit ourselves to seven pitches.
That demonstrates the need for increasing the number 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:
Factor1
D−
10/9
182		A
5/3
884		E
5/4
386		B
15/8
1088		F+
45/32
590		note
ratio
cents
1B−
16/9
996		F
4/3
498		C
1
0		G
3/2
702		D
9/8
204		note
ratio
cents
G−
64/45
610		D−
16/15
112		A
8/5
814		E
6/5
316		B
9/5
1018		note
ratio
cents

The factors listed in the first row and first column are powers of 3 and 5 respectively. Colors indicate couples of enharmonic notes with almost identical pitch. The ratios are all expressed relative to C in the centre of this diagram. They are computed in two steps:
  1. 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.
  2. 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. For instance, the base ratio for the lower left cell is multiplied by 26, 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 octaves. For instance, multiplying the frequency of a note by 26 means increasing it by 6 octaves. Moreover, each row of the table may be considered a sequence of fifths, and each column a sequence of major thirds. For instance, in the first row of the table, there is an ascending fifth from D and A, and another one from A to E. This suggests an alternative but equivalent method for computing the same ratios. For instance, you can obtain A, starting from C, by moving one cell to the left and one upward in the table, which means descending by one fifth and ascending by one major third :
Since this is below C, you need to move up by an octave to end up within the desired range of ratios :
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, 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. The third one, denoted as asymmetric scale, consists of discarding the first column. 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.
The asymmetric system has the advantage of having the "justest" ratios, nine pure fifths, eight pure major thirds by design, but also six pure minor thirds. However, it also contains two impure fifths and three impure minor thirds, which practically limits modulation 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, being over 21 cents. In equally tempered scales, the difference is eliminated by making all steps the same frequency ratio.
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.