List of intervals in 5-limit just intonation

The intervals of 5-limit just intonation (prime limit, not odd limit) are ratios involving only the powers of 2, 3, and 5. The fundamental intervals are the superparticular ratios 2/1 (the octave), 3/2 (the perfect fifth) and 5/4 (the major third). That is, the notes of the major triad are in the ratio 1:5/4:3/2 or 4:5:6.

In all tunings, the major third is equivalent to two major seconds. However, because just intonation does not allow the irrational ratio of 5/2, two different frequency ratios are used: the major tone (9/8) and the minor tone (10/9).

The intervals within the diatonic scale, as well as other marked* intervals with a maximum numerator and denominator of 1,000 inclusive [unfinished], are shown in the table below.

NamesRatioCents12ET CentsDefinition53ET commas

(syntonic, diaschisma,

magic, etc.)

53ET centsRepresentation (Makam)Complement
unison1/10.00000octave
syntonic comma81/8021.510c or T  t or X  x or

S  Λ or L  S or t  τ

122.64semi-diminished octave
diesis

minor diesis
diminished second

128/12541.060D or S  x or L  X245.28augmented seventh
maximal diesis*

porcupine comma

250/24349.17100x c or t L245.28
major diesis*

diminished comma

648/625 62.57 0 D + c or L x 3 67.92
lesser chromatic semitone
minor semitone
augmented unison
25/2470.67100x or t  S or T  L or

X  c or S  D or τ  Λ

367.92diminished octave
Pythagorean minor second
Pythagorean limma
256/24390.22100Λ490.57Pythagorean major seventh
greater chromatic semitone
wide augmented unison
135/12892.18100X or T  S490.57narrow diminished octave
major semitone
limma
minor second
16/15111.73100S5113.21major seventh
large limma
acute minor second
27/25133.24100L or T  x6135.85grave major seventh
classic neutral second (?)* 625/576 141.34 200 x + x or t D 6 135.85 greater neutral seventh
grave tone
grave major second
800/729160.90200τ or Λ + x or t  c7158.49acute minor seventh
minor tone
lesser major second
10/9182.40200t8181.13minor seventh
major tone
Pythagorean major second
greater major second
9/8203.91200T or t + c9203.77Pythagorean minor seventh
diminished third256/225223.46200S + S10226.42augmented sixth
acute major second* 729/640 225.42 200 T + c or L + X 10 226.42
classic diminished third* 144/125 244.97 200 T + D or L + S 11 249.06
semi-augmented second125/108253.08300t + x11249.06semi-augmented sixth
grave minor third (?)* 729/625 266.48 200 L + L 12 271.70 acute major sixth (?)
augmented second75/64274.58300T + x12271.70diminished seventh
Pythagorean minor third32/27294.13300T + Λ13294.34Pythagorean major sixth
minor third6/5315.64300T + S14316.98major sixth
acute minor third243/200333.18300T + L15339.62grave major sixth
classic neutral third* 625/512 345.25 400 T + x + x or T + t D or

t + X + x or τ + X + X

15 339.62
grave major third100/81364.81400T + τ16362.26acute minor sixth
major third5/4386.31400T + t17384.91minor sixth
Pythagorean major third81/64407.82400T + T18407.55Pythagorean minor sixth
classic diminished fourth32/25427.37400T + S + S19430.19classic augmented fifth
classic augmented third125/96456.99500T + t + x20452.83classic diminished sixth
wide augmented third675/512478.49500T + t + X21475.47narrow diminished sixth
perfect fourth4/3498.04500T + t + S22498.11perfect fifth
acute fourth[1]27/20519.55500T + t + L23520.75grave fifth
classic augmented fourth25/18568.72600T + t + t25566.04classic diminished fifth
augmented fourth45/32590.22600T + t + T26588.68diminished fifth
diminished fifth64/45609.78600T + t + S + S27611.32augmented fourth
classic diminished fifth36/25631.29600T + t + S + L28633.96classic augmented fourth
grave fifth[1]40/27680.45700T + t + S + t30679.25acute fourth
perfect fifth3/2701.96700T + t + S + T31701.89perfect fourth
narrow diminished sixth1024/675721.51700T + t + S + S + S32724.53wide augmented third
classic diminished sixth192/125743.01700T + t + S + L + S33747.17classic augmented third
classic augmented fifth25/16772.63800T + t + S + T + x34769.81classic diminished fourth
Pythagorean minor sixth128/81792.18800T + t + S + T + Λ35792.45Pythagorean major third
minor sixth8/5813.69800(T + t + S + T) + S36815.09major third
acute minor sixth81/50835.19800(T + t + S + T) + L37837.74grave major third
major sixth5/3884.36900(T + t + S + T) + t39883.02minor third
Pythagorean major sixth27/16905.87900(T + t + S + T) + T40905.66Pythagorean minor third
diminished seventh128/75925.42900(T + t + S + T) + S + S41928.30augmented second
augmented sixth225/128976.541000(T + t + S + T) + T + x43973.58diminished third
Pythagorean minor seventh16/9996.091000(T + t + S + T) + T + Λ44996.23Pythagorean major second
minor seventh9/51017.601000(T + t + S + T) + T + S451018.87lesser major second
acute minor seventh729/4001039.101000(T + t + S + T) + T + L461041.51grave major second
grave major seventh50/271066.761100(T + t + S + T) + T + τ471064.15acute minor second
major seventh15/81088.271100(T + t + S + T) + T + t481086.79minor second
narrow diminished octave256/1351107.821100(T + t + S + T) + t + S + S491109.43wide augmented unison
Pythagorean major seventh243/1281109.781100(T + t + S + T) + T + T491109.43Pythagorean minor second
diminished octave48/251129.331100(T + t + S + T) + T + S + S501132.08augmented unison
augmented seventh125/641158.941200(T + t + S + T) + T + t + x511154.72diminished second
semi-diminished octave160/811178.491200(T + t + S + T) + T + t + x + c521177.36syntonic comma
octave2/11200.001200(T + t + S + T) + (T + t + S)531200.00unison

(The Pythagorean minor second is found by adding 5 perfect fourths.)

The table below shows how these steps map to the first 31 scientific harmonics, transposed into a single octave.

HarmonicMusical NameRatioCents12ET Cents53ET Commas53ET Cents
1unison1/10.00000.00
2octave2/11200.001200531200.00
3perfect fifth3/2701.9670031701.89
5major third5/4386.3140017384.91
7augmented sixth§

harmonic seventh

7/4968.83100043973.58
9major tone9/8203.912009203.77
11undecimal semi-augmented fourth[2] 11/8551.32500 or 60024543.40
13acute minor sixth§

tridecimal neutral sixth

13/8840.5380037837.74
15major seventh15/81088.271100481086.79
17limma§

large septendecimal semitone

17/16104.961005113.21
19Pythagorean minor third§

Otonal minor third

19/16297.5130013294.34
21wide augmented third§

sub-fourth / narrow fourth

21/16470.7850021475.47
23classic diminished fifth§

23-limit superaugmented fourth

23/16628.2760028633.96
25classic augmented fifth25/16772.6380034769.81
27Pythagorean major sixth27/16905.8790040905.66
29minor seventh§

29-limit large minor seventh

29/161029.581000451018.87
31augmented seventh§

31-limit ultramajor seventh

31/161145.041100511154.72


§ These intervals also appear in the upper table, although with different ratios.

See also

References

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.