13/8: Difference between revisions

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{{Infobox Interval
{{Infobox Interval
| Ratio = 13/8
| Monzo = -3 0 0 0 0 1
| Cents = 840.52766
| Name = (lesser) tridecimal neutral sixth
| Name = (lesser) tridecimal neutral sixth
| Color name = 3o6, tho 6th
| Color name = 3o6, tho 6th
| FJS name = m6<sup>13</sup>
| Sound = jid_13_8_pluck_adu_dr220.mp3
| Sound = jid_13_8_pluck_adu_dr220.mp3
}}
}}


'''13/8''' is the '''(lesser) tridecimal neutral sixth''', which measures about 840.5¢, falling between the categories of minor sixth and major sixth. In [[13-limit]] [[just intonation]], 13/8, as an octave-reduced 13th harmonic, is treated as a basic component of harmony. In the harmonic series and in chords based on it, 13/8 sits between the more familiar consonances of [[3/2]] and [[7/4]], separated from each by the [[superparticular]] ratios [[13/12]] and [[14/13]], respectively. The word "lesser" is added when necessary to differentiate it from [[64/39]], another tridecimal neutral sixth. It may also be treated as a type of augmented fifth, as the sum of [[5/4]] and [[13/10]].
'''13/8''' is the '''(lesser) tridecimal neutral sixth''', which measures about 840.5¢, falling between the categories of minor sixth and major sixth. In [[13-limit]] [[just intonation]], 13/8, as the octave-reduced 13th harmonic, is treated as a basic component of harmony. In the harmonic series and in chords based on it, 13/8 sits between the more familiar consonances of [[3/2]] and [[7/4]], separated from each by the [[superparticular]] ratios [[13/12]] and [[14/13]], respectively. The word "lesser" is added when necessary to differentiate it from [[64/39]], another tridecimal neutral sixth. It may also be treated as a type of augmented fifth, as the sum of [[5/4]] and [[13/10]].


13/8 differs from the Pythagorean minor sixth [[128/81]] by [[1053/1024]], about 48¢, from the classic minor sixth [[8/5]] by [[65/64]], about 27¢, from the undecimal neutral sixth [[18/11]] by [[144/143]], about 12¢, and from the rastmic neutral sixth [[44/27]] by [[352/351]], about 4.9¢.  
13/8 differs from the Pythagorean minor sixth [[128/81]] by [[1053/1024]], about 48¢, from the classic minor sixth [[8/5]] by [[65/64]], about 27¢, from the undecimal neutral sixth [[18/11]] by [[144/143]], about 12¢, and from the rastmic neutral sixth [[44/27]] by [[352/351]], about 4.9¢.


== Approximation ==
== Approximation ==
Line 17: Line 13:


This interval is a ratio of two consecutive Fibonacci numbers, therefore it approximates the [[golden ratio]]. In this case, 13/8 is ~7.4 [[cent|¢]] sharp of the golden ratio.
This interval is a ratio of two consecutive Fibonacci numbers, therefore it approximates the [[golden ratio]]. In this case, 13/8 is ~7.4 [[cent|¢]] sharp of the golden ratio.
{{Interval edo approximation}}


== See also ==
== See also ==
Line 23: Line 20:
* [[Gallery of just intervals]]
* [[Gallery of just intervals]]


[[Category:13-limit]]
[[Category:Sixth]]
[[Category:Sixth]]
[[Category:Neutral sixth]]
[[Category:Neutral sixth]]
[[Category:Golden ratio approximations]]
[[Category:Golden ratio approximations]]
[[Category:Octave-reduced harmonics]]

Latest revision as of 17:13, 30 May 2026

Interval information
Ratio 13/8
Subgroup monzo 2.13 [-3 1⟩
Size in cents 840.5277¢
Name (lesser) tridecimal neutral sixth
Color name 3o6, tho 6th
FJS name [math]\displaystyle{ \text{m6}^{13} }[/math]
Special properties reduced,
reduced harmonic
Tenney norm (log2 nd) 6.70044
Weil norm (log2 max(n, d)) 7.40088
Wilson norm (sopfr(nd)) 19

[sound info]
Open this interval in xen-calc

13/8 is the (lesser) tridecimal neutral sixth, which measures about 840.5¢, falling between the categories of minor sixth and major sixth. In 13-limit just intonation, 13/8, as the octave-reduced 13th harmonic, is treated as a basic component of harmony. In the harmonic series and in chords based on it, 13/8 sits between the more familiar consonances of 3/2 and 7/4, separated from each by the superparticular ratios 13/12 and 14/13, respectively. The word "lesser" is added when necessary to differentiate it from 64/39, another tridecimal neutral sixth. It may also be treated as a type of augmented fifth, as the sum of 5/4 and 13/10.

13/8 differs from the Pythagorean minor sixth 128/81 by 1053/1024, about 48¢, from the classic minor sixth 8/5 by 65/64, about 27¢, from the undecimal neutral sixth 18/11 by 144/143, about 12¢, and from the rastmic neutral sixth 44/27 by 352/351, about 4.9¢.

Approximation

13/8 is a fraction of a cent away from the neutral sixth found in the 10n-edo family (7\10).

This interval is a ratio of two consecutive Fibonacci numbers, therefore it approximates the golden ratio. In this case, 13/8 is ~7.4 ¢ sharp of the golden ratio.

Edo approximations for 13/8 (840.53 ¢)
≤ 80edo, relative error ≤ 10%
Edo Step size Cents (¢) Absolute error (¢) Relative error (%)
7 5\7 857.14 +16.62 +9.69
10 7\10 840.00 -0.53 -0.44
17 12\17 847.06 +6.53 +9.25
20 14\20 840.00 -0.53 -0.88
27 19\27 844.44 +3.92 +8.81
30 21\30 840.00 -0.53 -1.32
37 26\37 843.24 +2.72 +8.37
40 28\40 840.00 -0.53 -1.76
47 33\47 842.55 +2.03 +7.93
50 35\50 840.00 -0.53 -2.20
57 40\57 842.11 +1.58 +7.49
60 42\60 840.00 -0.53 -2.64
67 47\67 841.79 +1.26 +7.05
70 49\70 840.00 -0.53 -3.08
77 54\77 841.56 +1.03 +6.61
80 56\80 840.00 -0.53 -3.52

See also

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