7/5: Difference between revisions
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{{Infobox Interval | {{Infobox Interval | ||
| | | Name = narrow tritone, lesser septimal tritone, Huygens' tritone | ||
| Color name = zg5, zogu 5th | |||
| | |||
| Sound = jid_7_5_pluck_adu_dr220.mp3 | | Sound = jid_7_5_pluck_adu_dr220.mp3 | ||
}} | }} | ||
{{Wikipedia|Septimal tritone}} | |||
In [[7-limit]] [[ | In [[7-limit]] [[just intonation]], '''7/5''' is a '''narrow [[tritone]]''' (or '''Huygens' tritone''') measuring about 582.5¢. It is a noticeable 17.5¢ away from the 600¢ half-octave (square root of 2) tritone of [[12edo]] and every even-numbered [[edo]]. It represents the difference between [[7/4]] and [[5/4]]. | ||
While in the context of the [[harmonic seventh chord]], it is rightly recognized as a type of diminished fifth, it can also be argued on the basis of the fact that 7/5 interval is smaller than 600 cents that it acts more as a type of augmented fourth than a diminished fifth – an analysis that is required in cases where this interval occurs in a [[5L 2s|diatonic scale]] that demonstrates [[Rothenberg propriety]]. This is one of the reasons why 7/4 can be argued to be a type of "sinth" – a cross between a sixth and a seventh – as opposed to merely a subminor seventh. | |||
7/5 is notable for its low [[harmonic entropy]], and is often reported to sound more consonant than the half-octave tritone; indeed it appears in the harmonic seventh chord that forms the basis of consonance in 7-limit JI. Its inversion is [[10/7]], which measures about 617.5¢, and these two septimal tritones differ by the [[superparticular]] interval [[50/49]], about 35.0¢. Systems which temper out 50/49 will equate 7/5 and [[10/7]], usually to the 600¢ half-octave. | |||
Another just tritone is [[729/512]] (611.7¢), and this is literally a tri-tone, since it is (9/8)<sup>3</sup>, or three "whole tones". Yet another is [[45/32]], about 590.2¢, which appears in the [[5-limit]] (inversion is [[64/45]]). See also [[13/9]], [[18/13]], [[17/12]], [[24/17]], [[25/18]] and [[36/25]]. | |||
== Approximation == | |||
{{Interval edo approximation|7/5}} | |||
== See also == | == See also == | ||
[[ | * [[10/7]] – its [[octave complement]] | ||
[[ | * [[15/14]] – its [[fifth complement]] | ||
[[ | * [[Gallery of just intervals]] | ||
[[ | |||
[[Category:Tritone]] | [[Category:Tritone]] | ||
[[Category:Over-5 intervals]] | |||
[[Category:Taxicab-2 intervals]] | |||
Latest revision as of 13:53, 8 January 2026
| Interval information |
lesser septimal tritone,
Huygens' tritone
[sound info]
In 7-limit just intonation, 7/5 is a narrow tritone (or Huygens' tritone) measuring about 582.5¢. It is a noticeable 17.5¢ away from the 600¢ half-octave (square root of 2) tritone of 12edo and every even-numbered edo. It represents the difference between 7/4 and 5/4.
While in the context of the harmonic seventh chord, it is rightly recognized as a type of diminished fifth, it can also be argued on the basis of the fact that 7/5 interval is smaller than 600 cents that it acts more as a type of augmented fourth than a diminished fifth – an analysis that is required in cases where this interval occurs in a diatonic scale that demonstrates Rothenberg propriety. This is one of the reasons why 7/4 can be argued to be a type of "sinth" – a cross between a sixth and a seventh – as opposed to merely a subminor seventh.
7/5 is notable for its low harmonic entropy, and is often reported to sound more consonant than the half-octave tritone; indeed it appears in the harmonic seventh chord that forms the basis of consonance in 7-limit JI. Its inversion is 10/7, which measures about 617.5¢, and these two septimal tritones differ by the superparticular interval 50/49, about 35.0¢. Systems which temper out 50/49 will equate 7/5 and 10/7, usually to the 600¢ half-octave.
Another just tritone is 729/512 (611.7¢), and this is literally a tri-tone, since it is (9/8)3, or three "whole tones". Yet another is 45/32, about 590.2¢, which appears in the 5-limit (inversion is 64/45). See also 13/9, 18/13, 17/12, 24/17, 25/18 and 36/25.
Approximation
| Edo | Step size | Cents (¢) | Absolute error (¢) | Relative error (%) |
|---|---|---|---|---|
| 2 | 1\2 | 600.00 | +17.49 | +2.91 |
| 4 | 2\4 | 600.00 | +17.49 | +5.83 |
| 6 | 3\6 | 600.00 | +17.49 | +8.74 |
| 29 | 14\29 | 579.31 | -3.20 | -7.74 |
| 31 | 15\31 | 580.65 | -1.87 | -4.82 |
| 33 | 16\33 | 581.82 | -0.69 | -1.91 |
| 35 | 17\35 | 582.86 | +0.34 | +1.01 |
| 37 | 18\37 | 583.78 | +1.27 | +3.92 |
| 39 | 19\39 | 584.62 | +2.10 | +6.84 |
| 41 | 20\41 | 585.37 | +2.85 | +9.75 |
| 62 | 30\62 | 580.65 | -1.87 | -9.65 |
| 64 | 31\64 | 581.25 | -1.26 | -6.73 |
| 66 | 32\66 | 581.82 | -0.69 | -3.82 |
| 68 | 33\68 | 582.35 | -0.16 | -0.90 |
| 70 | 34\70 | 582.86 | +0.34 | +2.01 |
| 72 | 35\72 | 583.33 | +0.82 | +4.93 |
| 74 | 36\74 | 583.78 | +1.27 | +7.84 |