15/8: Difference between revisions
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{{Infobox Interval | |||
| Name = just major seventh, classic(al) major seventh, ptolemaic major seventh | |||
| Color name = y7, yo 7th | |||
| Sound = jid_15_8_pluck_adu_dr220.mp3 | |||
}} | |||
{{Wikipedia|Major seventh}} | |||
In [[5-limit]] [[just intonation]], '''15/8''' is the '''just major seventh''', '''classic(al) major seventh''', or '''ptolemaic major seventh'''<ref>For reference, see [[5-limit]]. </ref> of about 1088.3¢. It is also the [[octave-reduced]] 15th [[harmonic]], and appears as a complex consonance in chords such as [[8:10:12:15]], a just version of a major seventh chord. Since 15/8 = [[3/2]] × [[5/4]], it can be seen as a perfect fifth above a major third or vice versa, and this understanding works in [[12edo]], as the sum of [[~]]3/2 and ~5/4 is 700{{c}} + 400{{c}} = 1100{{c}}, which 15/8 is mapped to. | |||
Since 15 is a perfect fifth above 10 (15/10 = [[3/2]]), seventh chords can be formed with the 10th harmonic as major third and 15th harmonic as major seventh. The simplest and most familiar example is the classical major seventh chord 8:10:12:15 with steps 5/4, 6/5 and 5/4. Another example replaces the 12 with 13, which leads to [[8:10:13:15]] with steps 5/4, 13/10 and 15/13, and contains the [[10:13:15]] barbados triad. A particularly uncommon but mentionable example is the [[23-limit]] seventh chord [[16:20:23:30]]. | |||
== Approximation == | |||
{{Interval edo approximation|15/8}} | |||
See | == See also == | ||
* [[16/15]] – its [[octave complement]] | |||
* [[8/5]] – its [[twelfth complement]] | |||
* [[Ed15/8]] | |||
* [[Gallery of just intervals]] | |||
== Notes == | |||
<references/> | |||
[[Category:Seventh]] | |||
[[Category:Major seventh]] | |||
Latest revision as of 21:53, 22 December 2025
| Interval information |
classic(al) major seventh,
ptolemaic major seventh
reduced harmonic
[sound info]
In 5-limit just intonation, 15/8 is the just major seventh, classic(al) major seventh, or ptolemaic major seventh[1] of about 1088.3¢. It is also the octave-reduced 15th harmonic, and appears as a complex consonance in chords such as 8:10:12:15, a just version of a major seventh chord. Since 15/8 = 3/2 × 5/4, it can be seen as a perfect fifth above a major third or vice versa, and this understanding works in 12edo, as the sum of ~3/2 and ~5/4 is 700 ¢ + 400 ¢ = 1100 ¢, which 15/8 is mapped to.
Since 15 is a perfect fifth above 10 (15/10 = 3/2), seventh chords can be formed with the 10th harmonic as major third and 15th harmonic as major seventh. The simplest and most familiar example is the classical major seventh chord 8:10:12:15 with steps 5/4, 6/5 and 5/4. Another example replaces the 12 with 13, which leads to 8:10:13:15 with steps 5/4, 13/10 and 15/13, and contains the 10:13:15 barbados triad. A particularly uncommon but mentionable example is the 23-limit seventh chord 16:20:23:30.
Approximation
| Edo | Step size | Cents (¢) | Absolute error (¢) | Relative error (%) |
|---|---|---|---|---|
| 10 | 9\10 | 1080.00 | -8.27 | -6.89 |
| 11 | 10\11 | 1090.91 | +2.64 | +2.42 |
| 21 | 19\21 | 1085.71 | -2.55 | -4.47 |
| 22 | 20\22 | 1090.91 | +2.64 | +4.84 |
| 32 | 29\32 | 1087.50 | -0.77 | -2.05 |
| 33 | 30\33 | 1090.91 | +2.64 | +7.26 |
| 42 | 38\42 | 1085.71 | -2.55 | -8.94 |
| 43 | 39\43 | 1088.37 | +0.10 | +0.37 |
| 44 | 40\44 | 1090.91 | +2.64 | +9.68 |
| 53 | 48\53 | 1086.79 | -1.48 | -6.52 |
| 54 | 49\54 | 1088.89 | +0.62 | +2.79 |
| 64 | 58\64 | 1087.50 | -0.77 | -4.10 |
| 65 | 59\65 | 1089.23 | +0.96 | +5.21 |
| 75 | 68\75 | 1088.00 | -0.27 | -1.68 |
| 76 | 69\76 | 1089.47 | +1.20 | +7.63 |