24/17: Difference between revisions
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{{Infobox Interval | {{Infobox Interval | ||
| Name = small septendecimal tritone | |||
| Color name = 17u4, su 4th | |||
| Name = | |||
| Color name = | |||
| Sound = jid_24_17_pluck_adu_dr220.mp3 | | Sound = jid_24_17_pluck_adu_dr220.mp3 | ||
}} | }} | ||
In [[17-limit]] [[just intonation]], '''24/17''' is the | In [[17-limit]] [[just intonation]], '''24/17''' is the '''small septendecimal tritone''', measuring very nearly 597¢. It is the [[mediant]] between [[7/5]] and [[17/12]], the "larger septendecimal tritone". The two septendecimal tritones are each 3¢ away from the 600¢ half-octave, and so they are well-represented in all even-numbered [[edo]] systems, including [[12edo]]. Indeed, the latter system, containing good approximations of the 3rd and 17th harmonics, can use the half-octave as 24/17 and 17/12 in close approximations to chords such as 8:12:17 and 16:17:24. [[22edo]] is another good edo system for using the half-octave in this way. | ||
''See also | == Terminology and notation == | ||
Conceptualization systems disagree on whether [[17/16]] should be a [[diatonic semitone]] or a [[chromatic semitone]], and as a result the disagreement propagates to all intervals of [[harmonic class|HC17]]. See [[17-limit]] for a detailed discussion. | |||
For 24/17 specifically: | |||
* In [[Functional Just System]], it is an augmented fourth, separated by [[4131/4096]] from the [[729/512|Pythagorean augmented fourth (729/512)]]. | |||
* In [[Helmholtz-Ellis notation]], it is a diminished fifth, separated by [[2187/2176]] from the [[1024/729|Pythagorean diminished fifth (1024/729)]]. | |||
The term ''small septendecimal tritone'' omits the distinction and only describes its melodic property i.e. the size. It is said in contrast to the large septendecimal tritone of [[17/12]]. | |||
== Approximation == | |||
{{Interval edo approximation|24/17}} | |||
== See also == | |||
* [[17/12]] – its [[octave complement]] | |||
* [[17/16]] – its [[fifth complement]] | |||
* [[Gallery of just intervals]] | |||
[[Category:Tritone]] | [[Category:Tritone]] | ||
Latest revision as of 13:15, 3 November 2025
| Interval information |
[sound info]
In 17-limit just intonation, 24/17 is the small septendecimal tritone, measuring very nearly 597¢. It is the mediant between 7/5 and 17/12, the "larger septendecimal tritone". The two septendecimal tritones are each 3¢ away from the 600¢ half-octave, and so they are well-represented in all even-numbered edo systems, including 12edo. Indeed, the latter system, containing good approximations of the 3rd and 17th harmonics, can use the half-octave as 24/17 and 17/12 in close approximations to chords such as 8:12:17 and 16:17:24. 22edo is another good edo system for using the half-octave in this way.
Terminology and notation
Conceptualization systems disagree on whether 17/16 should be a diatonic semitone or a chromatic semitone, and as a result the disagreement propagates to all intervals of HC17. See 17-limit for a detailed discussion.
For 24/17 specifically:
- In Functional Just System, it is an augmented fourth, separated by 4131/4096 from the Pythagorean augmented fourth (729/512).
- In Helmholtz-Ellis notation, it is a diminished fifth, separated by 2187/2176 from the Pythagorean diminished fifth (1024/729).
The term small septendecimal tritone omits the distinction and only describes its melodic property i.e. the size. It is said in contrast to the large septendecimal tritone of 17/12.
Approximation
| Edo | Step size | Cents (¢) | Absolute error (¢) | Relative error (%) |
|---|---|---|---|---|
| 2 | 1\2 | 600.00 | +3.00 | +0.50 |
| 4 | 2\4 | 600.00 | +3.00 | +1.00 |
| 6 | 3\6 | 600.00 | +3.00 | +1.50 |
| 8 | 4\8 | 600.00 | +3.00 | +2.00 |
| 10 | 5\10 | 600.00 | +3.00 | +2.50 |
| 12 | 6\12 | 600.00 | +3.00 | +3.00 |
| 14 | 7\14 | 600.00 | +3.00 | +3.50 |
| 16 | 8\16 | 600.00 | +3.00 | +4.00 |
| 18 | 9\18 | 600.00 | +3.00 | +4.50 |
| 20 | 10\20 | 600.00 | +3.00 | +5.00 |
| 22 | 11\22 | 600.00 | +3.00 | +5.50 |
| 24 | 12\24 | 600.00 | +3.00 | +6.00 |
| 26 | 13\26 | 600.00 | +3.00 | +6.50 |
| 28 | 14\28 | 600.00 | +3.00 | +7.00 |
| 30 | 15\30 | 600.00 | +3.00 | +7.50 |
| 32 | 16\32 | 600.00 | +3.00 | +8.00 |
| 34 | 17\34 | 600.00 | +3.00 | +8.50 |
| 36 | 18\36 | 600.00 | +3.00 | +9.00 |
| 38 | 19\38 | 600.00 | +3.00 | +9.50 |