10/9: Difference between revisions
Jump to navigation
Jump to search
m Added Wikipedia box, misc. edits, categories |
No edit summary |
||
| (7 intermediate revisions by 6 users not shown) | |||
| Line 1: | Line 1: | ||
{{Infobox Interval | {{Infobox Interval | ||
| Name = small whole tone, classic(al) whole tone, ptolemaic whole tone | |||
| Name = small whole tone | |||
| Color name = y2, yo 2nd | | Color name = y2, yo 2nd | ||
| Sound = jid_10_9_pluck_adu_dr220.mp3 | | Sound = jid_10_9_pluck_adu_dr220.mp3 | ||
}} | }} | ||
{{Wikipedia|Major second}} | {{Wikipedia|Major second}} | ||
In [[5-limit]] [[just intonation]], '''10/9''' is | In [[5-limit]] [[just intonation]], '''10/9''' is the '''small''', '''classic(al)''', or '''ptolemaic whole tone'''<ref>For reference, see [[5-limit]]. </ref> of about 182.4¢. It is a [[superparticular]] interval, as you can find it in the harmonic series between the 9th and the 10th overtones. It is one of two essential whole tones in the 5-limit; the other one is [[9/8]] (about 203.9¢), which is [[81/80]] (about 21.5¢) higher than 10/9. 9/8 is an octave-reduced overtone, and it is closer to [[12edo]]'s single whole step of 200¢. Thus, 9/8 is more familiar and less difficult to tune by ear than 10/9. | ||
The first three notes of a JI major scale – 1/1, 9/8, 5/4 – move by a step of 9/8 followed by a step of 10/9 (or alternatively 1/1, 10/9, 5/4 – move by a step of 10/9 followed by a step of 9/8). In systems where 81/80 is tempered out (in 12edo, [[19edo]], [[31edo]] and other [[meantone]] systems) that distinction is lost and this sounds like two equal-sized steps. In strict JI, the difference may be hard to notice at first. | The first three notes of a JI major scale – 1/1, 9/8, 5/4 – move by a step of 9/8 followed by a step of 10/9 (or alternatively 1/1, 10/9, 5/4 – move by a step of 10/9 followed by a step of 9/8). In systems where 81/80 is tempered out (in 12edo, [[19edo]], [[31edo]] and other [[meantone]] systems) that distinction is lost and this sounds like two equal-sized steps. In strict JI, the difference may be hard to notice at first. | ||
== Approximations == | |||
{{interval edo approximation}} | |||
== Temperaments == | |||
The following [[linear temperament]]s are [[generate]]d by a [[~]]10/9: | |||
* [[Porcupine]] | |||
* [[Minortone]] | |||
{{todo|complete list}} | |||
== See also == | == See also == | ||
| Line 21: | Line 26: | ||
* [[Gallery of just intervals]] | * [[Gallery of just intervals]] | ||
== Notes == | |||
<references/> | |||
[[Category:Second]] | [[Category:Second]] | ||
[[Category:Whole tone]] | [[Category:Whole tone]] | ||
[[Category: | [[Category:Tritave-reduced harmonics]] | ||
Latest revision as of 00:47, 27 November 2025
| Interval information |
classic(al) whole tone,
ptolemaic whole tone
reduced
[sound info]
In 5-limit just intonation, 10/9 is the small, classic(al), or ptolemaic whole tone[1] of about 182.4¢. It is a superparticular interval, as you can find it in the harmonic series between the 9th and the 10th overtones. It is one of two essential whole tones in the 5-limit; the other one is 9/8 (about 203.9¢), which is 81/80 (about 21.5¢) higher than 10/9. 9/8 is an octave-reduced overtone, and it is closer to 12edo's single whole step of 200¢. Thus, 9/8 is more familiar and less difficult to tune by ear than 10/9.
The first three notes of a JI major scale – 1/1, 9/8, 5/4 – move by a step of 9/8 followed by a step of 10/9 (or alternatively 1/1, 10/9, 5/4 – move by a step of 10/9 followed by a step of 9/8). In systems where 81/80 is tempered out (in 12edo, 19edo, 31edo and other meantone systems) that distinction is lost and this sounds like two equal-sized steps. In strict JI, the difference may be hard to notice at first.
Approximations
| Edo | Step size | Cents (¢) | Absolute error (¢) | Relative error (%) |
|---|---|---|---|---|
| 6 | 1\6 | 200.00 | +17.60 | +8.80 |
| 7 | 1\7 | 171.43 | -10.98 | -6.40 |
| 13 | 2\13 | 184.62 | +2.21 | +2.40 |
| 20 | 3\20 | 180.00 | -2.40 | -4.01 |
| 26 | 4\26 | 184.62 | +2.21 | +4.79 |
| 33 | 5\33 | 181.82 | -0.59 | -1.61 |
| 39 | 6\39 | 184.62 | +2.21 | +7.19 |
| 40 | 6\40 | 180.00 | -2.40 | -8.01 |
| 46 | 7\46 | 182.61 | +0.20 | +0.79 |
| 52 | 8\52 | 184.62 | +2.21 | +9.58 |
| 53 | 8\53 | 181.13 | -1.27 | -5.62 |
| 59 | 9\59 | 183.05 | +0.65 | +3.18 |
| 66 | 10\66 | 181.82 | -0.59 | -3.22 |
| 72 | 11\72 | 183.33 | +0.93 | +5.58 |
| 73 | 11\73 | 180.82 | -1.58 | -9.62 |
| 79 | 12\79 | 182.28 | -0.13 | -0.82 |
Temperaments
The following linear temperaments are generated by a ~10/9:
See also
- Decaononic - temperament which sets the tone to this interval, instead of to 9/8
- 9/5 – its octave complement
- 27/20 – its fifth complement
- 6/5 – its fourth complement
- Gallery of just intervals