28/27: Difference between revisions
Wikispaces>genewardsmith **Imported revision 244920009 - Original comment: ** |
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{{Infobox Interval | |||
| Name = septimal third-tone, small septimal chroma, subminor second, septimal minor second, septimal subminor second, trienstonic comma | |||
| Color name = z2, zo 2nd | |||
| Sound = jid_28_27_pluck_adu_dr220.mp3 | |||
| Comma = yes | |||
}} | |||
{{Wikipedia| Septimal third tone }} | |||
[[ | The [[superparticular]] interval '''28/27''', '''septimal third-tone''' has the seventh [[triangular number]] as a numerator and is the difference between [[15/14]] and [[10/9]], [[9/8]] and [[7/6]], [[9/7]] and [[4/3]], [[3/2]] and [[14/9]], [[12/7]] and [[16/9]], and [[9/5]] and [[28/15]]. | ||
== Terminology == | |||
28/27 is traditionally called the '''small septimal chroma''', perhaps for its proximity (and conflation in systems like [[septimal meantone]]) with the classic chroma, [[25/24]]. However, it is a ''[[diatonic semitone]]'' in just intonation notation systems such as [[Sagittal notation]], [[Helmholtz–Ellis notation]], and the [[Functional Just System]], viewed as the [[256/243|Pythagorean limma (256/243)]] altered by the [[64/63|septimal comma (64/63)]]. Hence, it may be described as the '''septimal minor second''' or '''septimal subminor second''' if treated as an interval in its own right. This is analogous to the septimal major second [[8/7]], which has the same relationship with [[9/8]], and such classification suggests the function of a strong leading tone added to the traditional harmony. | |||
== Approximation == | |||
This interval is very accurately approximated by [[19edo]] (1\19), and hence the [[enneadecal]] temperament. | |||
{{Interval edo approximation|28/27}} | |||
== Temperaments == | |||
If treated as a [[comma]] to be tempered out, 28/27 may be called the '''trienstonic comma''', which leads to the '''trienstonic temperaments'''. See [[Trienstonic clan]] for the rank-2 clan of temperaments where it is tempered out. | |||
== Notation == | |||
=== Sagittal notation === | |||
In the [[Sagittal]] system, this comma (possibly tempered) is represented (in a secondary role) by the sagittal {{sagittal| (|\ }} and is called the '''7 large diesis''', or '''7L''' for short, because the simplest interval it notates is 7/1 (equivalently, 7/4), as for example in C–A{{nbhsp}}{{sagittal | (|\ }}. The primary role of {{sagittal| (|\ }} is [[8505/8192 #Sagittal notation|8505/8192]] (35L). The downward version is called '''1/7L''' or '''7L down''' and is represented (in a secondary role) by {{sagittal| (!/ }}. | |||
== See also == | |||
* [[27/14]] – its [[octave complement]] | |||
* [[81/56]] – its [[fifth complement]] | |||
* [[9/7]] – its [[fourth complement]] | |||
* [[List of superparticular intervals]] | |||
* [[Gallery of just intervals]] | |||
* [[Trienstonoschisma]], the difference by which a stack of five 28/27's falls short of [[6/5]] | |||
[[Category:Second]] | |||
[[Category:Semitone]] | |||
[[Category:Third tone]] | |||
[[Category:Chroma]] | |||
[[Category:Trienstonic]] | |||
[[Category:Commas named for the intervals they stack]] | |||
{{Todo| improve synopsis }} | |||
Latest revision as of 02:56, 15 December 2025
| Interval information |
small septimal chroma,
subminor second,
septimal minor second,
septimal subminor second,
trienstonic comma
reduced
S4/S6
[sound info]
The superparticular interval 28/27, septimal third-tone has the seventh triangular number as a numerator and is the difference between 15/14 and 10/9, 9/8 and 7/6, 9/7 and 4/3, 3/2 and 14/9, 12/7 and 16/9, and 9/5 and 28/15.
Terminology
28/27 is traditionally called the small septimal chroma, perhaps for its proximity (and conflation in systems like septimal meantone) with the classic chroma, 25/24. However, it is a diatonic semitone in just intonation notation systems such as Sagittal notation, Helmholtz–Ellis notation, and the Functional Just System, viewed as the Pythagorean limma (256/243) altered by the septimal comma (64/63). Hence, it may be described as the septimal minor second or septimal subminor second if treated as an interval in its own right. This is analogous to the septimal major second 8/7, which has the same relationship with 9/8, and such classification suggests the function of a strong leading tone added to the traditional harmony.
Approximation
This interval is very accurately approximated by 19edo (1\19), and hence the enneadecal temperament.
| Edo | Step size | Cents (¢) | Absolute error (¢) | Relative error (%) |
|---|---|---|---|---|
| 18 | 1\18 | 66.67 | +3.71 | +5.56 |
| 19 | 1\19 | 63.16 | +0.20 | +0.31 |
| 20 | 1\20 | 60.00 | -2.96 | -4.93 |
| 37 | 2\37 | 64.86 | +1.90 | +5.87 |
| 38 | 2\38 | 63.16 | +0.20 | +0.62 |
| 39 | 2\39 | 61.54 | -1.42 | -4.62 |
| 40 | 2\40 | 60.00 | -2.96 | -9.87 |
| 56 | 3\56 | 64.29 | +1.32 | +6.18 |
| 57 | 3\57 | 63.16 | +0.20 | +0.94 |
| 58 | 3\58 | 62.07 | -0.89 | -4.31 |
| 59 | 3\59 | 61.02 | -1.94 | -9.56 |
| 75 | 4\75 | 64.00 | +1.04 | +6.49 |
| 76 | 4\76 | 63.16 | +0.20 | +1.25 |
| 77 | 4\77 | 62.34 | -0.62 | -4.00 |
| 78 | 4\78 | 61.54 | -1.42 | -9.25 |
Temperaments
If treated as a comma to be tempered out, 28/27 may be called the trienstonic comma, which leads to the trienstonic temperaments. See Trienstonic clan for the rank-2 clan of temperaments where it is tempered out.
Notation
Sagittal notation
In the Sagittal system, this comma (possibly tempered) is represented (in a secondary role) by the sagittal and is called the 7 large diesis, or 7L for short, because the simplest interval it notates is 7/1 (equivalently, 7/4), as for example in C–A . The primary role of is 8505/8192 (35L). The downward version is called 1/7L or 7L down and is represented (in a secondary role) by .
See also
- 27/14 – its octave complement
- 81/56 – its fifth complement
- 9/7 – its fourth complement
- List of superparticular intervals
- Gallery of just intervals
- Trienstonoschisma, the difference by which a stack of five 28/27's falls short of 6/5