21/20: Difference between revisions

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{{Infobox Interval
{{Infobox Interval
| Name = septimal minor semitone, septimal chromatic semitone, large septimal chroma
| Name = septimal chromatic semitone, aberschismic diatonic semitone, minor diatonic semitone
| Color name = zg2, zogu 2nd
| Color name = zg2, zogu 2nd
| Sound = jid_21_20_pluck_adu_dr220.mp3
| Sound = jid_21_20_pluck_adu_dr220.mp3
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{{Wikipedia|Septimal chromatic semitone}}
{{Wikipedia|Septimal chromatic semitone}}


'''21/20''' is a small semitone of about 85 cents. It may be found in [[7-limit]] [[just intonation]] as, for example, the difference between [[4/3]] and [[7/5]], [[8/7]] and [[6/5]], or [[5/3]] and [[7/4]].  
'''21/20''' is a small semitone in [[7-limit]] [[just intonation]] of about 84.5 [[cents]], traditionally called the '''septimal chromatic semitone''' for its proximity (and conflation in systems like septimal [[meantone]]) with the major chroma [[135/128]]. However, it functions as a ''[[diatonic semitone]]'', as is supported by [[Sagittal notation]], [[Helmholtz–Ellis notation]] and [[Functional Just System]], viewed as the [[Pythagorean limma]] altered by an [[aberschisma]]. This gives rise to the more precise name '''aberschismic diatonic semitone''', or as [[Marc Sabat]] has taken to call it, the '''minor diatonic semitone'''<ref>Marc Sabat. [https://masa.plainsound.org/pdfs/crystal-growth.pdf ''Three Crystal Growth Algorithms in 23-limit constrained Harmonic Space'']. Plainsound Music Edition, 2008.</ref>. It may be found as, for example, the difference between [[4/3]] and [[7/5]], between [[8/7]] and [[6/5]], or between [[5/3]] and [[7/4]].


== Terminology ==
In what is known as an authentic cadence, there is a resolution from the V chord to the I chord. If the V is a [[4:5:6:7|harmonic seventh chord]], its harmonic seventh ([[21/16]] above the tonic) resolves down to the major third of the I chord ([[5/4]]) by a step of 21/20.
21/20 is traditionally called a ''chroma'', perhaps for its proximity (and conflation in systems like septimal [[meantone]]) with the major chroma [[135/128]]. However, it is a ''diatonic semitone'' in both [[Helmholtz–Ellis notation]] and [[Functional Just System]], viewed as the Pythagorean minor second [[256/243]] altered by [[5120/5103]]. [[Marc Sabat]] has taken to call it the ''minor diatonic semitone'' in the same material where [[15/14]] is also named as the major chromatic semitone<ref>Marc Sabat. [https://masa.plainsound.org/pdfs/crystal-growth.pdf ''Three Crystal Growth Algorithms in 23-limit constrained Harmonic Space'']. Plainsound Music Edition, 2008.</ref>.  
 
== Approximation ==
{{Interval edo approximation|21/20}}
 
== Interval chain ==
An [[interval chain]] of 21/20's stacked on top of one another comes close to approximating some important [[JI]] intervals. The error between the approximation and the target JI interval may be tempered out in some [[regular temperaments]].
 
Some examples include:
* A stack of two 21/20's upwards is ~4{{c}} from [[11/10]];
* A stack of seven 21/20's upwards is ~9{{c}} from [[7/5]];
* A stack of ten 21/20's upwards is ~4{{c}} from [[13/8]];
* A stack of twelve 21/20's upwards is ~4{{c}} from [[9/5]];
and
* A stack of six 21/20's downwards is ~10{{c}} from [[3/2]];
* A stack of nine 21/20's downwards is ~5{{c}} from [[9/7]];
* A stack of eleven 21/20's downwards is ~4{{c}} from [[7/6]].  
 
When treated as a scale, this interval chain can be called the '''[[ambitonal sequence]] of 21/20''' ('''AS21/20''' or '''1ed21/20''').
 
1ed21/20 is equal to approximately 14.2067edo, and as a result of tethering between compressed 14 and heavily stretched 15. It is quite [[xenharmonic]] in its sound. It is related to the [[nautilus]], [[sextilifourths]] and [[floral]] temperaments.
 
1ed21/20 offers a possible approximation of the no-3s [[11-limit]], or alternatively of the 2.9.5.7.11.17 [[subgroup]].
 
{{Harmonics in equal|1|21|20|intervals=integer|columns=11}}
{{Harmonics in equal|1|21|20|intervals=integer|collapsed=1|start=12|columns=12}}


== See also ==
== See also ==
* [[40/21]] – its [[octave complement]]
* [[40/21]] – its [[octave complement]]
* [[10/7]] – its [[fifth complement]]
* [[10/7]] – its [[fifth complement]]
* [[1ed21/20]] – its [[ambitonal sequence]]
* [[List of superparticular intervals]]
* [[List of superparticular intervals]]
* [[Gallery of just intervals]]
* [[Gallery of just intervals]]
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[[Category:Septisemi]]
[[Category:Septisemi]]
[[Category:Commas named after their interval size]]
[[Category:Commas named after their interval size]]
{{todo|improve synopsis}}
{{Todo|improve synopsis}}

Latest revision as of 11:07, 1 August 2026

Interval information
Ratio 21/20
Factorization 2-2 × 3 × 5-1 × 7
Monzo [-2 1 -1 1⟩
Size in cents 84.46719¢
Names septimal chromatic semitone,
aberschismic diatonic semitone,
minor diatonic semitone
Color name zg2, zogu 2nd
FJS name [math]\displaystyle{ \text{m2}^{7}_{5} }[/math]
Special properties superparticular,
reduced
Tenney norm (log2 nd) 8.71425
Weil norm (log2 max(n, d)) 8.78463
Wilson norm (sopfr(nd)) 19
Comma size medium
S-expressions S6⋅S7,
S7⋅S8⋅S9

[sound info]
Open this interval in xen-calc
English Wikipedia has an article on:

21/20 is a small semitone in 7-limit just intonation of about 84.5 cents, traditionally called the septimal chromatic semitone for its proximity (and conflation in systems like septimal meantone) with the major chroma 135/128. However, it functions as a diatonic semitone, as is supported by Sagittal notation, Helmholtz–Ellis notation and Functional Just System, viewed as the Pythagorean limma altered by an aberschisma. This gives rise to the more precise name aberschismic diatonic semitone, or as Marc Sabat has taken to call it, the minor diatonic semitone[1]. It may be found as, for example, the difference between 4/3 and 7/5, between 8/7 and 6/5, or between 5/3 and 7/4.

In what is known as an authentic cadence, there is a resolution from the V chord to the I chord. If the V is a harmonic seventh chord, its harmonic seventh (21/16 above the tonic) resolves down to the major third of the I chord (5/4) by a step of 21/20.

Approximation

Edo approximations for 21/20 (84.47 ¢)
≤ 80edo, relative error ≤ 10%
Edo Step size Cents (¢) Absolute error (¢) Relative error (%)
13 1\13 92.31 +7.84 +8.49
14 1\14 85.71 +1.25 +1.45
15 1\15 80.00 -4.47 -5.58
27 2\27 88.89 +4.42 +9.95
28 2\28 85.71 +1.25 +2.91
29 2\29 82.76 -1.71 -4.13
42 3\42 85.71 +1.25 +4.36
43 3\43 83.72 -0.75 -2.67
44 3\44 81.82 -2.65 -9.71
56 4\56 85.71 +1.25 +5.82
57 4\57 84.21 -0.26 -1.22
58 4\58 82.76 -1.71 -8.26
70 5\70 85.71 +1.25 +7.27
71 5\71 84.51 +0.04 +0.24
72 5\72 83.33 -1.13 -6.80

Interval chain

An interval chain of 21/20's stacked on top of one another comes close to approximating some important JI intervals. The error between the approximation and the target JI interval may be tempered out in some regular temperaments.

Some examples include:

  • A stack of two 21/20's upwards is ~4 ¢ from 11/10;
  • A stack of seven 21/20's upwards is ~9 ¢ from 7/5;
  • A stack of ten 21/20's upwards is ~4 ¢ from 13/8;
  • A stack of twelve 21/20's upwards is ~4 ¢ from 9/5;

and

  • A stack of six 21/20's downwards is ~10 ¢ from 3/2;
  • A stack of nine 21/20's downwards is ~5 ¢ from 9/7;
  • A stack of eleven 21/20's downwards is ~4 ¢ from 7/6.

When treated as a scale, this interval chain can be called the ambitonal sequence of 21/20 (AS21/20 or 1ed21/20).

1ed21/20 is equal to approximately 14.2067edo, and as a result of tethering between compressed 14 and heavily stretched 15. It is quite xenharmonic in its sound. It is related to the nautilus, sextilifourths and floral temperaments.

1ed21/20 offers a possible approximation of the no-3s 11-limit, or alternatively of the 2.9.5.7.11.17 subgroup.


Approximation of harmonics in 1ed21/20
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -17.5 +40.8 -34.9 +1.1 +23.3 +9.9 +32.1 -2.9 -16.4 -12.4 +5.9
Relative (%) -20.7 +48.3 -41.3 +1.3 +27.6 +11.7 +38.0 -3.4 -19.4 -14.7 +7.0
Step 14 23 28 33 37 40 43 45 47 49 51
Approximation of harmonics in 1ed21/20
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +36.2 -7.6 +41.9 +14.6 -5.9 -20.3 -29.5 -33.8 -33.8 -29.9 -22.4 -11.6
Relative (%) +42.9 -9.0 +49.6 +17.3 -6.9 -24.1 -34.9 -40.0 -40.0 -35.4 -26.5 -13.7
Step 53 54 56 57 58 59 60 61 62 63 64 65

See also

References

  1. ↑ Marc Sabat. Three Crystal Growth Algorithms in 23-limit constrained Harmonic Space. Plainsound Music Edition, 2008.
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