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'''22/21''' is a small [[superparticular]] [[semitone (interval region)|semitone]] of about 80.5¢ that appears in [[11-limit]] [[just intonation]], commonly known as the '''small undecimal semitone''', or '''undecimal minor semitone'''. It is the difference between [[12/11]] and [[8/7]], or between [[7/6]] and [[11/9]]. | |||
'''22/21''' is a small [[superparticular]] [[semitone]] of about 80.5¢ that appears in [[11-limit]] [[just intonation]], commonly known as the '''small undecimal semitone''', or '''undecimal minor semitone'''. It | |||
In many notation systems (e.g. [[FJS]], [[HEJI]]), it is an imperfect unison, as it is the stack of an [[33/32|undecimal quartertone (33/32)]] and a [[64/63|septimal comma (64/63)]], neither of which changes the [[scale|scale degree]] or [[interval quality|quality]]. However, it is only flat of the [[256/243|Pythagorean minor second (256/243)]] by a [[896/891|pentacircle comma (896/891)]]. For this reason it could be called the '''pentacircle minor second'''. | In many notation systems (e.g. [[FJS]], [[HEJI]]), it is an imperfect unison, as it is the stack of an [[33/32|undecimal quartertone (33/32)]] and a [[64/63|septimal comma (64/63)]], neither of which changes the [[scale|scale degree]] or [[interval quality|quality]]. However, it is only flat of the [[256/243|Pythagorean minor second (256/243)]] by a [[896/891|pentacircle comma (896/891)]]. For this reason it could be called the '''pentacircle minor second'''. | ||
Furthermore, it is close in size to [[21/20]], a 7-limit superparticular interval most commonly treated as a minor second, differing from it by [[441/440]], about 3.9¢. The single degree of [[88cET]] can function as both 21/20 and 22/21, thus [[tempering out]] 441/440. | Furthermore, it is close in size to [[21/20]], a 7-limit superparticular interval most commonly treated as a minor second, differing from it by [[441/440]], about 3.9¢. The single degree of [[88cET]] can function as both 21/20 and 22/21, thus [[tempering out]] 441/440. | ||
== Approximation == | == Approximation == | ||
10 steps of [[149edo]] appoximate 22/21 with a precision of about 1 part in 15 million, or 1 part in 100000 when measured using [[relative cent]]s. | 10 steps of [[149edo]] appoximate 22/21 with a precision of about 1 part in 15 million, or 1 part in 100000 when measured using [[relative cent]]s. | ||
{{Interval edo approximation|22/21}} | |||
== See also == | == See also == | ||
Latest revision as of 13:01, 3 November 2025
| Interval information |
undecimal minor semitone,
pentacircle minor second
reduced
[sound info]
22/21 is a small superparticular semitone of about 80.5¢ that appears in 11-limit just intonation, commonly known as the small undecimal semitone, or undecimal minor semitone. It is the difference between 12/11 and 8/7, or between 7/6 and 11/9.
In many notation systems (e.g. FJS, HEJI), it is an imperfect unison, as it is the stack of an undecimal quartertone (33/32) and a septimal comma (64/63), neither of which changes the scale degree or quality. However, it is only flat of the Pythagorean minor second (256/243) by a pentacircle comma (896/891). For this reason it could be called the pentacircle minor second.
Furthermore, it is close in size to 21/20, a 7-limit superparticular interval most commonly treated as a minor second, differing from it by 441/440, about 3.9¢. The single degree of 88cET can function as both 21/20 and 22/21, thus tempering out 441/440.
Approximation
10 steps of 149edo appoximate 22/21 with a precision of about 1 part in 15 million, or 1 part in 100000 when measured using relative cents.
| Edo | Step size | Cents (¢) | Absolute error (¢) | Relative error (%) |
|---|---|---|---|---|
| 14 | 1\14 | 85.71 | +5.18 | +6.04 |
| 15 | 1\15 | 80.00 | -0.54 | -0.67 |
| 16 | 1\16 | 75.00 | -5.54 | -7.38 |
| 29 | 2\29 | 82.76 | +2.22 | +5.37 |
| 30 | 2\30 | 80.00 | -0.54 | -1.34 |
| 31 | 2\31 | 77.42 | -3.12 | -8.05 |
| 44 | 3\44 | 81.82 | +1.28 | +4.70 |
| 45 | 3\45 | 80.00 | -0.54 | -2.01 |
| 46 | 3\46 | 78.26 | -2.28 | -8.73 |
| 59 | 4\59 | 81.36 | +0.82 | +4.03 |
| 60 | 4\60 | 80.00 | -0.54 | -2.69 |
| 61 | 4\61 | 78.69 | -1.85 | -9.40 |
| 74 | 5\74 | 81.08 | +0.54 | +3.35 |
| 75 | 5\75 | 80.00 | -0.54 | -3.36 |