1216/1215: Difference between revisions
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{{Infobox Interval | {{Infobox Interval | ||
| Name = password, Eratosthenes' comma | |||
| Color name = s19og2, sanogu 2nd,<br>Sanogu comma | |||
| Comma = yes | |||
| Name = password, | |||
| Color name = | |||
| | |||
}} | }} | ||
'''1216/1215''', the '''password''' or '''Eratosthenes' comma''', is | '''1216/1215''', the '''password''' or '''Eratosthenes' comma''', is an [[unnoticeable comma|unnoticeable]] [[19-limit]] (also 2.3.5.19-[[subgroup]]) [[comma]]. It is the amount by which [[19/15]] exceeds the [[81/64|Pythagorean major third (81/64)]], or [[20/19]] falls short of the [[256/243|Pythagorean minor second (256/243)]]. It is also the difference between [[19/18]], the undevicesimal semitone and [[135/128]], the major chroma, and in addition, between the [[513/512|undevicesimal comma]] and the [[schisma]]. | ||
By tempering | == Commatic relations == | ||
This comma is the difference between the following superparticular pairs: | |||
* [[76/75]] and [[81/80]] * | |||
* [[136/135]] and [[153/152]] | |||
* [[190/189]] and [[225/224]] | |||
* [[256/255]] and [[324/323]] | |||
* [[352/351]] and [[495/494]] | |||
* [[361/360]] and [[513/512]] * | |||
* [[456/455]] and [[729/728]] | |||
* [[676/675]] and [[1521/1520]] | |||
* [[1156/1155]] and [[23409/23408]] | |||
<nowiki>*</nowiki> both of these commas are also within the 2.3.5.19 subgroup. | |||
It factors into the following superparticular pairs: | |||
* [[2431/2430]] and [[2432/2431]] | |||
* [[2080/2079]] and [[2926/2925]] | |||
* [[1729/1728]] and [[4096/4095]] | |||
* [[1540/1539]] and [[5776/5775]] | |||
* [[1225/1224]] and [[165376/165375]] | |||
== Temperaments == | |||
By tempering out this comma is defined the '''eratosthenes temperament''', which enables the [[eratosthenes chords]]. Edos supporting this temperament includes {{Optimal ET sequence| 12, 29, 41, 53, 65, 77, 87, 94, 99, 106, 111, 118, 130, 140, 152, 159, 183, 193, 205, 217, 270, 282, 311, 323, 364, 400, 422, 581, and 692 }}. | |||
== Etymology == | |||
This comma was named after {{w|Eratosthenes}}, the Ancient Greek polymath. According to an analysis by [[Joseph Monzo]], Eratosthenes made use of this comma by employing "standard" [[3-limit]] pitches in his diatonic [[genus]], but substituting 19-limit ratios which are very close by in pitch in his chromatic and enharmonic genera.<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_26618.html Yahoo! Tuning Group | ''ratios in ancient Greek tetrachords (was: Byzantine and Serb Church Music)'']</ref> | |||
== See also == | == See also == | ||
* [[List of superparticular intervals]] | * [[List of superparticular intervals]] | ||
== Notes == | |||
[[Category:Eratosthenes]] | [[Category:Eratosthenes]] | ||
[[Category:Commas named after polymaths]] | |||
[[Category:Commas with unknown etymology]]{{todo|add etymology|comment=add etymology for “password”}} | |||
Latest revision as of 02:16, 16 November 2024
| Interval information |
Eratosthenes' comma
Sanogu comma
reduced
1216/1215, the password or Eratosthenes' comma, is an unnoticeable 19-limit (also 2.3.5.19-subgroup) comma. It is the amount by which 19/15 exceeds the Pythagorean major third (81/64), or 20/19 falls short of the Pythagorean minor second (256/243). It is also the difference between 19/18, the undevicesimal semitone and 135/128, the major chroma, and in addition, between the undevicesimal comma and the schisma.
Commatic relations
This comma is the difference between the following superparticular pairs:
- 76/75 and 81/80 *
- 136/135 and 153/152
- 190/189 and 225/224
- 256/255 and 324/323
- 352/351 and 495/494
- 361/360 and 513/512 *
- 456/455 and 729/728
- 676/675 and 1521/1520
- 1156/1155 and 23409/23408
* both of these commas are also within the 2.3.5.19 subgroup.
It factors into the following superparticular pairs:
- 2431/2430 and 2432/2431
- 2080/2079 and 2926/2925
- 1729/1728 and 4096/4095
- 1540/1539 and 5776/5775
- 1225/1224 and 165376/165375
Temperaments
By tempering out this comma is defined the eratosthenes temperament, which enables the eratosthenes chords. Edos supporting this temperament includes 12, 29, 41, 53, 65, 77, 87, 94, 99, 106, 111, 118, 130, 140, 152, 159, 183, 193, 205, 217, 270, 282, 311, 323, 364, 400, 422, 581, and 692.
Etymology
This comma was named after Eratosthenes, the Ancient Greek polymath. According to an analysis by Joseph Monzo, Eratosthenes made use of this comma by employing "standard" 3-limit pitches in his diatonic genus, but substituting 19-limit ratios which are very close by in pitch in his chromatic and enharmonic genera.[1]