168edo: Difference between revisions

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'''168edo''' is the [[EDO|equal division of the octave]] into 168 parts of 7.1429 cents each. It is closely related to [[84edo]], but the patent vals differ on the mapping for 11 and 17. It is contorted in the 7-limit, tempering out 225/224, 1728/1715, and 78732/78125. Using the patent val, it tempers out 243/242, 2420/2401, 3025/3024, and 43923/43750 in the 11-limit; 351/350, 625/624, 640/637, 847/845, and 1573/1568 in the 13-limit; 375/374, 561/560, 715/714, 891/884, 936/935, and 1331/1326 in the 17-limit. Using the 168d val, it tempers out 3136/3125, 19683/19600, and 33075/32768 in the 7-limit; 243/242, 385/384, 3773/3750, and 9801/9800 in the 11-limit. Stacking alternating steps of 43 and 53 produces an optimal Whitewood [14] scale of 19 5 19 5 19 5 19 5 19 5 19 5 19 5 that spreads the overall flatness evenly between the major and minor thirds.

[[~~Category:Equal divisions~~ of the ~~octave~~]]

== Theory ==

168edo is closely related to [[84edo]], but the [[patent val]]s differ on the mapping for [[11/1|11]] and [[17/1|17]]. It is [[contorted]] in the 7-limit, [[tempering out]] [[225/224]], [[1728/1715]], and [[78732/78125]]. Using the patent val, it tempers out [[243/242]], 2420/2401, [[3025/3024]], and 43923/43750 in the 11-limit; [[351/350]], [[625/624]], [[640/637]], [[847/845]], and [[1573/1568]] in the 13-limit; [[375/374]], [[561/560]], [[715/714]], 891/884, [[936/935]], and 1331/1326 in the 17-limit. Using the 168d val, it tempers out [[3136/3125]], [[19683/19600]], and 33075/32768 in the 7-limit; 243/242, [[385/384]], 3773/3750, and [[9801/9800]] in the 11-limit.

Stacking alternating steps of 43 and 53 produces an optimal [[Whitewood]][14] scale of 19 5 19 5 19 5 19 5 19 5 19 5 19 5 that spreads the overall flatness evenly between the major and minor thirds. [[Substitute harmonic#Dotcom|Dotcom]] is also supported.

=== Odd harmonics ===

=== Subsets and supersets ===

Since 168 factors into 2<sup>3</sup> × 3 × 7, 168edo has subset edos {{EDOs| 2, 3, 4, 6, 7, 8, 12, 14, 21, 24, 28, 42, 56, and 84 }}.

== Notation ==

168edo can be notated using [[ups and downs notation]], in which [[Helmholtz–Ellis]] arrow accidentals can be used in combination with Stein–Zimmerman [[24edo#Notation|quarter tone]] accidentals:

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-'''168edo''' is the [[EDO|equal division of the octave]] into 168 parts of 7.1429 cents each. It is closely related to [[84edo]], but the patent vals differ on the mapping for 11 and 17. It is contorted in the 7-limit, tempering out 225/224, 1728/1715, and 78732/78125. Using the patent val, it tempers out 243/242, 2420/2401, 3025/3024, and 43923/43750 in the 11-limit; 351/350, 625/624, 640/637, 847/845, and 1573/1568 in the 13-limit; 375/374, 561/560, 715/714, 891/884, 936/935, and 1331/1326 in the 17-limit. Using the 168d val, it tempers out 3136/3125, 19683/19600, and 33075/32768 in the 7-limit; 243/242, 385/384, 3773/3750, and 9801/9800 in the 11-limit. Stacking alternating steps of 43 and 53 produces an optimal Whitewood [14] scale of 19 5 19 5 19 5 19 5 19 5 19 5 19 5 that spreads the overall flatness evenly between the major and minor thirds.
+{{Infobox ET}}
+{{ED intro}}
-[[Category:Equal divisions of the octave]]
+== Theory ==
+edo is closely related to [[84edo]], but the [[patent val]]s differ on the mapping for [[11/1|11]] and [[17/1|17]]. It is [[contorted]] in the 7-limit, [[tempering out]] [[225/224]], [[1728/1715]], and [[78732/78125]]. Using the patent val, it tempers out [[243/242]], 2420/2401, [[3025/3024]], and 43923/43750 in the 11-limit; [[351/350]], [[625/624]], [[640/637]], [[847/845]], and [[1573/1568]] in the 13-limit; [[375/374]], [[561/560]], [[715/714]], 891/884, [[936/935]], and 1331/1326 in the 17-limit. Using the 168d val, it tempers out [[3136/3125]], [[19683/19600]], and 33075/32768 in the 7-limit; 243/242, [[385/384]], 3773/3750, and [[9801/9800]] in the 11-limit.
+Stacking alternating steps of 43 and 53 produces an optimal [[Whitewood]][14] scale of 19 5 19 5 19 5 19 5 19 5 19 5 19 5 that spreads the overall flatness evenly between the major and minor thirds. [[Substitute harmonic#Dotcom|Dotcom]] is also supported.
+=== Odd harmonics ===
+{{Harmonics in equal|168}}
+=== Subsets and supersets ===
+Since 168 factors into 2<sup>3</sup> × 3 × 7, 168edo has subset edos {{EDOs| 2, 3, 4, 6, 7, 8, 12, 14, 21, 24, 28, 42, 56, and 84 }}.
+== Notation ==
+edo can be notated using [[ups and downs notation]], in which [[Helmholtz–Ellis]] arrow accidentals can be used in combination with Stein–Zimmerman [[24edo#Notation|quarter tone]] accidentals:
+{{sharpness-sharp14-qt1|168}}