172edo: Difference between revisions

(10 intermediate revisions by 8 users not shown)

Line 1:

'''172edo''' is the [[EDO|equal division of the octave]] into 172 parts of 6.9767 [[cent]]s each. It is inconsistent to the 5-limit and higher limit, with three mappings possible for the 5-limit: <172 273 399| (optimal patent val), <172 272 399| (172b), and <172 273 400| (172c). Using the patent val, it tempers out the semicomma, 2109375/2097152 and 1220703125/1162261467 in the 5-limit; 245/243, 3125/3087, and 2097152/2066715 in the 7-limit; 385/384, 896/891, 1331/1323, and 9375/9317 in the 11-limit. Using the 172b val, it tempers out the unicorn comma, 1594323/1562500 and 2197265625/2147483648 in the 5-limit; 1728/1715, 3645/3584, and 390625/388962 in the 7-limit; 441/440, 1944/1925, 4000/3993, and 4125/4096 in the 11-limit. Using the alternative 172bdee val, it tempers out 225/224, 118098/117649, and 3176523/3125000 in the 7-limit; 243/242, 1617/1600, 2079/2048, and 117649/117128 in the 11-limit. Using the 172c val, it tempers out the [[Diaschismic family|diaschisma]], 2048/2025 and |1 36 -25> in the 5-limit; 4375/4374, 50421/50000, and 110592/109375 in the 7-limit; 176/175, 896/891, and 1331/1323 in the 11-limit.

[[~~Category~~:~~Edo~~]]

172edo is in[[consistent]] to the [[5-odd-limit]] and higher limits, with three mappings possible for the 5-limit: {{val| 172 273 399 }} ([[patent val]]), {{val| 172 '''272''' 399 }} (172b), and {{val| 172 273 '''400''' }} (172c). Using the patent val, it [[tempering out|tempers out]] the [[semicomma]], 2109375/2097152 and 1220703125/1162261467 in the 5-limit; [[245/243]], [[3125/3087]], and 2097152/2066715 in the 7-limit, [[support]]ing the 7-limit [[bohpier]] temperament; [[385/384]], [[896/891]], 1331/1323, and 9375/9317 in the 11-limit; [[169/168]], [[352/351]], [[364/363]], and [[1716/1715]] in the 13-limit, supporting the [[leapweek]] temperament. Using the 172f val, [[275/273]], [[640/637]], [[847/845]], and [[1575/1573]] are tempered out in the 13-limit.

Using the 172c val, it tempers out the [[diaschisma]], 2048/2025 and {{monzo| 1 36 -25 }} in the 5-limit; [[4375/4374]], [[50421/50000]], and 110592/109375 in the 7-limit; [[176/175]], [[896/891]], and 1331/1323 in the 11-limit. Using the alternative 172cf val, [[196/195]], 1716/1715, [[2080/2079]], [[2197/2187]], and [[2200/2197]] are tempered out in the 13-limit. Using the alternative 172cef val, it tempers out [[441/440]], 1344/1331, and [[3388/3375]] in the 11-limit; 196/195, [[352/351]], [[832/825]], [[1001/1000]], and 2197/2187 in the 13-limit.

Using the 172b val, it tempers out the [[unicorn comma]], 1594323/1562500 and 2197265625/2147483648 in the 5-limit; [[1728/1715]], [[3645/3584]], and [[390625/388962]] in the 7-limit; [[441/440]], 1944/1925, [[4000/3993]], and 4125/4096 in the 11-limit; [[625/624]], 975/968, [[1188/1183]], [[1287/1280]], and [[1573/1568]] in the 13-limit. Using the alternative 172bdee val, it tempers out [[225/224]], 118098/117649, and 3176523/3125000 in the 7-limit; [[243/242]], 1617/1600, 2079/2048, and 117649/117128 in the 11-limit; [[351/350]], 625/624, 1188/1183, and 1573/1568 in the 13-limit.

Using the 172f val, every 7 steps is an excellent approximation of the ninth no-2 zeta peak - associated with [[39edt]] - in the 15-limit.

=== Odd harmonics ===

=== Subsets and supersets ===

Since 172 factors into {{factorization|172}}, 172edo has subset edos {{EDOs| 2, 4, 43, and 86 }}.

@@ Line 1: / Line 1: @@
-'''172edo''' is the [[EDO|equal division of the octave]] into 172 parts of 6.9767 [[cent]]s each. It is inconsistent to the 5-limit and higher limit, with three mappings possible for the 5-limit: &lt;172 273 399| (optimal patent val), &lt;172 272 399| (172b), and &lt;172 273 400| (172c). Using the patent val, it tempers out the semicomma, 2109375/2097152 and 1220703125/1162261467 in the 5-limit; 245/243, 3125/3087, and 2097152/2066715 in the 7-limit; 385/384, 896/891, 1331/1323, and 9375/9317 in the 11-limit. Using the 172b val, it tempers out the unicorn comma, 1594323/1562500 and 2197265625/2147483648 in the 5-limit; 1728/1715, 3645/3584, and 390625/388962 in the 7-limit; 441/440, 1944/1925, 4000/3993, and 4125/4096 in the 11-limit. Using the alternative 172bdee val, it tempers out 225/224, 118098/117649, and 3176523/3125000 in the 7-limit; 243/242, 1617/1600, 2079/2048, and 117649/117128 in the 11-limit. Using the 172c val, it tempers out the [[Diaschismic family|diaschisma]], 2048/2025 and |1 36 -25&gt; in the 5-limit; 4375/4374, 50421/50000, and 110592/109375 in the 7-limit; 176/175, 896/891, and 1331/1323 in the 11-limit.
+{{Infobox ET}}
+{{ED intro}}
-[[Category:Edo]]
+edo is in[[consistent]] to the [[5-odd-limit]] and higher limits, with three mappings possible for the 5-limit: {{val| 172 273 399 }} ([[patent val]]), {{val| 172 '''272''' 399 }} (172b), and {{val| 172 273 '''400''' }} (172c). Using the patent val, it [[tempering out|tempers out]] the [[semicomma]], 2109375/2097152 and 1220703125/1162261467 in the 5-limit; [[245/243]], [[3125/3087]], and 2097152/2066715 in the 7-limit, [[support]]ing the 7-limit [[bohpier]] temperament; [[385/384]], [[896/891]], 1331/1323, and 9375/9317 in the 11-limit; [[169/168]], [[352/351]], [[364/363]], and [[1716/1715]] in the 13-limit, supporting the [[leapweek]] temperament. Using the 172f val, [[275/273]], [[640/637]], [[847/845]], and [[1575/1573]] are tempered out in the 13-limit.
+Using the 172c val, it tempers out the [[diaschisma]], 2048/2025 and {{monzo| 1 36 -25 }} in the 5-limit; [[4375/4374]], [[50421/50000]], and 110592/109375 in the 7-limit; [[176/175]], [[896/891]], and 1331/1323 in the 11-limit. Using the alternative 172cf val, [[196/195]], 1716/1715, [[2080/2079]], [[2197/2187]], and [[2200/2197]] are tempered out in the 13-limit. Using the alternative 172cef val, it tempers out [[441/440]], 1344/1331, and [[3388/3375]] in the 11-limit; 196/195, [[352/351]], [[832/825]], [[1001/1000]], and 2197/2187 in the 13-limit.
+Using the 172b val, it tempers out the [[unicorn comma]], 1594323/1562500 and 2197265625/2147483648 in the 5-limit; [[1728/1715]], [[3645/3584]], and [[390625/388962]] in the 7-limit; [[441/440]], 1944/1925, [[4000/3993]], and 4125/4096 in the 11-limit; [[625/624]], 975/968, [[1188/1183]], [[1287/1280]], and [[1573/1568]] in the 13-limit. Using the alternative 172bdee val, it tempers out [[225/224]], 118098/117649, and 3176523/3125000 in the 7-limit; [[243/242]], 1617/1600, 2079/2048, and 117649/117128 in the 11-limit; [[351/350]], 625/624, 1188/1183, and 1573/1568 in the 13-limit.
+Using the 172f val, every 7 steps is an excellent approximation of the ninth no-2 zeta peak - associated with [[39edt]] - in the 15-limit.
+=== Odd harmonics ===
+{{Harmonics in equal|172}}
+=== Subsets and supersets ===
+Since 172 factors into {{factorization|172}}, 172edo has subset edos {{EDOs| 2, 4, 43, and 86 }}.