940edo: Difference between revisions

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~~The '''940 equal division''' divides the octave into 940 equal parts of 1.277 cents each. It is uniquely [[consistent~~|consistent]] through the 11-limit, tempering out 2401/2400 in the 7-limit and 5632/5625 and 9801/9800 in the 11-limit, which means it [[support]]s [[Breedsmic_temperaments#Decoid|decoid temperament]] and in fact gives an excellent tuning for it. In the 13-limit, it tempers out 676/675, 1001/1000, 1716/1715, 2080/2079, 4096/4095 and 4225/4224, so that it supports and gives the [[Optimal_patent_val|optimal patent val]] for 13-limit decoid. It also gives the optimal patent val for [[The_Archipelago#Rank tree temperaments|greenland]] and [[The_Archipelago#Rank tree temperaments|baffin]] temperaments, and for the rank five temperament temperament tempering out 676/675. The non-patent val <940 1491 2184 2638 3254 3481| gives a tuning almost identical to the POTE tuning for the 13-limit [[Hemifamity_family#Pele|pele temperament]] tempering out 196/195, 352/351 and 364/363.

940edo is uniquely [[consistent|consistent]] through the 11-limit, tempering out 2401/2400 in the 7-limit and 5632/5625 and 9801/9800 in the 11-limit, which means it [[support]]s [[Breedsmic_temperaments#Decoid|decoid temperament]] and in fact gives an excellent tuning for it. In the 13-limit, it tempers out 676/675, 1001/1000, 1716/1715, 2080/2079, 4096/4095 and 4225/4224, so that it supports and gives the [[Optimal_patent_val|optimal patent val]] for 13-limit decoid. It also gives the optimal patent val for [[The_Archipelago#Rank tree temperaments|greenland]] and [[The_Archipelago#Rank tree temperaments|baffin]] temperaments, and for the rank five temperament temperament tempering out 676/675.

The non-patent val {{val|940 1491 2184 2638 3254 3481}} gives a tuning almost identical to the POTE tuning for the 13-limit [[Hemifamity_family#Pele|pele temperament]] tempering out 196/195, 352/351 and 364/363.

In higher limits, it is a satisfactory no-13s 23-limit tuning. Another thing to note is that its 15th harmonic is just barely off of the sum of mappings for 3rd and 5th harmonics, which adds a peculiarity to 940edo's theory in 5-prime-limit where it has good 3/2 and 5/4, but 15/8 is one step off from the "expected" location.

=== Odd harmonics ===

=== Subsets and supersets ===

940edo has subset edos {{EDOs|1, 2, 4, 5, 10, 20, 47, 94, 188, 235, 470}}, of which [[94edo]] is notable.

[[1880edo]], which doubles 940edo, provides good correction for harmonics 13 and 15 and uses a new mapping for 5 which consistently leads to 15.

[[Category:Equal divisions of the octave|###]]

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 {{Infobox ET}}
-The '''940 equal division''' divides the octave into 940 equal parts of 1.277 cents each. It is uniquely [[consistent|consistent]] through the 11-limit, tempering out 2401/2400 in the 7-limit and 5632/5625 and 9801/9800 in the 11-limit, which means it [[support]]s [[Breedsmic_temperaments#Decoid|decoid temperament]] and in fact gives an excellent tuning for it. In the 13-limit, it tempers out 676/675, 1001/1000, 1716/1715, 2080/2079, 4096/4095 and 4225/4224, so that it supports and gives the [[Optimal_patent_val|optimal patent val]] for 13-limit decoid. It also gives the optimal patent val for [[The_Archipelago#Rank tree temperaments|greenland]] and [[The_Archipelago#Rank tree temperaments|baffin]] temperaments, and for the rank five temperament temperament tempering out 676/675. The non-patent val &lt;940 1491 2184 2638 3254 3481| gives a tuning almost identical to the POTE tuning for the 13-limit [[Hemifamity_family#Pele|pele temperament]] tempering out 196/195, 352/351 and 364/363.
+{{EDO intro|94}}
+edo is uniquely [[consistent|consistent]] through the 11-limit, tempering out 2401/2400 in the 7-limit and 5632/5625 and 9801/9800 in the 11-limit, which means it [[support]]s [[Breedsmic_temperaments#Decoid|decoid temperament]] and in fact gives an excellent tuning for it. In the 13-limit, it tempers out 676/675, 1001/1000, 1716/1715, 2080/2079, 4096/4095 and 4225/4224, so that it supports and gives the [[Optimal_patent_val|optimal patent val]] for 13-limit decoid. It also gives the optimal patent val for [[The_Archipelago#Rank tree temperaments|greenland]] and [[The_Archipelago#Rank tree temperaments|baffin]] temperaments, and for the rank five temperament temperament tempering out 676/675.
+The non-patent val {{val|940 1491 2184 2638 3254 3481}} gives a tuning almost identical to the POTE tuning for the 13-limit [[Hemifamity_family#Pele|pele temperament]] tempering out 196/195, 352/351 and 364/363.
+In higher limits, it is a satisfactory no-13s 23-limit tuning. Another thing to note is that its 15th harmonic is just barely off of the sum of mappings for 3rd and 5th harmonics, which adds a peculiarity to 940edo's theory in 5-prime-limit where it has good 3/2 and 5/4, but 15/8 is one step off from the "expected" location.
+=== Odd harmonics ===
+{{harmonics in equal|940}}
+=== Subsets and supersets ===
+edo has subset edos {{EDOs|1, 2, 4, 5, 10, 20, 47, 94, 188, 235, 470}}, of which [[94edo]] is notable.
+[[1880edo]], which doubles 940edo, provides good correction for harmonics 13 and 15 and uses a new mapping for 5 which consistently leads to 15.
 [[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->