2.3.35 and 2.3.49

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2.3.35 and 2.3.49 are two subsets of the 2.3.5.7 (septimal) group that involve many ratios further from 12edo than 2.3.7. The 35th harmonic (5*7) perfectly combines the deviations of the 5th and 7th harmonics to be about as far from 12edo as possible, whereas the 49th harmonic (7*7) overshoots slightly.

Intervals

2.3.35

2.3.35 intervals are the difference between a 5-over and 7-under interval, or vice versa.

36/35 septimal quarter tone (flat of 28/27 by 245/243)

35/32 septimal neutral second

81/70 septimal semifourth

Some of the best 2.3.5.7 scales with significant emphasis on 35 are a rank-3 scale consisting of either Meantone or Archy diatonic with a 35/32 offset. One small edo to do something like this is 25, which uses the notes of 5edo to approximate Archy. Stacking two offsets to get 10L5s is particularly useful.

2.3.5.49

2.3.49 intervals are the difference between a 7-over and 7-under interval. 49/5 is further from 12edo.

54/49 larger neutral second

49/45 smaller neutral second

49/40 neutral third

2.3.5.49 tripentatonic 5L2m5s3a (blackdye A Aeolian with added chain of 5/49 offset)

81/80

54/49

9/8

6/5

64/49

4/3

27/20

72/49

3/2

8/5

256/147

16/9

9/5

96/49

2/1

<adv>Tempering out 245/243 equates the two smallest steps, leaving 5L2m8s. Tempering out 1029/1024 equates 256/147 to 7/4. Tempering out both commas leads to Rodan temperament (2.3.5.7 41 & 46), an extension of Slendric which also notably tempers out 5120/5103.</adv>

Over-49 Chain

3^	/49	*245/243	*1029/1024
-1	256/147	1280/729	7/4
0	64/49	320/243	21/16
1	96/49	160/81	63/32
2	72/49	40/27	189/128
3	54/49	10/9	567/512
4	81/49	5/3	1701/1024
5	243/196	5/4	5103/4096