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Optimal bases for 7-limit planar temperaments

🔗Gene W Smith <genewardsmith@juno.com>

5/30/2002 8:02:28 AM

I found these by first taking the Hermite form of the generator matrix,
and then finding a basis which minimized the 7-limit complexity (sums of
squares of lattice distances.) These should be very useful in finding
scales associated to these temperaments.

25/24 <5/4, 21/20>

28/27 <4/3, 6/5>

250/243 <7/6, 10/9>

36/35 <4/3, 6/5>

49/48 <8/7, 35/32>

50/49 <5/4, 6/5>; 1/2 octave period

64/63 <4/3, 5/4>

875/874 <6/5, 25/24>

81/80 <4/3, 9/7>

2048/2025 <4/3, 8/7> 1/2 octave period

245/243 <4/3, 9/7> or <9/7, 7/6>

126/125 <5/4, 6/5>

4000/3969 <4/3, 80/63>

1728/1715 <7/6, 49/48>

1029/1024 <8/7, 35/32>

225/224 <4/3, 16/15> or <5/4, 16/15>

3136/3125 <6/5, 28/25>

5120/5103 <4/3, 27/20>

6144/6125 <6/5, 35/32>

2401/2400 <7/5, 60/49>

4375/4374 <6/5, 27/25>

250047/250000 <4/3, 6/5> or <4/3, 10/9> 1/3 octave period