7th-octave temperaments

From Xenharmonic Wiki
(Redirected from Nitrogen)
Jump to navigation Jump to search

a 7th-octave temperament can be described by temperament merging of edos whose greatest common divisor is 7. The most notable 7th-octave family is the whitewood family – tempering out 2187/2048 and associating 4\7 to 3/2.

A comma that frequently appears in 7th-octave temps is akjaysma, which sets 105/64 to be equal to 5\7.

Temperaments discussed elsewhere include:

Jamesbond

This temperament uses exactly the same 5-limit as 7et, but the harmonic 7 is mapped to an independent generator. It is so named because its wedgie starts with ⟨⟨0 0 7 …]].

Subgroup: 2.3.5.7

Comma list: 25/24, 81/80

Mapping[7 11 16 0], 0 0 0 1]]

Wedgie⟨⟨0 0 7 0 11 16]]

Optimal tuning (POTE): ~10/9 = 1\7, ~7/4 = 941.861

Optimal ET sequence7, 14c

Badness: 0.041714

11-limit

Subgroup: 2.3.5.7.11

Comma list: 25/24, 33/32, 45/44

Mapping: [7 11 16 0 24], 0 0 0 1 0]]

Optimal tuning (POTE): ~10/9 = 1\7, ~7/4 = 941.090

Optimal ET sequence7, 14c

Badness: 0.023524

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 25/24, 27/26, 33/32, 40/39

Mapping: [7 11 16 0 24 26], 0 0 0 1 0 0]]

Optimal tuning (POTE): ~10/9 = 1\7, ~7/4 = 949.236

Optimal ET sequence7, 14c

Badness: 0.023003

Septimal

Subgroup: 2.3.5.7.11.13

Comma list: 25/24, 33/32, 45/44, 65/63

Mapping: [7 11 16 0 24 6], 0 0 0 1 0 1]]

Optimal tuning (POTE): ~10/9 = 1\7, ~7/4 = 952.555

Optimal ET sequence7, 14cf

Badness: 0.022569

Akjaysmic (rank-3)

Subgroup: 2.3.5.7

Comma list: [47 -7 -7 -7

Mapping: [7 0 0 47], 0 1 0 -1], 0 0 1 -1]]

Mapping generators: ~1157625/1048576, ~3, ~5

POTE generators: ~3/2 = 701.965, ~5/4 = 386.330

Optimal ET sequence140, 224, 301, 441, 665, 742, 966, 1106, 1407, 1547, 1848, 2289, 2513, 2954, 3395, 4802

11-limit

Subgroup: 2.3.5.7.11

Comma list: 184549376/184528125, 199297406/199290375

Mapping: [7 0 0 47 -168], 0 1 0 -1 10], 0 0 1 -1 5]]

Mapping generators: ~29160/26411, ~3, ~5

POTE generators: ~3/2 = 701.968, ~5/4 = 386.332

Optimal ET sequence301, 364, 441, 742, 805, 1043, 1106, 1407, 1547, 1848, 2289, 2653, 2954, 3395, 4501, 5243, 6349, 8197

Nitrogen

Described as 140 & 1407 temperament in the 7-limit, named after the 7th element for being period-7 and also because 140 and 1407 only contain numbers 7 and 14, atomic number and atomic weight of nitrogen respectively. On top of this connection to the number 7, it also reaches 7th harmonic 7 generators down.

Subgroup: 2.3.5.7

Comma list: 3955078125/3954653486, 140737488355328/140710042265625

Mapping: [7 10 17 20], 0 22 -15 -7]]

Mapping generators: ~1157625/1048576, ~1029/1024

Optimal tuning (CTE): ~1157625/1048576 = 1\7, ~1029/1024 = 8.531

Optimal ET sequence140, 1407, 1547, ...

Jackpot

Subgroup: 2.3.29

Comma list: 17249876309/17179869184

Mapping[7 0 34], 0 1 0]]

mapping generators: ~32/29, ~3

Optimal tuning (CTE): ~32/29 = 1\7, ~3/2 = 701.955 (~24576/24389 = 16.239)

Supporting ETs: 7, 77, 70, 147, 224, 84, 63, 301, 217, 371, 56, 161, 91, 378