7th-octave temperaments
- This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.
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:
- Septant → Schismatic family
- Brahmagupta → Ragismic microtemperaments
- Absurdity → Syntonic–chromatic equivalence continuum
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" (a kind of mathematical object representing the temperament) starts with ⟨⟨ 0 0 7 … ]] (in fact, it is ⟨⟨ 0 0 7 0 11 16 ]])
Subgroup: 2.3.5.7
Comma list: 25/24, 81/80
Mapping: [⟨7 11 16 0], ⟨0 0 0 1]]
- mapping generators: ~10/9, ~7
- WE: ~10/9 = 172.790 ¢, ~7/4 = 949.343 ¢
- error map: ⟨+9.533 -1.261 -21.668 -0.418]
- CWE: ~10/9 = 171.429 ¢, ~7/4 = 948.499 ¢
- error map: ⟨-0.000 -16.241 -43.457 -20.327]
Optimal ET sequence: 7(d), 14c
Badness (Sintel): 1.06
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 tunings:
- WE: ~10/9 = 172.830 ¢, ~7/4 = 948.784 ¢
- CWE: ~10/9 = 171.429 ¢, ~7/4 = 946.554 ¢
Optimal ET sequence: 7(d), 14c
Badness (Sintel): 0.778
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 tunings:
- WE: ~10/9 = 172.390 ¢, ~7/4 = 954.559 ¢
- CWE: ~10/9 = 171.429 ¢, ~7/4 = 952.367 ¢
Optimal ET sequence: 7(d), 14c
Badness (Sintel): 0.951
Austinpowers
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 tunings:
- WE: ~10/9 = 172.873 ¢, ~7/4 = 960.581 ¢
- CWE: ~10/9 = 171.429 ¢, ~7/4 = 958.793 ¢
Optimal ET sequence: 7(df), 14cf
Badness (Sintel): 0.933
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
- WE: ~1157625/1048576 = 171.427811 ¢, ~3/2 = 701.962313 ¢, ~5/4 = 386.328628 ¢
- error map: ⟨-0.0053 +0.0020 +0.0043 +0.0062]
- CWE: ~1157625/1048576 = 171.428571 ¢, ~3/2 = 701.964859 ¢, ~5/4 = 386.330310 ¢
- error map: ⟨0.0000 +0.0099 +0.0166 +0.0218]
Optimal ET sequence: 56, 63, 77, 84, 140, 217, 224, 301, 441, 966, 1106, 1407, 1547, 1848, 2513, 2954, 6349, 9303, 11151, 14105, 17500, 20454
Badness (Sintel): 2.22
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
Optimal tunings:
- WE: ~29160/26411 = 171.427802 ¢, ~3/2 = 701.964561 ¢, ~5/4 = 386.329837 ¢
- CWE: ~29160/26411 = 171.428571 ¢, ~3/2 = 701.967291 ¢, ~5/4 = 386.331624 ¢
Optimal ET sequence: 301, 441, 665e, 742, 1106, 1547, 1848, 3395, 4501, 5243, 6349, 17941, 24290, 30639, 45185cde, 63126bcde, 69475bccdde, 75824bccddee
Badness (Sintel): 1.32
Nitrogen
Nitrogen may be described as the 140 & 1407 temperament in the 7-limit. It was named after the 7th element for having a 7th-octave period 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 the 7th harmonic 7 generators down.
Subgroup: 2.3.5.7
Comma list: 3955078125/3954653486, [47 -7 -7 -7⟩
Mapping: [⟨7 10 17 20], ⟨0 22 -15 -7]]
- mapping generators: ~1157625/1048576, ~1029/1024
- WE: ~1157625/1048576 = 171.4278 ¢, ~1029/1024 = 8.5308 ¢
- error map: ⟨-0.005 +0.001 -0.002 +0.015]
- CWE: ~1157625/1048576 = 171.4286 ¢, ~1029/1024 = 8.5308 ¢
- error map: ⟨0.000 +0.008 +0.010 +0.030]
Optimal ET sequence: 140, 847, 987, 1127, 1267, 1407, 1547, 2954
Badness (Sintel): 1.50