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.

Fractional-octave temperaments

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← 6th-octave • 7th-octave • 8th-octave →

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

Optimal tunings:

• 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

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

Optimal tunings:

• 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