2960edo: Difference between revisions
Rework theory; +subsets and supersets; mark todo (whether it supports mercury meantone must be thoroughly investigated) |
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{{ | {{ED intro}} | ||
== Theory == | == Theory == | ||
2960edo is in[[consistent]] to the [[5-odd-limit]] and [[harmonic]] [[3/1|3]] is about halfway between its steps. Otherwise it is excellent in approximating harmonics [[5/1|5]], [[9/1|9]], [[11/1|11]], [[17/1|17]], and [[19/1|19]], making it suitable for a 2.9.5.11.17.19 [[subgroup]] interpretation, with optional additions of [[7/1|7]] and [[23/1|23]], or [[21/1|21]] and [[13/1|13]]. | 2960edo is in[[consistent]] to the [[5-odd-limit]] and [[harmonic]] [[3/1|3]] is about halfway between its steps. Otherwise it is excellent in approximating harmonics [[5/1|5]], [[9/1|9]], [[11/1|11]], [[17/1|17]], and [[19/1|19]], making it suitable for a 2.9.5.11.17.19 [[subgroup]] interpretation, with optional additions of [[7/1|7]] and [[23/1|23]], or [[21/1|21]] and [[13/1|13]]. | ||
The 2960dh [[val]] {{val| 2960 4691 6873 '''8309''' 10240 10953 12099 '''12573''' }} is the unique mapping that supports both the 80th-octave temperament called [[mercury]], and the coincidentally similarly named [[ | The 2960dh [[val]] {{val| 2960 4691 6873 '''8309''' 10240 10953 12099 '''12573''' }} is the unique mapping that supports both the 80th-octave temperament called [[mercury]], and the coincidentally similarly named [[mercurial comma]], which is the difference between a stack of 5 [[19/17]] and 2 [[15/14]] with the octave. These can be arranged in [[diatonic]] pattern to sound like a [[meantone]] scale. In this case, 19/17 is mapped to 474 steps and 15/14 is mapped to 295 steps. | ||
From a regular temperament perspective, | From a regular temperament perspective, this in 2960edo can be potentially realized as [[893edo|893]] & 2960dh temperament in the 19-limit, as it maps two generators to 19/17 and 2955 generators to 15/14, which is circularly equivalent to 5 steps down in 2960edo ({{nowrap|2955 + 5 {{=}} 2960}}), corresponding to Phrygian and Locrian modes. Eliora proposes the name ''quicksilvertone'' for this regular temperament. | ||
=== Odd harmonics === | === Odd harmonics === | ||
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== Scales == | == Scales == | ||
* 474 474 295 474 474 474 295 | * 474 474 295 474 474 474 295 – mercury "meantone" (major scale) | ||
{{Todo| review }} | {{Todo| review }} | ||
Latest revision as of 12:20, 21 February 2025
| ← 2959edo | 2960edo | 2961edo → |
2960 equal divisions of the octave (abbreviated 2960edo or 2960ed2), also called 2960-tone equal temperament (2960tet) or 2960 equal temperament (2960et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 2960 equal parts of about 0.405 ¢ each. Each step represents a frequency ratio of 21/2960, or the 2960th root of 2.
Theory
2960edo is inconsistent to the 5-odd-limit and harmonic 3 is about halfway between its steps. Otherwise it is excellent in approximating harmonics 5, 9, 11, 17, and 19, making it suitable for a 2.9.5.11.17.19 subgroup interpretation, with optional additions of 7 and 23, or 21 and 13.
The 2960dh val ⟨2960 4691 6873 8309 10240 10953 12099 12573] is the unique mapping that supports both the 80th-octave temperament called mercury, and the coincidentally similarly named mercurial comma, which is the difference between a stack of 5 19/17 and 2 15/14 with the octave. These can be arranged in diatonic pattern to sound like a meantone scale. In this case, 19/17 is mapped to 474 steps and 15/14 is mapped to 295 steps.
From a regular temperament perspective, this in 2960edo can be potentially realized as 893 & 2960dh temperament in the 19-limit, as it maps two generators to 19/17 and 2955 generators to 15/14, which is circularly equivalent to 5 steps down in 2960edo (2955 + 5 = 2960), corresponding to Phrygian and Locrian modes. Eliora proposes the name quicksilvertone for this regular temperament.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -0.198 | +0.038 | +0.093 | +0.009 | +0.033 | -0.122 | -0.161 | +0.045 | +0.055 | -0.105 | +0.104 |
| Relative (%) | -48.9 | +9.3 | +22.9 | +2.2 | +8.2 | -30.2 | -39.6 | +11.0 | +13.5 | -26.0 | +25.7 | |
| Steps (reduced) |
4691 (1731) |
6873 (953) |
8310 (2390) |
9383 (503) |
10240 (1360) |
10953 (2073) |
11564 (2684) |
12099 (259) |
12574 (734) |
13001 (1161) |
13390 (1550) | |
Subsets and supersets
Since 2960 factors into 24 × 5 × 37, 2960edo has subset edos 2, 4, 5, 8, 10, 16, 20, 37, 40, 74, 80, 148, 185, 296, 370, 592, 740 and 1480.
Scales
- 474 474 295 474 474 474 295 – mercury "meantone" (major scale)