1ed33/32: Difference between revisions
Created page with "The '''equal multiplication of 33/32''', the Alpharabian quarter-tone, results in an interesting nonoctave tuning, equivalent to 22.5255 EDO. Lookalikes: 5ed7/6, 45ed4 =..." |
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{{todo|merge articles|inline=1|text=Merge into [[33/32]]?}} | |||
{{Infobox ET|1ed33/32}} | |||
'''1 equal division of 33/32''' ('''1ed33/32'''), also known as '''ambitonal sequence of 33/32''' ('''AS33/32''') or '''33/32 equal-step tuning''', is an equal multiplication of [[33/32]] (the Alpharabian quarter-tone), and results in a nonoctave tuning equivalent to 22.5255 EDO. | |||
Lookalikes: 5ed7/6, 45ed4 | Lookalikes: 5ed7/6, 45ed4 | ||
== Theory == | == Theory == | ||
{{Harmonics in equal|1|33|32|columns=11}} | {{Harmonics in equal|1|33|32|columns=11|intervals=prime}} | ||
In this tuning, 2 steps correspond to the parapotome [[1089/1024]], and | In this tuning, 2 steps by definition correspond to the parapotome [[1089/1024]]. | ||
Intervals with excellent approximation in this tuning are: 7/6 (5), 20/13 (14), 18/11 (16). Other intervals with good approximation are: 6/5, 7/5, 9/5, 13/7, 13/9, 11/10, 19/12, 17/16, 17/15, 16/15. | |||
In the 5-limit, 1ed33/32 tempers out the syntonic comma 81/80, making it meantone. | |||
In the 7-limit, as a consequence of representing 6/5 and 7/6 well, it's great at representing the 5:6:7 otonal tetrad. This means that 385/384 is tempered out. | |||
== Regular temperament properties == | |||
"Normal" subgroups calculated using the 23edo val that matches 33/32 equal step tuning patent val. | |||
{| class="wikitable" | |||
|+ | |||
!Subgroup | |||
!Comma list | |||
!Mapping | |||
|- | |||
|2.3.5 | |||
|81/80, 15625/12288 | |||
|23 36 52 | |||
|- | |||
|2.3.5.7 | |||
|35/32, 175/162, 625/588 | |||
|23 36 52 63 | |||
|- | |||
|33/32.7/6 | |||
|[[117440512/117406179]] | |||
|1 5 | |||
|} | |||
Latest revision as of 19:20, 1 August 2025
|
Todo: merge articles
Merge into 33/32? |
| ← 0ed33/32 | 1ed33/32 | 2ed33/32 → |
1 equal division of 33/32 (1ed33/32), also known as ambitonal sequence of 33/32 (AS33/32) or 33/32 equal-step tuning, is an equal multiplication of 33/32 (the Alpharabian quarter-tone), and results in a nonoctave tuning equivalent to 22.5255 EDO.
Lookalikes: 5ed7/6, 45ed4
Theory
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +25.3 | +15.9 | -16.1 | -12.6 | +4.0 | -18.9 | -3.8 | +16.7 | +5.6 | -22.8 | +21.5 |
| Relative (%) | +47.4 | +29.8 | -30.3 | -23.7 | +7.5 | -35.4 | -7.2 | +31.3 | +10.4 | -42.8 | +40.4 | |
| Step | 23 | 36 | 52 | 63 | 78 | 83 | 92 | 96 | 102 | 109 | 112 | |
In this tuning, 2 steps by definition correspond to the parapotome 1089/1024.
Intervals with excellent approximation in this tuning are: 7/6 (5), 20/13 (14), 18/11 (16). Other intervals with good approximation are: 6/5, 7/5, 9/5, 13/7, 13/9, 11/10, 19/12, 17/16, 17/15, 16/15.
In the 5-limit, 1ed33/32 tempers out the syntonic comma 81/80, making it meantone.
In the 7-limit, as a consequence of representing 6/5 and 7/6 well, it's great at representing the 5:6:7 otonal tetrad. This means that 385/384 is tempered out.
Regular temperament properties
"Normal" subgroups calculated using the 23edo val that matches 33/32 equal step tuning patent val.
| Subgroup | Comma list | Mapping |
|---|---|---|
| 2.3.5 | 81/80, 15625/12288 | 23 36 52 |
| 2.3.5.7 | 35/32, 175/162, 625/588 | 23 36 52 63 |
| 33/32.7/6 | 117440512/117406179 | 1 5 |