10ed7/4: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}}{{todo|expand}} | ||
'''10ED7/4''' is the [[Equal-step tuning|equal division]] of the [[7/4|harmonic seventh]] into ten parts of 96.8826 [[cent|cents]] each, corresponding to 12.3861 [[EDO]]. | '''10ED7/4''' is the [[Equal-step tuning|equal division]] of the [[7/4|harmonic seventh]] into ten parts of 96.8826 [[cent|cents]] each, corresponding to 12.3861 [[EDO]]. | ||
== Theory == | |||
This tuning tempers out 36/35 and 50/49 in the 7-limit; 55/54 in the 11-limit; 34/33 and 56/51 in the 17-limit; 31/30 in the 31-limit; and 38/37 in the 37-limit. | |||
== Intervals == | == Intervals == | ||
{| class="wikitable" | {| class="wikitable mw-collapsible" | ||
|+ Intervals of 10ed7/4 | |||
|- | |- | ||
! | | ! | Degree | ||
! | | ! | Cents value | ||
! | | ! | Ratio | ||
|- | |- | ||
| | 0 | | | 0 | ||
| Line 168: | Line 172: | ||
== Just approximation == | == Just approximation == | ||
Only very few intervals like the [[5/4|just major third]] and the [[7/5|Huygens' tritone]] are well approximated by 10ed7/4. | Only very few intervals like the [[5/4|just major third]] and the [[7/5|Huygens' tritone]] are well approximated by 10ed7/4. | ||
=== 15-odd-limit | === 15-odd-limit approximations === | ||
The following table shows how [[15-odd-limit intervals]] are represented in 10ed7/4 (can be ordered by absolute error). | The following table shows how [[15-odd-limit intervals]] are represented in 10ed7/4 (can be ordered by absolute error). | ||
{| class="wikitable sortable" | {| class="wikitable sortable mw-collapsible" | ||
|+ [[Direct approximation]] (even if [[inconsistent]]) | |||
|+ Direct | |||
! Interval(s) | ! Interval(s) | ||
! Error (abs, [[cent|¢]]) | ! Error (abs, [[cent|¢]]) | ||
| Line 180: | Line 182: | ||
|- | |- | ||
| [[7/4]] | | [[7/4]] | ||
|0.0 | | 0.0 | ||
|- | |- | ||
| [[2/1]] | | [[2/1]] | ||
| Line 186: | Line 188: | ||
|- | |- | ||
| [[3/2]] | | [[3/2]] | ||
|23.777 | | 23.777 | ||
|- | |- | ||
| [[5/4]] | | [[5/4]] | ||
|1.217 | | 1.217 | ||
|- | |- | ||
| [[9/8]] | | [[9/8]] | ||
|10.145 | | 10.145 | ||
|- | |- | ||
| [[11/8]] | | [[11/8]] | ||
|29.978 | | 29.978 | ||
|- | |- | ||
| [[13/8]] | | [[13/8]] | ||
|31.416 | | 31.416 | ||
|- | |- | ||
| [[15/8]] | | [[15/8]] | ||
|22.56 | | 22.56 | ||
|- | |- | ||
| [[14/9]] | | [[14/9]] | ||
|10.145 | | 10.145 | ||
|- | |- | ||
| [[28/15]] | | [[28/15]] | ||
|14.849 | | 14.849 | ||
|- | |- | ||
| [[10/7]] | | [[10/7]] | ||
|36.192 | | 36.192 | ||
|- | |- | ||
| [[16/11]] | | [[16/11]] | ||
|29.496 | | 29.496 | ||
|- | |- | ||
| [[13/10]] | | [[13/10]] | ||
|30.199 | | 30.199 | ||
|- | |- | ||
| [[9/5]] | | [[9/5]] | ||
|48.112 | | 48.112 | ||
|- | |- | ||
| [[10/9]] | | [[10/9]] | ||
|11.361 | | 11.361 | ||
|- | |- | ||
| [[26/15]] | | [[26/15]] | ||
|16.567 | | 16.567 | ||
|- | |- | ||
| [[13/11]] | | [[13/11]] | ||
|1.438 | | 1.438 | ||
|- | |- | ||
| [[13/7]] | | [[13/7]] | ||
|5.993 | | 5.993 | ||
|- | |- | ||
| [[16/13]] | | [[16/13]] | ||
|28.058 | | 28.058 | ||
|- | |- | ||
| [[7/6]] | | [[7/6]] | ||
|23.777 | | 23.777 | ||
|- | |- | ||
| [[5/3]] | | [[5/3]] | ||
|12.415 | | 12.415 | ||
|- | |- | ||
| [[20/13]] | | [[20/13]] | ||
|29.275 | | 29.275 | ||
|- | |- | ||
| [[11/10]] | | [[11/10]] | ||
|28.761 | | 28.761 | ||
|- | |- | ||
| [[8/5]] | | [[8/5]] | ||
|38.626 | | 38.626 | ||
|- | |- | ||
| [[9/7]] | | [[9/7]] | ||
|47.554 | | 47.554 | ||
|- | |- | ||
| [[11/9]] | | [[11/9]] | ||
|40.122 | | 40.122 | ||
|- | |- | ||
| [[18/11]] | | [[18/11]] | ||
|19.351 | | 19.351 | ||
|- | |- | ||
| [[24/13]] | | [[24/13]] | ||
|4.281 | | 4.281 | ||
|- | |- | ||
| [[22/15]] | | [[22/15]] | ||
|15.129 | | 15.129 | ||
|- | |- | ||
| [[15/13]] | | [[15/13]] | ||
|42.907 | | 42.907 | ||
|- | |- | ||
| [[15/11]] | | [[15/11]] | ||
|44.345 | | 44.345 | ||
|- | |- | ||
| [[16/9]] | | [[16/9]] | ||
|27.264 | | 27.264 | ||
|- | |- | ||
| [[12/7]] | | [[12/7]] | ||
|35.697 | | 35.697 | ||
|- | |- | ||
| [[7/5]] | | [[7/5]] | ||
|1.217 | | 1.217 | ||
|- | |- | ||
| [[12/11]] | | [[12/11]] | ||
|43.128 | | 43.128 | ||
|- | |- | ||
| [[4/3]] | | [[4/3]] | ||
|13.632 | | 13.632 | ||
|- | |- | ||
| [[11/6]] | | [[11/6]] | ||
|16.346 | | 16.346 | ||
|- | |- | ||
| [[13/12]] | | [[13/12]] | ||
|41.69 | | 41.69 | ||
|- | |- | ||
| [[8/7]] | | [[8/7]] | ||
|37.409 | | 37.409 | ||
|- | |- | ||
| [[20/11]] | | [[20/11]] | ||
|30.713 | | 30.713 | ||
|- | |- | ||
| [[14/13]] | | [[14/13]] | ||
|31.416 | | 31.416 | ||
|- | |- | ||
| [[6/5]] | | [[6/5]] | ||
|24.994 | | 24.994 | ||
|- | |- | ||
| [[18/13]] | | [[18/13]] | ||
|17.913 | | 17.913 | ||
|- | |- | ||
| [[15/14]] | | [[15/14]] | ||
|22.56 | | 22.56 | ||
|- | |- | ||
| [[11/7]] | | [[11/7]] | ||
|7.431 | | 7.431 | ||
|- | |- | ||
| [[13/9]] | | [[13/9]] | ||
|41.56 | | 41.56 | ||
|- | |- | ||
| [[14/11]] | | [[14/11]] | ||
|29.978 | | 29.978 | ||
|- | |- | ||
| [[22/13]] | | [[22/13]] | ||
|38.847 | | 38.847 | ||
|- | |- | ||
| [[16/15]] | | [[16/15]] | ||
|14.849 | | 14.849 | ||
|} | |} | ||
== Harmonics == | |||
{{Harmonics in equal | |||
| steps = 10 | |||
| num = 7 | |||
| denom = 4 | |||
}} | |||
{{Harmonics in equal | |||
| steps = 10 | |||
| num = 7 | |||
| denom = 4 | |||
| start = 12 | |||
| collapsed = 1 | |||
}} | |||
[[Category:Subminor seventh]] | [[Category:Subminor seventh]] | ||
[[Category:Equal-step tuning]] | [[Category:Equal-step tuning]] | ||