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== Theory ==
17 tone equal temperament, or 17-EDO, divides the octave in 17 equal steps, each 70.588 [[cent]]s in size. It is the seventh [[prime numbers|prime]] [[EDO]], following [[13edo]] and coming before [[19edo]].
17 tone equal temperament, or 17-EDO, divides the octave in 17 equal steps, each 70.588 [[cent]]s in size. It is the seventh [[prime numbers|prime]] [[EDO]], following [[13edo]] and coming before [[19edo]].


An introduction to 17-EDO theory, through the eyes of the [[SeventeenTonePianoProject]]: [[SeventeenTheory]].
== Introductory materials ==
 
* [[SeventeenTheory]], an introduction to 17-EDO theory, through the eyes of the [[SeventeenTonePianoProject]].  
Another introduction into 17-EDO theory: [http://anaphoria.com/Secor17puzzle.pdf The 17-tone Puzzle] by George Secor.
* [http://anaphoria.com/Secor17puzzle.pdf The 17-tone Puzzle] by George Secor, another introduction into 17-EDO theory.  
 
* [[17edo Solfege]]
[[17edo Solfege]]. [[17edo tetrachords]]. [http://microtonalismo.com/proyecto-xvii Proyect 17-Perú] {{forbidden}}
* [[17edo tetrachords]]
* [http://microtonalismo.com/proyecto-xvii Proyect 17-Perú] {{forbidden}}


== Theory ==
17-EDO can plausibly be treated as a 2.3.25.7.11.13.23 subgroup temperament, for which it is quite accurate (though the 7-limit ratios are generally not as well-represented as those of the other integers). Because the 3, 7, 11, and 13 are all sharp, it adapts well to octave shrinking; [[27edt]] (a variant of 17edo in which the octaves are flattened by ~2.5 cents) is a good alternative. Another one is [[44ed6]].
17-EDO can plausibly be treated as a 2.3.25.7.11.13.23 subgroup temperament, for which it is quite accurate (though the 7-limit ratios are generally not as well-represented as those of the other integers). Because the 3, 7, 11, and 13 are all sharp, it adapts well to octave shrinking; [[27edt]] (a variant of 17edo in which the octaves are flattened by ~2.5 cents) is a good alternative. Another one is [[44ed6]].


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! Cents
! Cents
! Names of Intervals
! Names of Intervals
! colspan="3" | [[Ups and Downs Notation|ups and downs notation]]
! colspan="3" | [[Ups and Downs Notation]]
! Approximate Ratios*
! Approximate Ratios*
! Temperament(s)generated
! Temperament(s) generated
|-
|-
| 0
| 0
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|  
|  
|}
|}
<nowiki>*</nowiki>Ratios based on treating 17edo as a 2.3.7.11.13.23.25 subgroup
<nowiki>*</nowiki> Ratios based on treating 17edo as a 2.3.7.11.13.23.25 subgroup


Combining ups and downs notation with [[color notation]], qualities can be loosely associated with colors:
Combining ups and downs notation with [[color notation]], qualities can be loosely associated with colors:
Line 314: Line 315:
| +7.0
| +7.0
|-
|-
! relative<ref>Relative error is measured in EDO steps (1\17).</ref> (%)
! [[Relative error|relative]] (%)
| 0
| 0
| +5.6
| +5.6
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| +6
| +6
|}
|}
<references/>


The following table shows how [[Just-24|some prominent just intervals]] are represented in 17edo (ordered by absolute error).
==== 15-odd-limit mappings ====
 
The following table shows how [[15-odd-limit intervals]] are represented in 17edo (ordered by absolute error). Prime harmonics are in '''bold'''; inconsistent intervals are in ''italic''.
=== Best direct mapping, even if inconsistent ===


{| class="wikitable sortable center-1 right-2"
{| class="wikitable sortable center-1 right-2"
|+Direct mapping (even if inconsistent)
|-
|-
! class="unsortable" | Interval, <br> complement
! class="unsortable" | Interval, complement
! Abs. error <br> (in [[cent]]s)
! Error (abs, [[Cent|¢]])
|-
|-
| [[18/13]], [[13/9]]
| [[18/13]], [[13/9]]
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| 2.604
| 2.604
|-
|-
| [[4/3]], [[3/2]]
| '''[[4/3]], [[3/2]]'''
| 3.927
| '''3.927'''
|-
|-
| [[11/9]], [[18/11]]
| [[11/9]], [[18/11]]
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| 6.021
| 6.021
|-
|-
| [[16/13]], [[13/8]]
| '''[[16/13]], [[13/8]]'''
| 6.531
| '''6.531'''
|-
|-
| [[13/11]], [[22/13]]
| [[13/11]], [[22/13]]
Line 380: Line 380:
| 12.878
| 12.878
|-
|-
| [[11/8]], [[16/11]]
| '''[[11/8]], [[16/11]]'''
| 13.388
| '''13.388'''
|-
|-
| [[7/6]], [[12/7]]
| [[7/6]], [[12/7]]
| 15.482
| 15.482
|-
|-
| [[7/5]], [[10/7]]
| ''[[7/5]], [[10/7]]''
| 17.806
| ''17.806''
|-
|-
| [[8/7]], [[7/4]]
| '''[[8/7]], [[7/4]]'''
| 19.409
| '''19.409'''
|-
|-
| [[15/14]], [[28/15]]
| ''[[15/14]], [[28/15]]''
| 21.734
| ''21.734''
|-
|-
| [[11/10]], [[20/11]]
| ''[[11/10]], [[20/11]]''
| 23.828
| ''23.828''
|-
|-
| [[15/11]], [[22/15]]
| ''[[15/11]], [[22/15]]''
| 27.755
| ''27.755''
|-
|-
| [[10/9]], [[9/5]]
| ''[[10/9]], [[9/5]]''
| 29.361
| ''29.361''
|-
|-
| [[16/15]], [[15/8]]
| [[16/15]], [[15/8]]
| 29.445
| 29.445
|-
|-
| [[13/10]], [[20/13]]
| ''[[13/10]], [[20/13]]''
| 30.685
| ''30.685''
|-
|-
| [[6/5]], [[5/3]]
| ''[[6/5]], [[5/3]]''
| 33.288
| ''33.288''
|-
|-
| [[5/4]], [[8/5]]
| '''[[5/4]], [[8/5]]'''
| 33.373
| '''33.373'''
|-
|-
| [[15/13]], [[26/15]]
| ''[[15/13]], [[26/15]]''
| 34.612
| ''34.612''
|}
|}
=== Patent val mapping ===


{| class="wikitable center-1 right-2"
{| class="wikitable center-1 right-2"
|+Patent val mapping
|-
|-
! Interval, <br> complement
! Interval, complement
! Abs. error <br> (in [[cent]]s)
! Error (abs, [[Cent|¢]])
|-
|-
| [[18/13]], [[13/9]]
| [[18/13]], [[13/9]]
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| '''33.373'''
| '''33.373'''
|-
|-
| [[15/13]], [[26/15]]
| ''[[15/13]], [[26/15]]''
| 35.976
| ''35.976''
|-
|-
| [[6/5]], [[5/3]]
| ''[[6/5]], [[5/3]]''
| 37.300
| ''37.300''
|-
|-
| [[13/10]], [[20/13]]
| ''[[13/10]], [[20/13]]''
| 39.904
| ''39.904''
|-
|-
| [[10/9]], [[9/5]]
| ''[[10/9]], [[9/5]]''
| 41.227
| ''41.227''
|-
|-
| [[15/11]], [[22/15]]
| ''[[15/11]], [[22/15]]''
| 42.833
| ''42.833''
|-
|-
| [[11/10]], [[20/11]]
| ''[[11/10]], [[20/11]]''
| 46.760
| ''46.760''
|-
|-
| [[15/14]], [[28/15]]
| ''[[15/14]], [[28/15]]''
| 48.855
| ''48.855''
|-
|-
| [[7/5]], [[10/7]]
| ''[[7/5]], [[10/7]]''
| 52.782
| ''52.782''
|}
|}


==== Selected 13-limit intervals ====
[[File:17ed2-001.svg|alt=alt : Your browser has no SVG support.]]
[[File:17ed2-001.svg|alt=alt : Your browser has no SVG support.]]
[[:File:17ed2-001.svg|17ed2-001.svg]]


== Commas ==
== Commas ==
Retrieved from "https://en.xen.wiki/w/17edo"