17 equal temperament

Musical tuning system with 17 pitches equally-spaced on a logarithmic scale upright=1.3|thumb|Figure 1: 17-ET on the regular diatonic tuning continuum at P5=705.88 cents.Milne Sethares Plamondon 2007 15–32 thumb|1 step in 17-ET In music, 17 equal temperament is the tempered scale derived by dividing the octave into 17 equal steps (equal frequency ratios). Each step represents a frequency ratio of , or 70.6 cents. 17-ET is the tuning of the regular diatonic tuning in which the tempered perfect fifth is equal to 705.88 cents, as shown in Figure 1 (look for the label "17-TET"). History and use Alexander J. Ellis refers to a tuning of seventeen tones based on perfect fourths and fifths as the Arabic scale.Ellis, Alexander J. (1863). "On the Temperament of Musical Instruments with Fixed Tones", Proceedings of the Royal Society of London, vol. 13. (1863–1864), pp. 404–422. In the thirteenth century, Middle-Eastern musician Safi al-Din Urmawi developed a theoretical system of seventeen tones to describe Arabic and Persian music, although the tones were not equally spaced. This 17-tone system remained the primary theoretical system until the development of the quarter tone scale.September 2015 Notation upright=2.5|thumb|Notation of Easley BlackwoodBlackwood Easley Easley Blackwood Jr. Summer 1991 833437 Modes and Chord Progressions in Equal Tunings Perspectives of New Music 29 2 166–200 (175) 10.2307/833437 for 17 equal temperament: intervals are notated similarly to those they approximate and enharmonic equivalents are distinct from those of 12 equal temperament (e.g., A#/Cb).File:17-tet scale on C.mid Easley Blackwood Jr. created a notation system where sharps and flats raised/lowered 2 steps, identical to ups and downs notation for 17-EDO. ((10*7) mod 17 = 2.) This yields the chromatic scale: C, Dflat, Csharp, D, Eflat, Dsharp, E, F, Gflat, Fsharp, G, Aflat, Gsharp, A, Bflat, Asharp, B, C Quarter tone sharps and flats can also be used, yielding the following chromatic scale: C, Ct/Dflat, Csharp/Dd, D, Dt/Eflat, Dsharp/Ed, E, F, Ft/Gflat, Fsharp/Gd, G, Gt/Aflat, Gsharp/Ad, A, At/Bflat, Asharp/Bd, B, C Interval size Below are some intervals in 17 EDO compared to just. thumb|upright=1.3| Major chord on C in 17 EDO : All notes are within 37 cents of just intonation (rather than 14 cents for 12 EDO). 17 EDO File:Major chord on C in 17 equal temperament.mid just File:Major chord on C in just intonation.mid 12 EDO File:Major chord on C.mid thumb|center|upright=1.4|I–IV–V–I chord progression in 17 EDO.Milne Sethares Plamondon 2007 29 File:Simple_I-IV-V-I_isomorphic_17-TET.mid Whereas in 12 EDO, Bnatural is 11 steps, in 17 EDO, Bnatural is 16 steps. {| class="wikitable sortable" style="vertical-align:center;text-align:center;" |- style="vertical-align:bottom;" ! interval name ! size(steps) ! size(cents) ! midiaudio ! justratio ! just(cents) ! midiaudio ! error |- style="text-align:center;" | octave | 17 | 1200 | | 2:1 | 1200 | | 0 |- style="text-align: center;" |Major seventh |16 |1129.41 | |27:14 |1137.04 | | -7.62 |- style="text-align: center;" |Major seventh |16 |1129.41 | |15:8 |1088.27 | | +41.14 |- style="text-align: center;" |Neutral seventh |15 |1058.82 | |11:6 |1049.36 | | +9.46 |- style="text-align:center;" | minor seventh | 14 | 988.23 | | 16:9 | 996.09 | | −7.77 |- style="text-align:center;background:#D4D4D4;" | harmonic seventh | 14 | 988.23 | | 7:4 | 968.83 | | +19.41 |- style="text-align: center;" |major sixth |13 |917.65 | |27:16 |905.87 | | +11.78 |- style="text-align: center;" |neutral sixth |12 |847.06 | |13:8 |840.53 | | +6.53 |- style="text-align: center;" |minor sixth |11 |776.47 | |14:9 |764.92 | | +11.55 |- style="text-align:center;" | perfect fifth | 10 | 705.88 | 120px | 3:2 | 701.96 | 120px | +3.93 |- style="text-align: center;" |augmented fourth (notational) |9 |635.29 | |16:11 |648.68 | | -13.39 |- style="text-align:center;background:#D4D4D4;" | septimal tritone | 8 | 564.71 | 120px | 7:5 | 582.51 | 120px | −17.81 |- style="text-align:center;" | tridecimal narrow tritone | 8 | 564.71 | 120px | 18:13 | 563.38 | 120px | +1.32 |- style="text-align:center;" | undecimal super-fourth | 8 | 564.71 | 120px | 11:8 | 551.32 | 120px | +13.39 |- style="text-align:center;" | perfect fourth | 7 | 494.12 | 120px | 4:3 | 498.04 | 120px | −3.93 |- style="text-align:center;" | septimal major third | 6 | 423.53 | 120px | 9:7 | 435.08 | 120px | −11.55 |- style="text-align:center;" | undecimal major third | 6 | 423.53 | 120px | 14:11 | 417.51 | 120px | +6.02 |- style="text-align:center;background:#D4D4D4;" | major third | 5 | 352.94 | 120px | 5:4 | 386.31 | 120px | −33.37 |- style="text-align:center;" | tridecimal neutral third | 5 | 352.94 | 120px | 16:13 | 359.47 | 120px | −6.53 |- style="text-align:center;" | undecimal neutral third | 5 | 352.94 | 120px | 11:9 | 347.41 | 120px | +5.53 |- style="text-align:center;background:#D4D4D4;" | minor third | 4 | 282.35 | 120px | 6:5 | 315.64 | 120px | −33.29 |- style="text-align:center;" | tridecimal minor third | 4 | 282.35 | 120px | 13:11 | 289.21 | 120px | −6.86 |- style="text-align:center;" | septimal minor third | 4 | 282.35 | 120px | 7:6 | 266.87 | 120px | +15.48 |- style="text-align:center;background:#D4D4D4;" | septimal whole tone | 3 | 211.76 | 120px | 8:7 | 231.17 | 120px | −19.41 |- style="text-align:center;" | greater whole tone | 3 | 211.76 | 120px | 9:8 | 203.91 | 120px | +7.85 |- style="text-align:center;background:#D4D4D4;" | lesser whole tone | 3 | 211.76 | 120px | 10:9 | 182.40 | 120px | +29.36 |- style="text-align:center;" | neutral second, lesser undecimal | 2 | 141.18 | 120px | 12:11 | 150.64 | 120px | −9.46 |- style="text-align:center;" | greater tridecimal  2  3 -tone | 2 | 141.18 | 120px | 13:12 | 138.57 | 120px | +2.60 |- style="text-align:center;" | lesser tridecimal  2  3 -tone | 2 | 141.18 | 120px | 14:13 | 128.30 | 120px | +12.88 |- style="text-align:center;background:#D4D4D4;" | septimal diatonic semitone | 2 | 141.18 | 120px | 15:14 | 119.44 | 120px | +21.73 |- style="text-align:center;background:#D4D4D4;" | diatonic semitone | 2 | 141.18 | 120px | 16:15 | 111.73 | 120px | +29.45 |- style="text-align:center;" | septimal chromatic semitone | 1 | 70.59 | 120px | 21:20 | 84.47 | 120px | −13.88 |- style="text-align:center;" | chromatic semitone | 1 | 70.59 | 120px | 25:24 | 70.67 | 120px | −0.08 |} Relation to 34 EDO 17 EDO is a subset of 34 EDO, equivalent to every other step in the 34 EDO scale. References 25em Sources Milne Andrew Sethares William William Sethares Plamondon James Winter 2007 Isomorphic controllers and dynamic tuning: Invariant fingering over a tuning continuum Computer Music Journal 31 4 15–32 27906745 10.1162/comj.2007.31.4.15 free https://www.mitpressjournals.org/doi/pdf/10.1162/comj.2007.31.4.15 mitpressjournals.org External links 17edo on Xenharmonic Wiki "The 17-tone Puzzle — And the Neo-medieval Key that Unlocks It" by George Secor Libro y Programa Tonalismo, heptadecatonic system applications (in Spanish) Georg Hajdu's 1992 ICMC paper on the 17-tone piano projectmedic otABmneuKV4 "Crocus", 17 equal temperament, 9 tone mode, by Wongi Hwang Category:Equal temperaments Category:Microtonality