17 equal temperament
![](/w/images/thumb/4/4c/Rank-2_temperaments_with_the_generator_close_to_a_fifth_and_period_an_octave.jpg/250px-Rank-2_temperaments_with_the_generator_close_to_a_fifth_and_period_an_octave.jpg)
In music, 17 tone 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 21/17, or 70.6 cents (<phonos file="1 step in 17-et on C.mid">play</phonos>). Alexander J. Ellis refers to a tuning of seventeen tones based on perfect fourths and fifths as the Arabic scale.[2] 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.[citation needed]
17-ET is the tuning of the syntonic temperament in which the tempered perfect fifth is equal to 705.88 cents, as shown in Figure 1 (look for the label "17-TET"). On an isomorphic keyboard, the fingering of music composed in 17-ET is precisely the same as it is in any other syntonic tuning (such as 12-ET), so long as the notes are spelled properly -- that is, with no assumption of enharmonicity.
History
![](/w/images/thumb/2/26/17-tet_scale_on_C.png/573px-17-tet_scale_on_C.png)
![](/w/images/thumb/7/79/Major_chord_on_C.png/292px-Major_chord_on_C.png)
![](/w/images/thumb/a/a0/Simple_I-IV-V-I_isomorphic_17-TET.png/328px-Simple_I-IV-V-I_isomorphic_17-TET.png)
Interval size
interval name | size (steps) | size (cents) | midi | just ratio | just (cents) | midi | error |
---|---|---|---|---|---|---|---|
perfect fifth | 10 | 705.88 | <phonos file="10 steps in 17-et on C.mid">Play</phonos> | 3:2 | 701.96 | <phonos file="Just perfect fifth on C.mid">Play</phonos> | +3.93 |
septimal tritone | 8 | 564.71 | <phonos file="8 steps in 17-et on C.mid">Play</phonos> | 7:5 | 582.51 | <phonos file="Lesser septimal tritone on C.mid">Play</phonos> | −17.81 |
tridecimal narrow tritone | 8 | 564.71 | <phonos file="8 steps in 17-et on C.mid">Play</phonos> | 18:13 | 563.38 | +1.32 | |
undecimal super-fourth | 8 | 564.71 | <phonos file="8 steps in 17-et on C.mid">Play</phonos> | 11:8 | 551.32 | <phonos file="Eleventh harmonic on C.mid">Play</phonos> | +13.39 |
perfect fourth | 7 | 494.12 | <phonos file="7 steps in 17-et on C.mid">Play</phonos> | 4:3 | 498.04 | <phonos file="Just perfect fourth on C.mid">Play</phonos> | −3.93 |
septimal major third | 6 | 423.53 | <phonos file="6 steps in 17-et on C.mid">Play</phonos> | 9:7 | 435.08 | <phonos file="Septimal major third on C.mid">Play</phonos> | −11.55 |
undecimal major third | 6 | 423.53 | <phonos file="6 steps in 17-et on C.mid">Play</phonos> | 14:11 | 417.51 | <phonos file="Undecimal major third on C.mid">Play</phonos> | +6.02 |
major third | 5 | 352.94 | <phonos file="5 steps in 17-et on C.mid">Play</phonos> | 5:4 | 386.31 | <phonos file="Just major third on C.mid">Play</phonos> | −33.37 |
tridecimal neutral third | 5 | 352.94 | <phonos file="5 steps in 17-et on C.mid">Play</phonos> | 16:13 | 359.47 | <phonos file="Tridecimal neutral third on C.mid">Play</phonos> | −6.53 |
undecimal neutral third | 5 | 352.94 | <phonos file="5 steps in 17-et on C.mid">Play</phonos> | 11:9 | 347.41 | <phonos file="Undecimal neutral third on C.mid">Play</phonos> | +5.53 |
minor third | 4 | 282.35 | <phonos file="4 steps in 17-et on C.mid">Play</phonos> | 6:5 | 315.64 | <phonos file="Just minor third on C.mid">Play</phonos> | −33.29 |
tridecimal minor third | 4 | 282.35 | <phonos file="4 steps in 17-et on C.mid">Play</phonos> | 13:11 | 289.21 | <phonos file="Tridecimal minor third on C.mid">play</phonos> | −6.86 |
septimal minor third | 4 | 282.35 | <phonos file="4 steps in 17-et on C.mid">Play</phonos> | 7:6 | 266.87 | <phonos file="Septimal minor third on C.mid">Play</phonos> | +15.48 |
septimal whole tone | 3 | 211.76 | <phonos file="3 steps in 17-et on C.mid">Play</phonos> | 8:7 | 231.17 | <phonos file="Septimal major second on C.mid">Play</phonos> | −19.41 |
whole tone | 3 | 211.76 | <phonos file="3 steps in 17-et on C.mid">Play</phonos> | 9:8 | 203.91 | <phonos file="Major tone on C.mid">Play</phonos> | +7.85 |
neutral second, lesser undecimal | 2 | 141.18 | <phonos file="2 steps in 17-et on C.mid">Play</phonos> | 12:11 | 150.64 | <phonos file="Lesser undecimal neutral second on C.mid">Play</phonos> | −9.46 |
greater tridecimal 2/3-tone | 2 | 141.18 | <phonos file="2 steps in 17-et on C.mid">Play</phonos> | 13:12 | 138.57 | +2.60 | |
lesser tridecimal 2/3-tone | 2 | 141.18 | <phonos file="2 steps in 17-et on C.mid">Play</phonos> | 14:13 | 128.30 | +12.88 | |
septimal diatonic semitone | 2 | 141.18 | <phonos file="1_step_in_17-et_on_C.mid">Play</phonos> | 15:14 | 119.44 | <phonos file="Just chromatic semitone on C.mid">Play</phonos> | +21.73 |
diatonic semitone | 2 | 141.18 | <phonos file="2 steps in 17-et on C.mid">Play</phonos> | 16:15 | 111.73 | <phonos file="Just diatonic semitone on C.mid">Play</phonos> | +29.45 |
septimal chromatic semitone | 1 | 70.59 | <phonos file="1_step_in_17-et_on_C.mid">Play</phonos> | 21:20 | 84.47 | <phonos file="Septimal chromatic semitone on C.mid">Play</phonos> | −13.88 |
chromatic semitone | 1 | 70.59 | <phonos file="1_step_in_17-et_on_C.mid">Play</phonos> | 25:24 | 70.67 | <phonos file="Just chromatic semitone on C.mid">Play</phonos> | −0.08 |
Relation to 34-ET
17-ET is where every other step in the 34-ET scale is included, and the others are not accessible. Conversely 34-ET is a subdivision of 17-ET.
External links
- Secor, George. "The 17-tone Puzzle — And the. Neo-medieval Key That Unlocks It".
- Microtonalismo Heptadecatonic System Applications
- Georg Hajdu's 1992 ICMC paper on the 17-tone piano project
- ProyectoXVII Heptadecatonic System Applications project XVII - Peruvian
Sources
- ↑ Milne, A., Sethares, W.A. and Plamondon, J.,"Isomorphic Controllers and Dynamic Tuning: Invariant Fingerings Across a Tuning Continuum", Computer Music Journal, Winter 2007, Vol. 31, No. 4, Pages 15-32.
- ↑ 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.
- ↑ Blackwood, Easley (Summer, 1991). "Modes and Chord Progressions in Equal Tunings", p.175, Perspectives of New Music, Vol. 29, No. 2, pp. 166-200.
- ↑ Andrew Milne, William Sethares, and James Plamondon (2007). "Isomorphic Controllers and Dynamic Tuning: Invariant Fingering over a Tuning Continuum", p.29. Computer Music Journal, 31:4, pp.15–32, Winter 2007.