17edo

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17 tone equal temperament

17-EDO divides the octave in 17 equal steps, each 70.588 cents in size. It is the seventh prime edo, following 13edo and coming before 19edo.

Theory

An introduction to 17-EDO theory, through the eyes of the SeventeenTonePianoProject: SeventeenTheory.

Another introduction into 17-EDO theory: The 17-tone Puzzle by George Secor.

17edo Solfege. 17edo tetrachords. Proyect 17-Perú

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.

As a no-fives system, it is best used with timbres in which harmonic multiples of 5 are attenuated or absent. Also, the standard major chord (4:5:6) cannot be used since it includes the fifth harmonic.

Instead, the tonic chords of 17-EDO could be considered to be the tetrad 6:7:8:9 and its utonal inversion, the former of which is a subminor chord with added fourth, and the latter a supermajor chord with added second (resembling the famous mu major chord of Steely Dan fame). These are realized in 17-EDO as 0-4-7-10 and 0-3-6-10, respectively. Both of these have distinct moods, and are stable and consonant, if somewhat more sophisticated than their classic 5-limit counterparts. To this group we could also add the 0-3-7-10 (which is a sus4 with added second, or sus2 with added fourth). These three chords comprise the three ways to divide the 17-EDO perfect fifth into two whole tones and one subminor third. Chromatic alterations of them also exist, for example, the 0-3-7-10 chord may be altered to 0-2-7-10 (which approximates 12:13:16:18) or 0-3-8-10 (which approximates 8:9:11:12). The 0-3-8-10 chord is impressive-sounding, resembling a sus4 but with even more tension; it resolves quite nicely to 0-3-6-10.

Intervals

Degree	Cents	Names ofIntervals	ups and downs notation			Approximate Ratios*	Temperament(s)generated
0	0	Unison	unison	P1	C	1/1
1	70.59	Super Unison/Minor Second	minor 2nd	m2	Db	25/24, 26/25, 33/32
2	141.18	Augmented Unison/Neutral Second	mid 2nd	~2	Dv	13/12, 12/11, 14/13	Bleu
3	211.765	Major Second/Sub Third	major 2nd	M2	D	9/8, 25/22, 8/7, 28/25	Machine
4	282.35	Minor Third/Super Second	minor 3rd	m3	Eb	13/11, 75/64, 7/6	Huxley
5	352.94	Augmented Second/Neutral Third/Diminished Fourth	mid 3rd	~3	Ev	11/9, 16/13	Maqamic/hemif
6	423.53	Major Third/Sub Fourth	major 3rd	M3	E	32/25, 33/26, 9/7, 14/11, 51/40	Skwares
7	494.11	Perfect Fourth	perfect 4th	P4	F	4/3	Supra
8	564.71	Super FourthDiminished Fifth	up 4th,diminished 5th	^4, d5	F^, Gb	11/8, 18/13	Progress
9	635.29	Augmented Fourth/Sub Fifth	augmented 4th,down 5th	A4, v5	F#, Gv	16/11, 13/9, 23/16	Progress
10	705.88	Perfect Fifth	perfect 5th	P5	G	3/2	Supra
11	776.47	Super Fifth/Minor Sixth	minor 6th	m6	Ab	25/16, 52/33, 14/9, 11/7	Skwares
12	847.06	Augmented Fifth/Neutral Sixth/Diminished Seventh	mid 6th	~6	Av	13/8, 18/11	Maqamic/hemif
13	917.65	Major Sixth/Sub Seventh	major 6th	M6	A	17/10, 22/13, 128/75, 12/7	Huxley
14	988.235	Minor Seventh/Super Sixth	minor 7th	m7	Bb	16/9, 44/25, 7/4, 25/14	Machine
15	1058.82	Augmented Sixth/Neutral Seventh/Diminished Octave	mid 7th	~7	Bv	11/6, 24/13, 13/7	Bleu
16	1129.41	Major Seventh/Sub Octave	major 7th	M7	B	25/13, 48/25, 64/33
17	1200	Perfect Octave	octave	P8	C	2/1

*Ratios based on treating 17edo as a 2.3.7.11.13.25 subgroup

Combining ups and downs notation with color notation, qualities can be loosely associated with colors:

quality	color	monzo format	examples
minor	blue	{a, b, 0, 1}	7/6, 7/4
"	fourthward white	{a, b}, b < -1	32/27, 16/9
mid	jade	{a, b, 0, 0, 1}	11/9, 11/6
"	amber	{a, b, 0, 0, -1}	12/11, 18/11
major	fifthward white	{a, b}, b > 1	9/8, 27/16
"	red	{a, b, 0, -1}	9/7, 12/7

Chord Names

All 17edo chords can be named using ups and downs. Here are the blue, jade and red triads:

color of the 3rd	JI chord	notes as edosteps	notes of C chord	written name	spoken name
blue	6:7:9	0-4-10	C Eb G	Cm	C minor
jade	18:22:27	0-5-10	C Ev G	C~	C mid
red	14:18:27	0-6-10	C E G	C	C major or C

0-4-9 = C Eb Gv = Cm(v5) = C minor down-five

0-5-9 = C Ev G = C~(v5) = C mid down-five

0-6-11 = C E G^ = C(^5) = C up-five

0-4-10-14 = C Eb G Bb = Cm7 = C minor seven

0-5-10-14 = C Ev G Bb = C7(~3) = C seven mid-three

0-6-10-15 = C E G Bv = C,~7 = C mid seven

0-5-10-15 = C Ev G Bv = C.~7 = C dot mid seven

For a more complete list, see Ups and Downs Notation - Chord names in other EDOs.

Selected just intervals by error

The following table shows how some prominent just intervals are represented in 17edo (ordered by absolute error).

Best direct mapping, even if inconsistent

Interval, complement	Error (abs., in cents)
18/13, 13/9	1.324
13/12, 24/13	2.604
4/3, 3/2	3.927
11/9, 18/11	5.533
14/11, 11/7	6.021
16/13, 13/8	6.531
13/11, 22/13	6.857
9/8, 16/9	7.855
12/11, 11/6	9.461
9/7, 14/9	11.555
14/13, 13/7	12.878
11/8, 16/11	13.388
7/6, 12/7	15.482
7/5, 10/7	17.806
8/7, 7/4	19.409
15/14, 28/15	21.734
11/10, 20/11	23.828
15/11, 22/15	27.755
10/9, 9/5	29.361
16/15, 15/8	29.445
13/10, 20/13	30.685
6/5, 5/3	33.288
5/4, 8/5	33.373
15/13, 26/15	34.612

Patent val mapping

Interval, complement	Error (abs., in cents)
18/13, 13/9	1.324
13/12, 24/13	2.604
4/3, 3/2	3.927
11/9, 18/11	5.533
14/11, 11/7	6.021
16/13, 13/8	6.531
13/11, 22/13	6.857
9/8, 16/9	7.855
12/11, 11/6	9.461
9/7, 14/9	11.555
14/13, 13/7	12.878
11/8, 16/11	13.388
7/6, 12/7	15.482
8/7, 7/4	19.409
16/15, 15/8	29.445
5/4, 8/5	33.373
15/13, 26/15	35.976
6/5, 5/3	37.300
13/10, 20/13	39.904
10/9, 9/5	41.227
15/11, 22/15	42.833
11/10, 20/11	46.760
15/14, 28/15	48.855
7/5, 10/7	52.782

17ed2-001.svg

Commas

17 EDO tempers out the following commas. (Note: This assumes val < 17 27 39 48 59 63 |, cent values ​​rounded to 5 digits.)

Comma	Monzo	Value (Cents)	Name 1	Name 2	Name 3
	| 27 -17 >	66.765	17-Comma
25/24	|-3 -1 2>	70.762	Chromatic semitone	Dicot comma
32805/32768	| -15 8 1 >	1.9537	Schisma
64/63	| 6 -2 0 -1 >	27.264	Septimal Comma	Archytas' Comma	Leipziger Komma
245/243	| 0 -5 1 2 >	14.191	Sensamagic
1728/1715	| 6 3 -1 -3 >	13.074	Orwellisma	Orwell Comma
	| -6 -8 2 5 >	1.1170	Wizma
99/98	| -1 2 0 -2 1 >	17.576	Mothwellsma
896/891	| 7 -4 0 1 -1 >	9.6880	Pentacircle
243/242	| -1 5 0 0 -2 >	7.1391	Rastma
385/384	| -7 -1 1 1 1 >	4.5026	Keenanisma
525/512	| -9 1 2 1 >	43.408	Avicennma	Avicennma's Enharmonic Diesis

Note that despite their relatively large size, the 17-comma, the avicennma and the chromatic semitone are all tempered out by the 13-limit patent val, as stated.

Scales

• Otonal 17
• Blues Peruvian 17edo
• 17edo neutral scale
• Scorp
• Screamapillar
• Hydra
• MOS scales of 17edo (horograms)

List of 17edo rank two temperaments by badness

List of edo-distinct 17c rank two temperaments

Music

Compositions

• Demanding Two Faces (xen-pop) by Stephen Weigel
• Where were you at the Apocalypse? (xen-pop) by Stephen Weigel
• Prelude by Daniel Wolf
• Heptadecatonic Drops by Georg Hajdu
• Klangmoraste by Georg Hajdu
• Charles Loli 17edo music for guitar heptadecatonic (2001) and armony inductive microtonaly (1993)
• microtonalismo Heptadecatonic Peruvian
• Multiverse by Jacob Barton

Sound files

• Shanidar Cave is a piece in 17 edo that features an electric 17 edo guitar and what is essentially an electric tanpura which ends up making this a sort of fusion of middle eastern and Indian music in a sense. by Chris Vaisvil
• Puhlops and Laugua's Big Adventure by |Aaron Krister Johnson
• Adagio for Margo by Aaron Krister Johnson
• Transformation by Herman Miller
• Waltz by Christopher Bailey
• Lostfound by Christopher Bailey
• excerpt from Balladei (in 17, 29 and 12) by Christopher Bailey , CD available here .
• Two-Part Invention in 17ET by Aaron Andrew Hunt
• Overture to A Midsummer Night's Dream by Rick McGowan
• Fairy Lullaby from A Midsummer Night's Dream by Rick McGowan
• A Calamitous Simultaneity by Igliashon Jones (17edo and 22edo)
• First Impressions by Igliashon Jones
• I Insist by Igliashon Jones
• Etude no1 for 2 Pianos in 17 Equal Temperament and Etude no2 for 2 Pianos in 17 Equal Temperament Jon Lyle Smith
• A number of compositions from seventeen-tone piano project phase I , seventeen-tone piano project phase II, seventeen-tone piano project phase III.
• 17edo - 17edo-tagged compositions on www.archive.org
• sing a blue by Andrew Heathwaite (composed 2008, recorded 2010). This and the other pieces below by Andrew for cümbüş, steel tubes & voice.
• stringfinger it everybean by Andrew Heathwaite (composed 2008, recorded 2010).
• cat feet belly by Andrew Heathwaite (composed 2008, recorded 2010).
• 17 Tone Jam by Marmalade Man
• youtube videos tagged with 17edo
• On the Shores of the Dead Sea blog video of On the Shores of the Dead Sea by Chris Vaisvil
• Only in Disneyland 17 blog edo guitar solo play Only in Disneyland by Chris Vaisvil
• 17 Reasons I Hate the Blues blog play 17 Reasons I Hate the Blues by Chris Vaisvil
• Klingon Opera Overture blog play Klingon Opera Overture by Chris Vaisvil
• Seventeen Selfless Notes blog play Seventeen Selfless Notes by Chris Vaisvil
• 17ET Jazz blog play 17et Jazz by Chris Vaisvil (60 x 60 winner)
• 17 Pink Tuxedos blog play 17 Pink Tuxedos by Chris Vaisvil
• Devil in the Deep Blue Sea blog play Devil in the Deep Blue sea blues collaboration between The Two Regs (vocals / lyrics) and Norm Harris (percussion) and Chris Vaisvil (17 note per octave electric guitar and fretless bass).
• Seventeen Years in the Sixties blog play Seventeen Years in the Sixties by Chris Vaisvil
• CT Scan blog play video CT Scan by Chris Vaisvil
• Fish and a Grenade blog play Fish and a Grenade (parental advisory language) by Chris Vaisvil
• Seventeen Unsteady Hands blog play video of performance of Seventeen Unsteady Hands by Chris Vaisvil
• The Pond blog by Chris Vaisvil and video version of The Pond with Japanese Garden
• Graveyard blog play Graveyard by Chris Vaisvil
• For Brass and Voice Choirs in 17 edo by Chris Vaisvil more on the composition
• And I Became One With My Per Fungi by Chris Vaisvil => more on composition
• counterintuitive (17 edo guitar solo) by Chris Vaisvil => more on composition
• No Love by Gregory Sanchez
• Micro7 by Ralph Jarzombek
• Flying Into O'Hare by Chris Vaisvil
• Helas, pitié by Alex Ness (in 17edo with stretched octaves)

Instruments

• Guitar Heptadecatonic from Peruvian - Charles Loli and Antonio Huamani

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• Bass Heptadecatonic from Peruvian - Charles Loli and Antonio Huamani

  External image: http://sphotos.ak.fbcdn.net/hphotos-ak-ash2/hs382.ash2/66019_167001006650538_100000219181856_601987_48585_n.jpg
  

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• 17 note per octave conversion from a "standard" stratocastor copy - conversion by Brad Smith