17edo

← 16edo

17edo

18edo →

Prime factorization	17
Step size	70.5882 ¢
Fifth	10\17 (705.882 ¢) (semiconvergent)
Semitones (A1:m2)	2:1 (141.2 ¢ : 70.59 ¢)
Consistency limit	3
Distinct consistency limit	3

17 tone equal temperament, or 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

		prime 2	prime 3	prime 5	prime 7	prime 11	prime 13	prime 17	prime 19	prime 23
Error	absolute (¢)	0	+3.93	-33.4	+19.4	+13.4	+6.5	-34.3	-15.2	+7.0
	relative (%)	0	+6	-47	+27	+19	+9	-49	-21	+10
nearest edomapping		17	10	5	14	8	12	1	4	9
fifthspan		0	+1	-8	-2	-6	+8	-5	-3	+6

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 mu 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

Edo steps	Cents	Names of Intervals, extended pythagorean note names		Ups and Downs Notation			Approximate Ratios*	Temperament(s) generated
0	0.00	Unison	C	unison	P1	C	1/1
1	70.59	Super Unison/Minor Second	Db (B#)	minor 2nd	m2	^C	25/24, 26/25, 33/32, 24/23
2	141.18	Augmented Unison/Neutral Second	C#	mid 2nd	~2	vD	13/12, 12/11, 14/13, 25/23	Bleu
3	211.76	Major Second/Sub Third	D	major 2nd	M2	D	9/8, 8/7, 28/25, 25/22, 26/23	Machine
4	282.35	Minor Third/Super Second	Eb	minor 3rd	m3	^D	13/11, 7/6	Huxley/Lovecraft
5	352.94	Augmented Second/Neutral Third/Diminished Fourth	D# (Fb)	mid 3rd	~3	vE	11/9, 16/13, 28/23	Maqamic/Hemif
6	423.53	Major Third/Sub Fourth	E	major 3rd	M3	E	32/25, 9/7, 14/11, 33/26, 23/18	Skwares
7	494.12	Perfect Fourth	F	perfect 4th	P4	F	4/3	Supra
8	564.71	Super Fourth/Diminshed Fifth	Gb (E#)	mid 4th, diminished 5th	~4, d5	^F	11/8, 18/13, 32/23	Progress
9	635.29	Augmented Fourth/Sub Fifth	F#	augmented 4th, mid 5th	A4, ~5	vG	16/11, 13/9, 23/16	Progress
10	705.88	Perfect Fifth	G	perfect 5th	P5	G	3/2	Supra
11	776.47	Super Fifth/Minor Sixth	Ab	minor 6th	m6	^G	25/16, 14/9, 11/7, 52/33, 36/23	Skwares
12	847.06	Augmented Fifth/Neutral Sixth/Diminished Seventh	G#	mid 6th	~6	vA	13/8, 18/11, 23/14	Maqamic/hemif
13	917.65	Major Sixth/Sub Seventh	A	major 6th	M6	A	17/10, 22/13,12/7	Huxley
14	988.24	Minor Seventh/Super Sixth	Bb	minor 7th	m7	^A	16/9, 7/4, 25/14, 44/25, 23/13	Machine
15	1058.82	Augmented Sixth/Neutral Seventh/Diminished Octave	A# (Cb)	mid 7th	~7	vB	11/6, 24/13, 13/7, 46/25	Bleu
16	1129.41	Major Seventh/Sub Octave	B	major 7th	M7	B	25/13, 48/25, 64/33, 23/12
17	1200.00	Perfect Octave	C	octave	P8	C	2/1

* Ratios based on treating 17edo as a 2.3.7.11.13.23.25 subgroup

In 17edo, ups and downs can respectively be substituted with half-sharps and half-flats, since sharps and flats each span two edo steps. Using half-sharps and half-flats may be preferable for compatibility with the ups-and-downs notation in 34edo, in which an up or down respectively constitute a quarter-sharp or quarter-flat.

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

quality	color	monzo format	examples
minor	zo	{a, b, 0, 1}	7/6, 7/4
"	fourthward wa	{a, b}, b < -1	32/27, 16/9
mid	ilo	{a, b, 0, 0, 1}	11/9, 11/6
"	lu	{a, b, 0, 0, -1}	12/11, 18/11
major	fifthward wa	{a, b}, b > 1	9/8, 27/16
"	ru	{a, b, 0, -1}	9/7, 12/7

Chord Names

All 17edo chords can be named using ups and downs. Here are the zo, ilo and ru triads:

color of the 3rd	JI chord	notes as edosteps	notes of C chord	written name	spoken name
zo	6:7:9	0-4-10	C Eb G	Cm	C minor
ilo	18:22:27	0-5-10	C vE G	C~	C mid
ru	14:18:21	0-6-10	C E G	C	C major or C

Alterations are always enclosed in parentheses, additions never are. An up, down or mid immediately after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord 6-1-3-5-7-9-11-13).

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

0-5-9 = C vE vG = 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 vE G Bb = C~,7 = C mid add seven

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

0-5-10-15 = C vE G vB = C~7 = C mid-seven

For a more complete list, see Ups and Downs Notation - Chords and Chord Progressions.

Just approximation

Selected just intervals by 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.

Direct mapping (even if inconsistent)

Interval, complement	Error (abs, ¢)
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, ¢)
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

Selected 13-limit intervals

Temperament measures

The following table shows TE temperament measures (RMS normalized by the rank) of 17et.

		3-limit	7-limit no-5	11-limit no-5	13-limit no-5
Octave stretch (¢)		-1.24	-3.13	-3.31	-3.00
Error	absolute (¢)	1.24	2.85	2.49	2.31
	relative (%)	1.76	4.05	3.54	3.28

• 17et has a lower relative error than any previous ETs in the no-5 11- and 13-limit. The next ET that does better in these subgroups is 41 and 207, respectively.

Commas

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

Prime Limit	Comma	Monzo	Cents	Color Name	Name(s)
3		[27 -17⟩	66.765	Sasawa	17-comma
5	25/24	[-3 -1 2⟩	70.762	Yoyo	Chromatic semitone, dicot comma
"	32805/32768	[-15 8 1⟩	1.9537	Layo	Schisma
7	525/512	[-9 1 2 1⟩	43.408	Zoyoyo	Avicennma, Avicennma's enharmonic diesis
"	64/63	[6 -2 0 -1⟩	27.264	Ru	Septimal comma, Archytas' comma, Leipziger Komma
"	245/243	[0 -5 1 2⟩	14.191	Zozoyo	Sensamagic
"	1728/1715	[6 3 -1 -3⟩	13.074	Triru-agu	Orwellisma, orwell comma
"		[-6 -8 2 5⟩	1.1170	Quinzo-ayoyo	Wizma
11	99/98	[-1 2 0 -2 1⟩	17.576	Loruru	Mothwellsma
"	896/891	[7 -4 0 1 -1⟩	9.6880	Saluzo	Pentacircle
"	243/242	[-1 5 0 0 -2⟩	7.1391	Lulu	Rastma
"	385/384	[-7 -1 1 1 1⟩	4.5026	Lozoyo	Keenanisma
13	1352/1331	[3 0 0 0 -3 2⟩	27.101	Bithotrilu	Lovecraft comma

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)

Temperaments

Rank-two temperaments

• List of 17edo rank two temperaments by badness
• List of edo-distinct 17c rank two temperaments

Well temperaments

• George Secor’s well temperament of this tuning
• Sam's 17-note Well Temperament
• Flora's 17-note well temperament

Introductory materials

• SeventeenTheory, an introduction to 17-EDO theory, through the eyes of the SeventeenTonePianoProject.
• The 17-tone Puzzle by George Secor, another introduction into 17-EDO theory.
• 17edo Solfege
• 17edo tetrachords
• Proyect 17-Perú ^[forbidden]

Music

Scores

• Prelude (PDF) by Daniel Wolf
• Heptadecatonic Drops by Georg Hajdu
• Klangmoraste by Georg Hajdu
• Charles Loli 17edo ^[forbidden] music for guitar heptadecatonic (2001) and armony inductive microtonally (1993)
• microtonalismo Heptadecatonic Peruvian
• Multiverse by Jacob Barton
• Balladei by Christopher Bailey

Sound files

• Arm's Length by Diamond Doll (xen-pop)
• Spidermilk (prog album) by The Mercury Tree
• Demanding Two Faces (xen-pop) by Stephen Weigel
• Where were you at the Apocalypse? (xen-pop) by Stephen Weigel*
• 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 ^{[dead link]}
• Waltz by Christopher Bailey
• Lost & Found Things #2 (studio) (live) by Christopher Bailey
• Balladei (live) (in 17, 29 and 12) by Christopher Bailey , CD available here .
• Two-Part Invention in 17ET ^{[dead link]} 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 ^{[dead link]} and Etude no2 for 2 Pianos in 17 Equal Temperament ^{[dead link]} by 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.
• 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
• No Love by Gregory Sanchez
• Micro7 by Ralph Jarzombek
• Helas, pitié by Alex Ness (in 17edo with stretched octaves)
• March of the Mushrooms by Jutomi
• by Chris Vaisvil:
  • On the Shores of the Dead Sea: blog | video
  • Only in Disneyland: blog | MP3 (guitar solo)
  • 17 Reasons I Hate the Blues: blog | MP3
  • Klingon Opera Overture: blog | MP3
  • Seventeen Selfless Notes: blog | MP3
  • 17et Jazz: blog | MP3 (60 x 60 winner)
  • 17 Pink Tuxedos: blog | MP3
  • Devil in the Deep Blue Sea: blog | MP3 (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 | MP3
  • CT Scan: blog | MP3
  • Fish and a Grenade: blog | MP3 (parental advisory: language)
  • Seventeen Unsteady Hands: blog | video of performance
  • The Pond: blog | video
  • Graveyard: blog | MP3
  • For Brass and Voice Choirs in 17 edo: blog | MP3
  • And I Became One With My Pet Fungi: blog | MP3
  • Counterintuitive: blog | MP3 (guitar solo)
  • Flying Into O'Hare: blog | MP3

Instruments

• Guitar Heptadecatonic from Peruvian - Charles Loli and Antonio Huamani ^[forbidden]

External image: http://sphotos.ak.fbcdn.net/hphotos-ak-snc4/hs883.snc4/71639_167001659983806_100000219181856_601995_1526184_n.jpg ^{[dead link]}

WARNING: MediaWiki doesn't have very good support for external images.

Furthermore, since external images can break, we recommend that you replace the above with a local copy of the image.

• Bass Heptadecatonic from Peruvian - Charles Loli and Antonio Huamani ^[forbidden]

External image: http://sphotos.ak.fbcdn.net/hphotos-ak-ash2/hs382.ash2/66019_167001006650538_100000219181856_601987_48585_n.jpg ^{[dead link]}

WARNING: MediaWiki doesn't have very good support for external images.

Furthermore, since external images can break, we recommend that you replace the above with a local copy of the image.

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