17edo

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17edo

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Prime factorization	17 (prime)
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 17edo, 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 (¢)	0.0	+3.9	-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#)	up 1sn, minor 2nd	^1, m2	^C, Db	25/24, 26/25, 33/32, 24/23
2	141.18	Augmented Unison/Neutral Second	C#	aug 1sn, mid 2nd	A1, ~2	C#, 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	Eb	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, Gb	11/8, 18/13, 32/23	Progress
9	635.29	Augmented Fourth/Sub Fifth	F#	augmented 4th, mid 5th	A4, ~5	F#, 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	Ab	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	Bb	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

Notations

From the appendix to The Sagittal Songbook by Jacob A. Barton, a diagram of how to notate 17-EDO in the Revo flavor of Sagittal:

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.

Scales

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

Tuning 17edo by ear

17edo is very close to a circle of seventeen 25/24 chromatic semitones: (25/24)^17 is only 1.43131 cents sharp of an octave. This means that if you can tune seventeen 25/24's accurately (by say, tuning 5/4 up, 3/2 down and 5/4 up, taking care to minimize the error at each step), you have a shot at approximating 17edo within melodic just noticeable difference.

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	Ratio	Monzo	Cents	Color name	Name(s)
3	(18 digits)	[27 -17⟩	66.765	Sasawa	17-comma
5	25/24	[-3 -1 2⟩	70.762	Yoyo	Chromatic semitone, dicot comma
5	32805/32768	[-15 8 1⟩	1.9537	Layo	Schisma
7	525/512	[-9 1 2 1⟩	43.408	Zoyoyo	Avicennma, Avicennma's enharmonic diesis
7	64/63	[6 -2 0 -1⟩	27.264	Ru	Septimal comma, Archytas' comma, Leipziger Komma
7	245/243	[0 -5 1 2⟩	14.191	Zozoyo	Sensamagic
7	1728/1715	[6 3 -1 -3⟩	13.074	Triru-agu	Orwellisma, orwell comma
7	(12 digits)	[-6 -8 2 5⟩	1.1170	Quinzo-ayoyo	Wizma
11	99/98	[-1 2 0 -2 1⟩	17.576	Loruru	Mothwellsma
11	896/891	[7 -4 0 1 -1⟩	9.6880	Saluzo	Pentacircle
11	243/242	[-1 5 0 0 -2⟩	7.1391	Lulu	Rastma
11	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
13	364/363	[2 -1 0 1 -2 1⟩	4.763	Tholuluzo	Gentle 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.

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

Important MOSes include:

• diatonic (leapfrog/archy) 5L2s 3331331 (10\17, 1\1)
• maqamic 3L4s 3232322 (5\17, 1\1)
• maqamic 7L3s 2221221221 (5\17, 1\1)
• squares 3L5s 1141414 (6\17, 1\1)
• squares 3L8s 13113113 (6\17, 1\1)
• Pathological squares 3L11s 11211121112 (6\17, 1\1)
• lovecraft 4L5s 313131311 (4\17, 1\1)
• Pathological 1L 13s 4 1 1 1 1 1 1 1 1 1 1 1 1 (1\17, 1\1)
• Pathological 1L 14s 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 (1\17, 1\1)
• Pathological 2L 13s 2 1 1 1 1 1 1 1 2 1 1 1 1 1 1 (8\17, 1\1)
• Pathological 1L 15s 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 (1\17, 1\1)

Introductory materials

• File:17edo 1MC.mp3 17-edo example composition by Schrodingasdawg (score)
• 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
• Sarabande from the Locrian Suite by Inthar

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 ^{[dead link]} by Igliashon Jones
• I Insist ^{[dead link]} 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]}

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• 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