17edo: Difference between revisions

17edo's perfect fifth is around 4 cents sharp of just, and around 6 cents sharp of [[12edo]]'s, lending itself to a [[5L 2s|diatonic]] scale with more constrasting large and small steps. Meanwhile, it approximates [[harmonic]]s [[7/1|7]], [[11/1|11]], [[13/1|13]], and [[23/1|23]] to reasonable degrees, despite completely missing harmonic [[5/1|5]]. Thus it 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 these harmonics are all tempered 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]].

The standard major triad is quite dissonant as the major third is closer to [[9/7]] than the traditional [[5/4]]. Instead, the tonic chords of 17edo could be considered to be the tetrad 6:7:8:9 and its utonal inversion (representing 14:16:18:21 as [[64/63]] is tempered out), the former of which is a subminor chord with added fourth, and the latter a supermajor chord with added second (resembling the {{w|mu chord}} of Steely Dan fame). These are realized in 17edo 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 17edo 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.  

{{Harmonics in equal|17|intervals=odd|prec=2}}

{{Harmonics in equal|17|intervals=odd|prec=2|start=12|collapsed=true|title=Approximation of odd harmonics in 17edo (continued)}}

17edo is the seventh [[prime edo]], following [[13edo]] and coming before [[19edo]]. [[34edo]], which doubles it, provides a good correction to harmonics 5 and 17; while [[68edo]], which quadruples it, provides additionally the primes 7, 19, and 31.

{{See also|17edo solfege}}

! #

! colspan="2" | [[Circle-of-fifths notation]]^†

| 0.00

| 1/1

| 70.59

| Minor 2nd<br />(Semiaugmented 1sn)

| Eb<br />(Dt)

| up unison, <br />minor 2nd

| uber unison, <br />minor 2nd

| 141.18

| Augmented 1sn<br />(Neutral 2nd)

| D#<br />(Ed)

| augmented 1sn, <br />mid 2nd

| 211.76

| 282.35

| 352.94

| Diminished 4th<br />(Neutral 3rd)

| Gb<br />(Ft)

| 423.53

| Major 3rd<br />(Semidiminished 4th)

| F#<br />(Gd)

| 494.12

| 564.71

| Diminished 5th<br />(Semiaugmented 4th)

| Ab<br />(Gt)

| mid 4th, <br />diminished 5th

| uber 4th/<br />neutral 4th

| 635.29

| Augmented 4th<br />(Semidiminished 5th)

| G#<br />(Ad)

| augmented 4th, <br />mid 5th

| unter 5th/<br />neutral 5th

| 705.88

| 776.47

| Minor 6th<br />(Semiaugmented 5th)

| Bb<br />(At)

| 847.06

| Augmented 5th<br />(Neutral 6th)

| A#<br />(Bd)

| 917.65

| 988.24

| 1058.82

| Diminished 8ve<br />(Neutral 7th)

| Db<br />(Ct)

| 1129.41

| Major 7th<br />(Semidiminished 8ve)

| C#<br />(Dd)

| major 7th,<br />down 8ve

| major 7th,<br />unter octave

| 1200.00

^&dagger; Half-sharps and half-flats (denoted "t" and "d", respectively) can be used to alter the note by a single step, since sharps and flats each span two edosteps. 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.

! Monzo Format

| (a, b), b &lt; -1

| (a, b), b &gt; 1

! [[Kite's color notation|Color of the 3rd]]

! JI Chord

! Notes as Edosteps

! Notes of C Chord

! Written Name

! Spoken Name

For a more complete list, see [[Ups and downs notation #Chords and Chord Progressions]].

{{sharpness-sharp2a}}

{{sharpness-sharp2}}

This notation uses the same sagittal sequence as EDOs [[24edo#Sagittal notation|24]], [[31edo#Sagittal notation|31]], and [[38edo#Sagittal notation|38]], and is a subset of the notation for [[34edo#Sagittal notation|34-EDO]].

Because it contains no Sagittal symbols, this Evo-SZ Sagittal notation is also a Stein-Zimmerman notation.

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.

| {{monzo| 27 -17 }}

| {{mapping| 17 27 }}

| {{mapping| 17 27 48 }}

| {{mapping| 17 27 48 59 }}

| {{mapping| 17 27 48 59 63 }}

* [[Secor wt17|George Secor’s well temperament of this tuning]]

* [[User:CritDeathX/Sam's 17-note Well Temperament|Nicolai's 17-note Well Temperament]]