53 equal temperament

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Template:Short description

Figure 1: 53 TET on the syntonic temperament's tuning continuum at 701.89 cents, from Template:Harvp[1]

In music, 53 equal temperament, called 53 TET, 53 EDO, or 53 ET, is the tempered scale derived by dividing the octave into 53 equal steps (equal frequency ratios) (Template:Audio). Each step represents a frequency ratio of Template:Math or 22.6415 cents (Template:Audio), an interval sometimes called the Holdrian comma.

53 TET is a tuning of equal temperament in which the tempered perfect fifth is 701.89 cents wide, as shown in Figure 1, and sequential pitches are separated by 22.642 cents.

The 53-TET tuning equates to the unison, or tempers out, the intervals Template:Math known as the schisma, and Template:Math known as the kleisma. These are both 5 limit intervals, involving only the primes 2, 3, and 5 in their factorization, and the fact that 53 TET tempers out both characterizes it completely as a 5 limit temperament: It is the only regular temperament tempering out both of these intervals, or commas, a fact which seems to have first been recognized by Japanese music theorist Shohé Tanaka. Because it tempers these out, 53 TET can be used for both schismatic temperament, tempering out the schisma, and Hanson temperament (also called kleismic), tempering out the kleisma.

The interval of Template:Math is closest to the 43rd note (counting from 0) and Template:Math is only 4.8 cents sharp from the harmonic 7th Template:Math in 53 TET, and using it for 7-limit harmony means that the septimal kleisma, the interval Template:Sfrac, is also tempered out.

History and use

Theoretical interest in this division goes back to antiquity. Jing Fang (78–37 BCE), a Chinese music theorist, observed that a series of 53 just fifths Template:Math Template:Big is very nearly equal to 31 octaves (Template:Math). He calculated this difference with six-digit accuracy to be Template:Sfrac.[2][3] Later the same observation was made by the mathematician and music theorist Nicholas Mercator (Template:Circa), who calculated this value precisely as Template:MathTemplate:Verification needed which is known as Mercator's comma.[4] Mercator's comma is of such small value to begin with Template:Nobr but 53 equal temperament flattens each fifth by only Template:Math of that comma (Template:Nobr Template:Nobr Template:Nobr Thus, 53 tone equal temperament is for all practical purposes equivalent to an extended Pythagorean tuning.

After Mercator, William Holder published a treatise in 1694 which pointed out that 53 equal temperament also very closely approximates the just major third (to within 1.4 cents), and consequently 53 equal temperament accommodates the intervals of 5 limit just intonation very well.[5][6] This property of 53 TET may have been known earlier; Isaac Newton's unpublished manuscripts suggest that he had been aware of it as early as 1664–1665.[7]

Music

In the 19th century, people began devising instruments in 53 TET, with an eye to their use in playing near-just 5-limit music. Such instruments were devised by R.H.M. Bosanquet[8]Template:Rp and the American tuner J.P. White.[8]Template:Rp Subsequently, the temperament has seen occasional use by composers in the west, and by the early 20th century, 53 TET had become the most common form of tuning in Ottoman classical music, replacing its older, unequal tuning. Arabic music, which for the most part bases its theory on quartertones, has also made some use of it; the Syrian violinist and music theorist Twfiq Al-Sabagh proposed that instead of an equal division of the octave into 24 parts a 24 note scale in 53 TET should be used as the master scale for Arabic music.Template:Citation needed

Croatian composer Josip Štolcer-Slavenski wrote one piece, which has never been published, which uses Bosanquet's Enharmonium during its first movement, entitled Music for Natur-ton-system.[9][10][11]Template:Efn


Furthermore, General Thompson worked in league with the London-based guitar maker Louis Panormo to produce the Enharmonic Guitar.[12]

Notation

Notation used in Ottoman classical music, where the whole notes are divided into 9 commas.

Attempting to use standard notation, seven-letter notes plus sharps or flats, can quickly become confusing. This is unlike the case with 19 TET and 31 TET where there is little ambiguity. By not being meantone, it adds some problems that require more attention. Specifically, the Pythagorean major third (ditone) and just major third are distinguished, as are the Pythagorean minor third (semiditone) and just minor third. The fact that the syntonic comma is not tempered out means that notes and intervals need to be defined more precisely. Ottoman classical music uses a notation of flats and sharps for the 9 comma tone.

Furthermore, since 53 is not a multiple of 12, notes such as GTemplate:Music and ATemplate:Music are not enharmonically equivalent, nor are the corresponding key signatures. As a result, many key signatures will require the use of double sharps (such as GTemplate:Music major / ETemplate:Music minor), double flats (such as FTemplate:Music major / DTemplate:Music minor), or microtonal alterations.

Extended pythagorean notation, using only sharps and flats, gives the following chromatic scale:

Unfortunately, the notes run out of letter-order, and up to quadruple sharps and flats are required. As a result, a just major 3rd must be spelled as a diminished 4th.

Ups and downs notation[13] keeps the notes in order and also preserves the traditional meaning of sharp and flat. It uses up and down arrows, written as a caret or a lower-case "v", usually in a sans-serif font. One arrow equals one step of 53-TET. In note names, the arrows come first, to facilitate chord naming. The many enharmonic equivalences allow great freedom of spelling.

Chords of 53 equal temperament

Since 53-TET is a Pythagorean system, with nearly pure fifths, justly-intonated major and minor triads cannot be spelled in the same manner as in a meantone tuning. Instead, the major triads are chords like C-FTemplate:Music-G (using the Pythagorean-based notation), where the major third is a diminished fourth; this is the defining characteristic of schismatic temperament. Likewise, the minor triads are chords like C-DTemplate:Music-G. In 53-TET, the dominant seventh chord would be spelled C-FTemplate:Music-G-BTemplate:Music, but the otonal tetrad is C-FTemplate:Music-G-CTemplate:Music, and C-FTemplate:Music-G-ATemplate:Music is still another seventh chord. The utonal tetrad, the inversion of the otonal tetrad, is spelled C-DTemplate:Music-G-GTemplate:Music.

Further septimal chords are the diminished triad, having the two forms C-DTemplate:Music-GTemplate:Music and C-FTemplate:Music-GTemplate:Music, the subminor triad, C-FTemplate:Music-G, the supermajor triad C-DTemplate:Music-G, and corresponding tetrads C-FTemplate:Music-G-BTemplate:Music and C-DTemplate:Music-G-ATemplate:Music. Since 53-TET tempers out the septimal kleisma, the septimal kleisma augmented triad C-FTemplate:Music-BTemplate:Music in its various inversions is also a chord of the system. So is the Orwell tetrad, C-FTemplate:Music-DTemplate:MusicTemplate:Music-GTemplate:Music in its various inversions.

Ups and downs notation permits more conventional spellings. Since it also names the intervals of 53 TET,[14] it provides precise chord names too. The pythagorean minor chord with a Template:Sfrac third is still named Cm and still spelled C–ETemplate:Music–G. But the 5-limit upminor chord uses the upminor 3rd 6/5 and is spelled C–^ETemplate:Music–G. This chord is named C^m. Compare with ^Cm (^C–^ETemplate:Music–^G).

Interval size

7-Limit just intonation intervals approximated in 53 TET

Because a distance of 31 steps in this scale is almost precisely equal to a just perfect fifth, in theory this scale can be considered a slightly tempered form of Pythagorean tuning that has been extended to 53 tones. As such the intervals available can have the same properties as any Pythagorean tuning, such as fifths that are (practically) pure, major thirds that are wide from just (about Template:Math opposed to the purer Template:Math and minor thirds that are conversely narrow (Template:Math compared to Template:Math).

However, 53 TET contains additional intervals that are very close to just intonation. For instance, the interval of 17 steps is also a major third, but only 1.4 cents narrower than the very pure just interval Template:Math 53 TET is very good as an approximation to any interval in 5 limit just intonation. Similarly, the pure just interval Template:Math is only 1.3 cents wider than 14 steps in 53 TET.

The matches to the just intervals involving the 7th harmonic are slightly less close (43 steps are 4.8 cents sharp for [[harmonic seventh|Template:Math]]), but all such intervals are still quite closely matched with the highest deviation being the Template:Math tritone. The 11th harmonic and intervals involving it are less closely matched, as illustrated by the undecimal neutral seconds and thirds in the table below. 7-limit ratios are colored light gray, and 11- and 13-limit ratios are colored dark gray.

Size
(steps) 		Size
(cents) 		Interval name Template:Small Just
(cents) 		Error
(cents) 		Limit
53 1200 perfect octave Template:Math 1200 0 2
52 1177.36 grave octave Template:Math 1178.49 −1.14 5
51 1154.72 just augmented seventh Template:Math 1158.94 −4.22 5
50 1132.08 diminished octave Template:Math 1129.33 +2.75 5
48 1086.79 just major seventh Template:Math 1088.27 −1.48 5
45 1018.87 just minor seventh Template:Math 1017.60 +1.27 5
44 996.23 Pythagorean minor seventh Template:Math 996.09 +0.14 3
43 973.59 accute augmented sixth Template:Math 976.54 −2.95 5
43 973.59 harmonic seventh Template:Math 968.83 +4.76 7
43 973.59 accute diminished seventh Template:Math 968.43 +5.15 5
42 950.94 just augmented sixth Template:Math 955.03 −4.09 5
42 950.94 just diminished seventh Template:Math 946.92 +4.02 5
39 883.02 major sixth Template:Math 884.36 −1.34 5
37 837.73 tridecimal neutral sixth Template:Math 840.53 −2.8 13
36 815.09 minor sixth Template:Math 813.69 +1.40 5
31 701.89 perfect fifth Template:Math 701.96 −0.07 3
30 679.25 grave fifth Template:Math 680.45 −1.21 5
28 633.96 just diminished fifth
Template:Small 		Template:Math 631.28 +2.68 5
27 611.32 Pythagorean augmented fourth Template:Math 611.73 −0.41 3
27 611.32 greater ‘classic’ tritone Template:Math 609.78 +1.54 5
26 588.68 lesser ‘classic’ tritone Template:Math 590.22 −1.54 5
26 588.68 septimal tritone Template:Math 582.51 +6.17 7
25 566.04 just augmented fourth
Template:Small 		Template:Math 568.72 −2.68 5
24 543.40 undecimal major fourth Template:Math 551.32 −7.92 11
24 543.40 double diminished fifth Template:Math 539.10 +4.30 5
24 543.40 undecimal augmented fourth Template:Math 536.95 +6.45 11
23 520.76 acute fourth Template:Math 519.55 +1.21 5
22 498.11 perfect fourth Template:Math 498.04 +0.07 3
21 475.47 grave fourth Template:Math 476.54 −1.07 5
21 475.47 septimal narrow fourth Template:Math 470.78 +4.69 7
20 452.83 just augmented third Template:Math 456.99 −4.16 5
20 452.83 tridecimal augmented third Template:Math 454.21 −1.38 13
19 430.19 septimal major third Template:Math 435.08 −4.90 7
19 430.19 just diminished fourth Template:Math 427.37 +2.82 5
18 407.54 Pythagorean ditone Template:Math 407.82 −0.28 3
17 384.91 just major third Template:Math 386.31 −1.40 5
16 362.26 grave major third Template:Math 364.80 −2.54 5
16 362.26 neutral third, tridecimal Template:Math 359.47 +2.79 13
15 339.62 neutral third, undecimal Template:Math 347.41 −7.79 11
15 339.62 acute minor third Template:Math 337.15 +2.47 5
14 316.98 just minor third Template:Math 315.64 +1.34 5
13 294.34 Pythagorean semiditone Template:Math 294.13 +0.21 3
12 271.70 just augmented second Template:Math 274.58 −2.88 5
12 271.70 septimal minor third Template:Math 266.87 +4.83 7
11 249.06 just diminished third Template:Math 244.97 +4.09 5
10 226.41 septimal whole tone Template:Math 231.17 −4.76 7
10 226.41 diminished third Template:Math 223.46 +2.95 5
9 203.77 whole tone, major tone,
greater tone, just second 		Template:Math 203.91 −0.14 3
8 181.13 grave whole tone, minor tone,
lesser tone, Template:Nobr 		Template:Math 182.40 −1.27 5
7 158.49 neutral second, greater undecimal Template:Math 165.00 −6.51 11
7 158.49 doubly grave whole tone Template:Math 160.90 −2.41 5
7 158.49 neutral second, lesser undecimal Template:Math 150.64 +7.85 11
6 135.85 accute diatonic semitone Template:Math 133.24 +2.61 5
5 113.21 greater Pythagorean semitone Template:Math 113.69 −0.48 3
5 113.21 just diatonic semitone,
just minor second 		Template:Math 111.73 +1.48 5
4 90.57 major limma Template:Math 92.18 −1.61 5
4 90.57 lesser Pythagorean semitone Template:Math 90.22 +0.34 3
3 67.92 just chromatic semitone Template:Math 70.67 −2.75 5
3 67.92 greater diesis Template:Math 62.57 +5.35 5
2 45.28 just diesis Template:Math 41.06 +4.22 5
1 22.64 syntonic comma Template:Math 21.51 +1.14 5
0 0 perfect unison Template:Math 0 0 1

Scale diagram

The following are 21 of the 53 notes in the chromatic scale. The rest can easily be added.

Interval (steps) 3 2 4 3 2 3 2 1 2 4 1 4 3 2 4 3 2 3 2 1 2
Interval (cents) 68 45 91 68 45 68 45 23 45 91 23 91 68 45 91 68 45 68 45 23 45
Note name (Pythagorean notation) C ETemplate:Music CTemplate:Music D FTemplate:Music DTemplate:Music FTemplate:Music DTemplate:Music CTemplate:MusicTemplate:Music/ATemplate:MusicTemplate:Music F GTemplate:Music FTemplate:Music G BTemplate:Music GTemplate:Sharp BTemplate:Music CTemplate:Music ATemplate:Sharp CTemplate:Music ATemplate:Music GTemplate:MusicTemplate:Music/ETemplate:MusicTemplate:Music C
Note name (ups and downs notation) C vvCTemplate:Sharp/vDTemplate:Flat CTemplate:Sharp/^DTemplate:Flat D vvDTemplate:Sharp/vETemplate:Flat DTemplate:Sharp/^ETemplate:Flat vE ^E ^^E/vvF F vFTemplate:Sharp/GTemplate:Flat FTemplate:Sharp/^GTemplate:Flat G vvGTemplate:Sharp/vATemplate:Flat GTemplate:Sharp/^ATemplate:Flat vA vvATemplate:Sharp/vBTemplate:Flat ATemplate:Sharp/^BTemplate:Flat vB ^B ^^B/vvC C
Note (cents)   0    68  113 204 272 317 385 430 453 498 589 611 702 770 815 883 974 1018 1087 1132 1155 1200
Note (steps) 0 3 5 9 12 14 17 19 20 22 26 27 31 34 36 39 43 45 48 50 51 53

Holdrian comma

In music theory and musical tuning the Holdrian comma, also called Holder's comma, and rarely the Arabian comma,[15] is a small musical interval of approximately 22.6415 cents,[15] equal to one step of 53 equal temperament, or  253  (Template:Audio). The name "comma", however, is technically misleading, since this interval is an irrational number and it does not describe a compromise between intervals of any tuning system. The interval gets the name "comma" because it is a close approximation of several commas, most notably the syntonic comma (21.51 cents) (Template:Audio), which was widely used as a unit of tonal measurement during Holder's time.

The origin of Holder's comma resides in the fact that the Ancient Greeks (or at least to the Roman BoethiusTemplate:Efn) believed that in the Pythagorean tuning the tone could be divided in nine commas, four of which forming the diatonic semitone and five the chromatic semitone. If all these commas are exactly of the same size, there results an octave of Template:Nobr semitones, Template:Nobr commas. Holder[16] attributes the division of the octave in 53 equal parts to Nicholas Mercator,Template:Efn who himself had proposed that Template:Nobr of the octave be named the "artificial comma".

Mercator's comma and the Holdrian comma

Mercator's comma is a name often used for a closely related interval because of its association with Nicholas Mercator.Template:Efn One of these intervals was first described by Jing Fang in Template:Nobr[15] Mercator applied logarithms to determine that  255  (≈ 21.8182 cents) was nearly equivalent to a syntonic comma of ≈ 21.5063 cents (a feature of the prevalent meantone temperament of the time). He also considered that an "artificial comma" of  253  might be useful, because 31 octaves could be practically approximated by a cycle of 53 just fifths. Holder, for whom the Holdrian comma is named, favored this latter unit because the intervals of 53 equal temperament are closer to just intonation than to 55 TET. Thus Mercator's comma and the Holdrian comma are two distinct but nearly equal intervals.

Use in Turkish makam theory

The Holdrian comma has been employed mainly in Ottoman/Turkish music theory by Kemal Ilerici, and by the Turkish composer Erol Sayan. The name of this comma is Template:Lang in Turkish.

Name of interval Commas Cents Symbol
Koma 1 22.64 F
Bakiye 4 90.57 B
Küçük Mücennep 5 113.21 S
Büyük Mücennep 8 181.13 K
Tanini 9 203.77 T
Artık Aralık (12) 12 271.70 A (12)
Artık Aralık (13) 13 294.34 A (13)

For instance, the Rast makam (similar to the Western major scale, or more precisely to the justly-tuned major scale) may be considered in terms of Holdrian commas:

where Template:Music denotes a Holdrian comma flat,Template:Efn while in contrast, the Nihavend makam (similar to the Western minor scale):

where Template:Music denotes a five-comma flat, has medium seconds between d–eTemplate:Music, e–f, g–aTemplate:Music, aTemplate:Music–bTemplate:Music, and bTemplate:Music–c′, a medium second being somewhere in between 8 and 9 commas.[15]

Notes

Template:Notelist

References

Template:Reflist

Template:Refbegin

Template:Refend

Template:Microtonal music Template:Musical tuning

fr:Tempérament par division multiple

  1. ↑ Template:Cite journal
  2. ↑ Template:Cite journal
  3. ↑ Template:Cite web
  4. ↑ Template:Cite web
  5. ↑ Template:Harvp
  6. ↑ Template:Cite book — see also Template:Harvp
  7. ↑ Template:Cite book
  8. ↑ 8.0 8.1 Template:Cite book
  9. ↑ Template:Cite web
  10. ↑ Template:Cite web
  11. ↑ Template:Cite web
    Template:Cite web
  12. ↑ Template:Cite magazine
  13. ↑ Template:Cite web
  14. ↑ Template:Cite web
  15. ↑ 15.0 15.1 15.2 15.3 Template:Cite book
  16. ↑ Template:Cite book
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