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19-tone equal-temperament and 1/3-comma meantone

[Joe Monzo]

19edo on standard 5-line musical staff

19-edo on 5-line musical staff

Below is a list of pitches in 19-edo, with letter nominals using "C" as the reference (as in the music notation illustrated above), showing the enharmonic equivalences which occur in 19-edo.

letter   19-edo
nominal  degree    ~cents        exact cents

 C        19    1200            1200
 B#/Cb    18    1136.842105     1136 + 16/19
 B        17    1073.684211     1073 + 13/19
 Ax/Bb    16    1010.526316     1010 + 10/19
 A#/Bbb   15     947.3684211     947 +  7/19
 A        14     884.2105263     884 +  4/19
 Gx/Ab    13     821.0526316     821 +  1/19
 G#/Abb   12     757.8947368     757 + 17/19
 G        11     694.7368421     694 + 14/19
 Fx/Gb    10     631.5789474     631 + 11/19
 F#/Gbb    9     568.4210526     568 +  8/19
 F         8     505.2631579     505 +  5/19
 E#/Fb     7     442.1052632     442 +  2/19
 E         6     378.9473684     378 + 18/19
 Dx/Eb     5     315.7894737     315 + 15/19
 D#/Ebb    4     252.6315789     252 + 12/19
 D         3     189.4736842     189 +  9/19
 Cx/Db     2     126.3157895     126 +  6/19
 C#/Dbb    1      63.15789474     63 +  3/19
 C         0       0               0
			

An early and very simple proposal for a meantone tuning was 1/3-comma meantone and its close relative 19-edo. The earliest reference to a tuning which may be 19-edo is in the 5th Treatise of the Berkeley Manuscript -- although it is not clear exactly how the tuning described here worked, it does state that the tone is divided into 3 parts, which are apparently equal. The earliest alleged use of this tuning is by Guillaume Costeley in 1558, in his chanson Seigneur Dieu ta pitié. The most well-known early reference, to both tunings, was by Salinas in 1577.

1/4-comma meantone is the only tuning of the meantone family which provides a "tone" which is the exact mean between the two sizes of 5-limit just-intonation "tones" -- the "tone" of 1/3-comma meantone and 19-edo is therefore smaller than the true meantone, and closer in size to the 10:9 ratio, the smaller of the two just-intonation tones.

Meantone family tunings are historically very important to the development of Western music, as the paradigms postulated by "common-practice" music-theory to a great extent depend on the elimination or tempering-out of the syntonic-comma, which is probably the most prominent feature of all meantones.

Note that while in the familiar 12-edo (which also belongs to the meantone family) there is a complete set of enharmonic equivalents, such that all 7 notes which have "flats" can also be "spelled" as 7 different notes which have "sharps", in all other meantones the "flats" are higher in pitch than supposedly enharmonically-equivalent "sharps", which is the opposite of the case in the much older pythagorean tuning, and also the opposite of the case in the "expressive intonation" which has been widely taught in Eurocentric schools of playing since Beethoven's time (around 1800).

Tunings of the meantone family were the closest thing to a "standard" tuning in most of Europe from approximately 1500 to 1700, and were still commonly found on keyboards (especially organs) until about 1850. It would be fair to say that most instrumental music of the Renaissance and Baroque periods was intended to be played in some form of meantone, and even after the growth in popularity of well-temperaments for keyboards after 1700, some form of meantone (generally more like 1/6-comma or 55-edo) was usually still intended for orchestral music.

During Mozart's lifetime (late 1700s) orchestral players began using an "expressive intonation" which veered back towards Pythagorean, and Beethoven's musical language certainly encouraged the spread of 12-edo, but to some extent meantone persisted in orchestral playing until approximately around the time of Wagner (mid-to-late 1800s). After the nearly universal adoption of 12-edo, the loss of meantone was lamented by Mahler in the early 1900s. (see Monzo, A Century of New Music in Vienna.)

. . . . . . . . .

The 1/3-comma meantone "5th" -- the generator -- is (3/2) / ((81/80)(1/3)). Using vector addition, that's:

       2^    3^    5^

   [ -1     1     0   ]    =  Pythagorean ("pure") "perfect 5th" of 3:2 ratio
 - [ -4/3   4/3  -1/3 ]    =  (2-4345-1)(1/3)  =  (81/80)(1/3)
 -----------------------
   [  1/3  -1/3   1/3 ]    =  1/3-comma meantone "5th"


 = ~694.7862377 cents.
				
This has the effect of tempering out the syntonic-comma so that it vanishes, thus making 4 "5ths" minus 2 "8ves" fairly close to the just "major 3rd":
       2^    3^    5^

   [  4/3  -4/3   4/3 ]    =  4  1/3-comma meantone "5ths"
 - [  2     0     0   ]    =  2 "8ves"
 -----------------------
   [ -2/3  -4/3   4/3 ]    =  1/3-comma meantone "major 3rd"  =  ~379.1449507 cents.
				
the difference between the just-intonation and the 1/3-comma meantone "major-3rds" is:
       2^    3^    5^
   [ -2/3  -4/3   4/3 ]    =  1/3-comma meantone "major 3rd"
 - [ -2     0     1   ]    =  just-intonation "major 3rd" of 5:4 ratio
   -----------------------
   [  4/3  -4/3   1/3 ]    =  error of 19-edo from just-intonation "major-3rd" =  - ~7.168763 cents.

				
This amount is exactly 1/3 of a syntonic-comma.

Assuming "octave"-equivalence (i.e., the exponents of 2 are irrelevant to the construction of the scale, so i've added 2-2 to the vector here to put the note into the reference "octave"), the next note in the cycle after the "5th", that of +2 generators, is the "whole-tone" 2(2/3) 3-(2/3) 5(2/3) = ~189.5724753 cents. If we compare this to the two just-intonation "whole-tones", subtracting the meantone from the larger pythagorean 9/8 and subtracting the smaller 5-limit 10/9 from the meantone, we find where the meantone lies between the two just-intonation "whole-tones":

       2^    3^    5^

   [ -3     2     0   ]    =  9/8
 - [  2/3  -2/3   2/3 ]    =  1/3-comma meantone
 -----------------------
   [ -2     2    -2/4 ]    =  ~14.3375264 cents  =  exactly 2/3-comma smaller than 9/8.



       2^    3^    5^

   [  2/3  -2/3   2/3 ]    =  meantone
 - [  1    -2     1   ]    =  10/9
 -----------------------
   [ -1/3   4/3  -1/3 ]    =  ~7.168763199 cents  =  exactly 1/3-comma larger than 10/9.
				

. . . . . . . . .

The 7-tone diatonic scale in 1/3-comma meantone contains only two "step" sizes: the ~189.5724753-cent "whole-tone" described above, between C:D, D:E, F:G, G:A, and A:B, and the ~126.0688117-cent "diatonic semitone" between E:F and B:C:

generator    8ves     2     3     5      ~cents

   -1         1  * [-1/3   1/3  -1/3]   505.2137623   "F"
    4     -  -2  * [ 4/3  -4/3   4/3]   379.1449507   "E"
          ---------------------------
   -5         3  * [-5/3   5/3  -5/3]   126.0688117    1/3-comma meantone diatonic semitone ("minor-2nd")
				

7-tone 1/3-comma meantone diatonic scale

7-tone 1/3-comma meantone diatonic scale

Introducing "Bb" into the scale causes a new between-degree interval to appear: the ~63.50366367-cent "chromatic semitone" between Bb:B :

generator    8ves     2     3     5      ~cents

    5        -2  * [ 5/3  -5/3   5/3]  1073.931188    "B"
   -2     -   2  * [-2/3   2/3  -2/3]  1010.427525    "Bb"
          ---------------------------
    7        -4  * [ 7/3  -7/3   7/3]   63.50366367   1/3-comma meantone chromatic semitone ("augmented prime")
				

1/3-comma meantone diatonic scale with Bb

8-tone 1/3-comma meantone diatonic scale with Bb

Continuing to add pitches at either end of the chain, we eventually come to the "typical" 12-tone chromatic scale used in Europe during the meantone era, from Eb to G#. This scale has as between-degree intervals only the two sizes of semitones, chromatic and diatonic:

12-tone 1/3-comma meantone chromatic scale 12-tone 1/3-comma meantone chromatic scale

Adding one more note to either end results in another new between-degree interval, of ~62.565148 cents, as between G#:Ab in my example here:

generator    8ves      2     3     5      ~cents

   -4         3  * [ -4/3   4/3  -4/3]  820.8550493   "Ab"
    8     -  -4  * [  8/3  -8/3   8/3]  758.2899013   "G#"
          ---------------------------
  -12         7  * [-12/3  12/3 -12/3]   62.565148     1/3-comma meantone "great" (enharmonic) diesis ("diminished-2nd")
				
13-tone 1/3-comma meantone chromatic scale 13-tone 1/3-comma meantone chromatic scale

It can be seen both from the numbers and from the graph that this interval is almost the same size as the last one derived; thus, there is essentially no difference in 1/3-comma meantone between the chromatic semitone and the enharmonic diesis (-- and in 19-edo, they do in fact become the same size) :

generator    8ves      2     3     5      ~cents

    7        -4  * [  7/3  -7/3   7/3]  63.50366367   1/3-comma meantone chromatic semitone ("augmented prime")
  -12     -   7  * [-12/3  12/3 -12/3]  62.565148     1/3-comma meantone "great" (enharmonic) diesis ("diminished-2nd")
          ---------------------------
   19       -11  * [ 19/3 -19/3  19/3]   0.938515663  1/3-comma meantone "small diesis" ("magic-comma")
				

One may continue to add 6 more notes in this manner without encountering a step-size which is radically different, thus because of the similarity of the 1/3-comma meantone chromatic semitone and enharmonic diesis, the division of the "8ve" tends to become evened-out:

19-tone 1/3-comma meantone chromatic scale

19-tone 1/3-comma meantone chromatic scale

It can be seen from the graph directly above that this divides the "8ve" into 19 nearly-identical steps, and so it can be easily inferred that other new notes will be quite similar to those already produced, and that the system is thus in effect closed at 19 notes.

If a 20th note is added, for example -7 (Cb), there occurs a new, tiny between-degree interval, which is in fact the same "small diesis" calculated above: ~0.938515663 (= ~15/16) cent, ocurring between Cb and B# :

generator    8ves     2     3     5      ~cents

   12        -6  * [12/3 -12/3  12/3]  1137.434852     "B#"
   -7     -   5  * [-7/3   7/3  -7/3]  1136.496336    "Cb"
          ---------------------------
   19       -11  * [19/3 -19/3  19/3]    0.938515663   1/3-comma meantone small diesis ("magic-comma")
				

For all intents and purposes, this difference may be disregarded, so that 19-edo can be considered identical to a 19-tone chain of 1/3-comma meantone. ~0.938515663 / 19 = ~0.049395561 cent: this miniscule amount is the difference between the 1/3-comma meantone "5th" and the 19-edo "5th".

20-tone 1/3-comma meantone scale

20-tone 1/3-comma meantone scale

Below is a graph showing the pitch-height of this 20-tone chain of 1/3-comma meantone. The red line connects the two pitches which are close together.

20-tone 1/3-comma meantone chain

20-tone 1/3-comma meantone chain

. . . . . . . . .

19-edo is audibly indistinguishable from 1/3-comma meantone. 2(11/19) = 696.7741935 cents. Using vector addition again to compare the 1/3-comma meantone "5th" with the 19-EDO "5th", we get a difference between the two of:

         2^     3^    5^

   [   1/3   -1/3  1/3 ]    =  ~694.7862377 cents  =  1/3-comma meantone "5th"
 - [  11/19   0    0   ]    =  ~694.7368421 cents  =  19-EDO "5th"
 -----------------------
   [ -14/57  -1/3  1/3 ]    =  19-edo "5th" "-" 1/3-comma meantone "5th"


 = ~0.049395561 cent  = ~1/20 cent
				

Thus, 2(19/3) * 3-(19/3) * 5(19/3) acts as a unison-vector which is not tempered out in 1/3-comma meantone, and it acts as a unison-vector which is tempered out in 19-edo. Because this interval is so small, it really makes no difference whether or not it is "officially" tempered out: it will sound like a unison in either case.

Salinas in 1577 (De Musica, book 3, chapter 16) described 1/3-comma meantone with mathematical exactitude for the first time. First he constructed a 24-tone just-intonation system, which had duplicate pairs of some pitches a syntonic-comma apart, the higher of which was labeled "superius" and the lower "inferius". Then he explained the amount of tempering for each of the meantone pitches. By tempering out the full comma which exists between the 5 pairs of "superius/inferius" pitches, he reduced the number of pitches from 24 to 19.

Below is a lattice which places Salinas's 1/3-comma meantone in prime-space and shows its relationship to his just-intonation system, as he describes it; the slanted arrows represent the syntonic-comma:

Salinas - 1/3-comma meantone Salinas: explanation of 1/3-comma meantone temperament

It can be seen that all pitches of the meantone are either exactly those of Salinas's just-intonation system, or are 1/3, 2/3, or a full comma higher or lower than those in his just-intonation system.

He goes on to explain how to temper the 24-note just-intonation system into 19 notes of 2/7-comma meantone, and then also into 19 notes of 1/4-comma meantone, the latter of which he finally declares to be the best of the three temperaments.

Salinas did not explicitly mention the equal nature of the 8ve-division in his 1/3-comma meantone, but he would have known about it himself and it can be inferred from the measurements he described. The 19-tone systems of the 2/7-comma and 1/4-comma meantone are less equally-spaced, being rather closer to subsets of 50-edo and 31-edo, respectively.

Below is a 5-limit triangular lattice-diagram of just-intonation ratios, showing an example 19-tone periodicity-block which is defined and bounded by 2 unison-vectors the syntonic-comma and the magic-comma, which can form a basis to be tempered into 19-edo. Data given for each lattice-point are the note's ratio representation, 19-edo degree, and letter-name (with accidental where needed).


19-edo

5-limit meantone

unison-vectors:

  monzo      ratio        name
 [ 4, -1>    81:80     syntonic-comma
 [-1,  5>  3125:3072   magic-comma


C = 1:1

                 125:96   125:64  375:256
                    7 ----- 18 ---- 10
                    E#      B#      Fx
                   /  \    /  \    /  \
                  /    \  /    \  /    \
              25:24   25:16    75:64  225:128
                1 ----- 12 ----- 4 ---- 15
               C#       G#      D#      A#
              /  \    /  \    /  \    /
             /    \  /    \  /    \  /
         5:3      5:4    15:8     45:32
          14 ----- 6 ----- 17 ----- 9
           A       E       B       F#
         /  \    /  \    /  \    /
        /    \  /    \  /    \  /
     4:3     1:1     3:2     9:8
      8 ----- 0 ----- 11 ---- 3
      F       C       G       D
    /  \    /  \    /  \    /
   /    \  /    \  /    \  /
16:15   8:5    6:5     9:5
  2 ---- 13 --- 5 ----- 16
  Db     Ab     Eb      Bb

			

In fact this structure perfectly describes Salinas's just-intonation structure as described above.

Below are two circular graphs of 19-edo in its typical meantone usage (be aware that 19-edo also belongs to other temperament families!). One graph arranges the pitches in scalar order by 1/3-tone scale degree generators around the circle, and the other arranges the pitches in order of diatonic-5th generators; both show the enharmonic equivalences which occur in 19-edo.

19-edo circle-of-degrees

19-edo circle of scale-degrees

19-edo circle-of-5ths

19-edo circle-of-5ths (generators)
. . . . . . . . .

Below is a table showing how some of the most common just-intonation ratios are mapped to 19-edo, using the closest-fit mappings:

====================================
edomap.py (c)2008-12-23 by Joe Monzo
====================================

Finds the EDO mappings for a set of ratios

=============== new run =====================
please enter edo: 19 -edo 

floating-point mappings:
 2 = 19.0          ... displacement? (0 if none): 0
 3 = 30.1142875137 ... displacement? (0 if none): 0
 5 = 44.1166338029 ... displacement? (0 if none): 0
 7 = 53.3397435191 ... displacement? (0 if none): 0
11 = 65.7292007541 ... displacement? (0 if none): 0

integer (i.e., true) mappings, compared with cents-value of actual prime
map  2 = 19 = 1200.0 cents        (::) 1200.0
map  3 = 30 = 1894.73684211 cents (::) 1901.95500087
map  5 = 44 = 2778.94736842 cents (::) 2786.31371386
map  7 = 53 = 3347.36842105 cents (::) 3368.82590647
map 11 = 66 = 4168.42105263 cents (::) 4151.31794236

--------------

examples:
             ratio =        19 -edo mapping:
      ratio       cents         edo    cents    name
      2	:1      = 1200.0 -->  19 /19 = 1200.0  (octave)	
     15	:8      = 1088.3 -->  17 /19 = 1073.7  (just major-7th)	
     11	:6      = 1049.4 -->  17 /19 = 1073.7  (undecimal submajor-7th)	
     20	:11     = 1035.0 -->  16 /19 = 1010.5  (undecimal superminor-7th)	
      9	:5      = 1017.6 -->  16 /19 = 1010.5  (just minor-7th)	
     16	:9      =  996.1 -->  16 /19 = 1010.5  (pythagorean minor-7th)	
      7	:4      =  968.8 -->  15 /19 =  947.4  (harmonic-7th)	
     12	:7      =  933.1 -->  15 /19 =  947.4  (septimal supermajor-6th)	
      5	:3      =  884.4 -->  14 /19 =  884.2  (just major-6th)	
      8	:5      =  813.7 -->  13 /19 =  821.1  (just minor-6th)	
     14	:9      =  764.9 -->  12 /19 =  757.9  (septimal subminor-9th)	
      3	:2      =  702.0 -->  11 /19 =  694.7  (perfect-5th)	
     16	:11     =  648.7 -->  10 /19 =  631.6  (11th-subharmonic)	
     10	:7      =  617.5 -->  10 /19 =  631.6  (septimal large-tritone)	
      7	:5      =  582.5 -->   9 /19 =  568.4  (septimal small-tritone)	
     11	:8      =  551.3 -->   9 /19 =  568.4  (harmonic-11th)	
      4	:3      =  498.0 -->   8 /19 =  505.3  (perfect-4th)	
      9	:7      =  435.1 -->   7 /19 =  442.1  (septimal supermajor-3rd)	
      5	:4      =  386.3 -->   6 /19 =  378.9  (just major-3rd)	
      6	:5      =  315.6 -->   5 /19 =  315.8  (just minor-3rd)	
      7	:6      =  266.9 -->   4 /19 =  252.6  (septimal subminor-7th)	
      8	:7      =  231.2 -->   4 /19 =  252.6  (septimal tone)	
      9	:8      =  203.9 -->   3 /19 =  189.5  (tone)	
     10	:9      =  182.4 -->   3 /19 =  189.5  (just minor-tone)	
     11	:10     =  165.0 -->   3 /19 =  189.5  (undecimal small-tone)	
     12	:11     =  150.6 -->   2 /19 =  126.3  (undecimal large-semitone)	
     16	:15     =  111.7 -->   2 /19 =  126.3  (just diatonic-semitone)	
     25	:24     =   70.7 -->   1 /19 =   63.2  (just chromatic-semitone)	
      1	:1      =    0.0 -->   0 /19 =    0.0  (prime)	
 531441	:524288 =   23.5 -->  -1 /19 =  -63.2  (pythagorean-comma)	aristoxenean
    648	:625    =   62.6 -->   1 /19 =   63.2  (major-diesis)	diminished
    250	:243    =   49.2 -->   1 /19 =   63.2  (maximal-diesis)	porcupine
    128	:125    =   41.1 -->   1 /19 =   63.2  (enharmonic-diesis)	augmented
   3125	:3072   =   29.6 -->   0 /19 =    0.0  (magic-comma)	magic
     81	:80     =   21.5 -->   0 /19 =    0.0  (syntonic-comma)	meantone
   2048	:2025   =   19.6 -->   1 /19 =   63.2  (diaschisma)	pajara
  15625	:15552  =    8.1 -->   0 /19 =    0.0  (kleisma)	kleismic
  32805	:32768  =    2.0 -->  -1 /19 =  -63.2  (skhisma)	schismic
     36	:35     =   48.8 -->   1 /19 =   63.2  (septimal-diesis)	
     49	:48     =   35.7 -->   0 /19 =    0.0  (septimal-comma-3)	
     50	:49     =   35.0 -->   1 /19 =   63.2  (septimal-comma-2)	
     64	:63     =   27.3 -->   1 /19 =   63.2  (septimal-comma)	
    225	:224    =    7.7 -->   0 /19 =    0.0  (septimal-kleisma)	
     22	:21     =   80.5 -->   2 /19 =  126.3  ()	
     33	:32     =   53.3 -->   1 /19 =   63.2  (undecimal-diesis)	
     45	:44     =   38.9 -->   0 /19 =    0.0  ()	
     55	:54     =   31.8 -->   1 /19 =   63.2  ()	
     56	:55     =   31.2 -->   0 /19 =    0.0  ()	
-------------------- end of coded printout ----------------

run again? - enter 1 for yes, 0 for no: 
				

Below is a graph showing the degrees of 19-edo as the Y-axis grid, with the pitch of the JI ratios plotted against it:

Below is a graph showing the degrees of 19-edo along the X-axis, with the Y-value of zero representing the pitch of the 19-edo degrees, and the deviation of the JI ratios from the 19-edo degrees plotted against that:

Below is a Tonescape® Lattice illustrating some of the ratios approximated by this mapping:

. . . . . . . . .

Below are graphs of the error of 12-edo and 19-edo for approximations to the prime-factors up to 43, given as a percentage of one degree of each tuning. It can easily be seen that the big improvements of 19-edo over 12-edo are the approximations to prime-factors 11, 13, 23, 31, and 37, and that 19-edo's approximations to 3, 17, and 19 are considerably worse than 12-edo's.

. . . . . . . . .
Some rational convergents to the 19-edo degrees
19=0 2:1 = 1:1
18 25:13 , 27:14
17 13:7 , 119:64 , 251:135
16 9:5 , 43:24 , 52:29, 95:53 , 147:82
15 19:11 , 121:70 , 140:81 , 401:232
14 5:3
13 8:5 , 45:28 , 143:89 , 188:117
12 14:9 , 17:11 , 31:20 , 79:51 , 110:71
11 3:2 , 118:79 , 121:81 , 239:160
10 13:9 , 36:25 , 229:159
9 7:5 , 18:13 , 25:18 , 318:229
8 4:3 , 79:59 , 241:180 , 320:239
7 9:7 , 31:24 , 71:55
6 5:4 , 56:45 , 61:49 , 117:94
5 6:5
4 7:6 , 15:13 , 22:19 , 59:51 , 81:70
3 9:8 , 10:9 , 19:17 , 29:26 , 135:121 , 299:268 , 328:294
2 13:12 , 14:13 , 55:53 , 71:66 , 199:18 5 , 270:251
1 26:25 , 27:26 , 28:27 , 307:296
0 1:1
progressing to the right = greater accuracy of approximation and greater complexity

Below is a proposal for a new staff-notation based on 19-edo, where the naturals (nominals) are notated on thin lines and the sharps and flats on thick lines.

19-edo-staff notation
[Joe Monzo]