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méride / 43-edo / 43-tone equal-temperament

[Joe Monzo]

Méride is a term coined by Joseph Saveur to designate the larger of his two interval measurements, the smaller being the heptaméride.

A meride is calculated as the 43rd root of 2, or 2(1/43), with a ratio of approximately 1:1.016250325. It is an irrational number.

A meride is exactly 2739/43 or 27.906976744186046511627... cents, or just a bit larger than several of the most common commas.

The formula for calculating the meride-value of any ratio is: merides = log10(ratio) * [43 / log10(2)] or merides = log2(ratio) * 43.

This interval therefore divides the octave, which is assumed to have the ratio 2:1, into 43 equal parts. Thus a meride represents one degree in 43-edo tuning.

One potential defect of using merides is that the familiar 12-edo semitone does not come out with an integer number of merides, since 43 is a prime number and therefore does not divide evenly by 12. Thus, the 12-EDO semitone is ~3.583333333, or exactly 3 & 7/12, merides.

Note that 7 * 43 = 301, so both of Saveur's units, the merides and heptamerides, did divide evenly.

The interval which functions as the 5th in 43-edo is nearly identical to the 1/5-comma meantone "5th":

43-edo "5th"  =  2(25/43)  =  25 merides  =  ~697.6744186 cents.


1/5-comma meantone "5th"  =  (3/2) / ( (81/80)^(1/5) )
=
  2-(5/5) * 3(5/5) * 5(0/5)    Pythagorean "perfect 5th", ratio 3:2
- 2-(4/5) * 3(4/5) * 5-(1/5)   1/5 syntonic comma
----------------------------
  2-(1/5) * 3(1/5) * 5(1/5)    =  ~24.99925912 (= 24 ~1349/1350) merides  =  ~697.6537429 cents.


Difference between 43-edo and 1/5-comma "5ths":
=
  2-(125/215)                       43-edo "5th"
- 2-(43/215) * 3(1/5)  * 5(1/5)     1/5-comma meantone "5th"
-------------------------------
  2(168/215) * 3-(1/5) * 5-(1/5)    =  ~0.020675659 (= ~1/50) cent  =  ~1/2 jot.
			

Here is a graph of a 44-tone cycle of 1/5-comma meantone, centered on "C" as the reference (= generator 0):

Note that the pitches represented as Fbbb (-22 generators) and C### (+21 generators) are nearly identical:

Fbbb = ( (3/2)-22 / ( (81/80)(-22/5) ) ) * 213

     =  2(87/5) * 3-(22/5) * 5-(22/5)

     =  ~251.6176552 cents


C### = ( (3/2)21 / ( (81/80)(21/5) ) ) / 212

     =  2-(81/5) * 3(21/5) * 5(21/5)

     =  ~250.7286019 cents


  2(87/5) * 3-(22/5) * 5-(22/5)    Fbbb
- 2-(81/5) * 3(21/5) * 5(21/5)     C###
----------------------------
  2(168/5) * 3(-43/5) * 5(-43/5)    =  ~0.88905332 (= ~8/9) cent.


 0.88905332 / 43  =  0.020675659 --> compare with above.
			

Thus, limiting the meantone cycle to 43 pitches and distributing this difference equally among them, results in 43-edo. The chromatic semitone = 3 merides, the diatonic semitone = 4 merides, and the whole-tone = 7 merides.

These are Gene Ward Smith's 7-limit MT reduced bases for 43-edo: 43: [81/80, 126/125, 12288/12005], which can also be written as a matrix of monzos thus:

2,3,5,7-monzo

[-4  4, -1  0 >   =     81/80
[ 1  2, -3  1 >   =    126/125
[12  1, -1 -4 >   =  12288/12005
			

1/5-comma meantone was first described by Abraham Verheyen in a letter to Simon Stevin, around 1600.

Below is a lattice which shows one pattern by which the plane of the [3 5] lattice is tiled by 43-edo:

Below is a table of intervals, including all 23 intervals which occur in a 12-tone subset of 1/5-comma meantone, and some of the 43-edo equivalents of those, as well as several others in JI and in various other meantones.

                     --- prime-factor vector ---    cents     merides
                        2     3     5    7   11

8ve                    1                          1200            43
1/5cMT dim-1me         7/5  -7/5  -7/5           ~1116.423799    ~40
1/5-cMT maj-7th       -1     1     1             ~1088.268715    ~39
1/5-cMT min-7th        2/5  -2/5  -2/5           ~1004.692514    ~36
12edo min-7th          5/6                        1000           ~35 5/6
43-edo aug-6th         35/43                       ~976.744186     35
1/5-cMT aug-6th       -2     2     2              ~976.5374295   ~35
7:4                   -2     0     0     1        ~968.8259065   ~34 5/7
1/5-cMT dim-7th        9/5  -9/5  -9/5            ~921.1163135   ~33
1/5-cMT maj-6th       -3/5   3/5   3/5            ~892.9612288   ~32
1/5-cMT min-6th        4/5  -4/5  -4/5            ~809.3850282   ~29
1/5-cMT aug-5th       -8/5   8/5   8/5            ~781.2299436   ~28
25:16 aug-5th         -4     0     2              ~772.6274277   ~27 2/3
1/5-cMT dim-6th       11/5 -11/5 -11/5            ~725.8088276   ~26
3:2                   -1     1                    ~701.9550009   ~25 1/7
12edo 5th              7/12                        700           ~25
1/6-cMT 5th           -1/3   1/3   1/6            ~698.3706193   ~25
43-edo 5th             25/43                       ~697.6744186    25
1/5-c MT 5th          -1/5   1/5   1/5            ~697.6537429   ~25
1/4-c MT 5th           0     0     1/4            ~696.5784285   ~25
2/7-cMT 5th            1/7  -1/7   2/7            ~695.8103467   ~25
10:7 tritone           1     0     1    -1        ~617.4878074   ~22 1/8
1/5-cMT dim-5th        6/5  -6/5  -6/5            ~614.0775423   ~22
1/5cMT aug-4th        -6/5   6/5   6/5            ~585.9224577   ~21
7:5 tritone            0     0    -1     1        ~582.5121926   ~20 7/8
11:8                  -3     0     0     0    1   ~551.3179424   ~19 3/4
1/5-cMT p4th           1/5  -1/5  -1/5            ~502.3462571   ~18
4:3                    2    -1                    ~498.0449991   ~17 6/7
1/5-cMT aug-3rd      -11/5  11/5  11/5            ~474.1911724   ~17
1/5-cMT dim-4th        8/5  -8/5  -8/5            ~418.7700564   ~15
12edo maj-3rd          1/3                         400           ~14 1/3
43-edo major-3rd       14/43                       ~390.6976744    14
1/5-c MT major-3rd    -4/5   4/5   4/5            ~390.6149718   ~14
5:4                   -2     0     1              ~386.3137139   ~13 5/6
6:5                    1     1    -1              ~315.641287    ~11 1/3
1/5-cMT min-3rd        3/5  -3/5  -3/5            ~307.0387712   ~11
12edo min-3rd          1/4                         300           ~10 3/4
1/5-cMT aug-2nd       -9/5   9/5   9/5            ~278.8836865   ~10
7:6                   -1    -1     0     1        ~266.8709056    ~9 4/7
8:7                    3     0     0    -1        ~231.1740935    ~8 2/7
1/5-cMT dim-3rd        2    -2    -2              ~223.4625705    ~8
9:8                   -3     2                    ~203.9100017    ~7 1/3
43-edo tone             7/43                        195.3488372     7
1/5-c MT tone         -2/5   2/5   2/5            ~195.3074859    ~7
1/4-c MT tone                      1/2            ~193.1568569    ~7
10:9                   1    -2     1              ~182.4037121    ~6 1/2
1/5-cMT min-2nd        1    -1    -1              ~111.7312853    ~4
12edo semitone         1/12                       ~100            ~3 3/5
1/5-cMT chr semitone  -7/5   7/5   7/5             ~83.57620062   ~3
chromatic semitone    -3    -1     2               ~70.67242686   ~2 1/2
43-edo meride           1/43                        ~27.90697674    1
Pythagorean comma    -19    12                     ~23.46001038     ~5/6
syntonic comma        -4     4    -1               ~21.5062896      ~7/9
			

Below is a graphic showing a 7-limit "closest to 1/1" (by the taxicab metric) periodicity-block for 43-edo, in which three 5-limit planes are shown side-by-side; exponents of 7 are 0, 1, and -1 from left to right.

Below is a plot of the points in a 3-dimensional Monzo lattice of the same 7-limit "closest to 1/1" periodicity-block for 43-edo. In addition to the usual rectangular vertices, here i have also drawn lines connecting the doubled and trebled pitches (shown in darker grey in the "5-limit sheets" lattices above).

Below is a table and graphs which show the error of 43-edo in representing all ratios in the 11-odd-limit (i.e., those in Partch's tonality diamond), both the absolute error in cents, and the relative error as a percentage of one degree of 43-edo.

It can be seen that 43-edo's success at representing the 11-limit ratios can be categorized as follows:

REFERENCES

Ellis, Alexander.  1885.
Appendix XX, in his translation of
Helmholtz, On the Sensations of Tone, p 437.
Dover reprint 1954.
			
. . . . . . . . .

mérides calculator

Ratio may be entered as fraction or floating-point decimal number.
(value must be greater than 1)

For EDOs (equal-temperaments), type: "a/b" (without quotes)
where "a" = EDO degree and "b" = EDO cardinality.
(value must be less than 1)

Enter ratio: = mérides

. . . . . . . . .

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