# heptaméride

[Joe Monzo]

A term coined by Joseph Saveur to designate the small interval which represents 1000 * log10(2), thus obviating the need to calculate logarithms.

A heptaméride is calculated as the 301st root of 2, or 2(1/301), with a ratio of approximately 1:1.002305468. It is an irrational number. A heptaméride is ~3.986710963 cents, or just under 4 cents.

The formula for calculating the heptaméride-value of any ratio is: heptamérides = log10(ratio) * [301 / log10(2)] or heptamérides = log2(ratio) * 301

The heptaméride therefore divides the octave, which is assumed to have the ratio 2:1, into 301 equal parts. Thus a heptaméride represents one degree of 301-edo tuning.

One potential defect of using heptamérides is that the familiar 12-edo semitone does not come out with an integer number of heptamérides, since 301 does not divide evenly by 12. Thus, the 12-edo semitone is ~25.08333333, or exactly 25 & 1/12, heptamérides.

Note that Sauveur's "heptamérides" are related to "jots", being simply a less accurate rounding. Saveur chose this measurement partly because he also used "mérides" of 2(1/43), and 7 * 43 = 301, so both of his units divided evenly.

The "savart" was originally identical to the heptaméride, but was later "rationalized" to be 2(1/300). (Many thanks to John Chalmers for clarifying the history of savarts.)

Below is a table of heptaméride values for some 7-limit JI intervals, all 23 intervals which occur in a 12-tone version of 1/5-comma meantone and its close relative 43-edo, and some of the "5ths" of other meantones, with cents-values given for comparison:

```                       ------ 2,3,5,7,11-monzo -------
2      3  ,   5     7   11     ~cents     heptamérides

octave                [  1      0  ,   0     0    0 >   1200            301
1/5cMT dim-prime      [ 32/5   -7/5,  -7/5   0    0 >   1116.423799    ~280
1/5-cMT maj-7th       [ -3      1  ,   1     0    0 >   1088.268715    ~273     =  15:8   ratio
1/5-cMT min-7th       [ 12/5   -2/5,  -2/5   0    0 >   1004.692514    ~252

43edo aug-6th         [ 35/43   0  ,   0     0    0 >    976.744186     245
1/5-cMT aug-6th       [ -7      2  ,   2     0    0 >    976.5374295   ~245     = 225:128 ratio
7:4 harmonic 7th      [ -2      0  ,   0     1    0 >    968.8259065   ~243

1/5-cMT dim-7th       [ 39/5   -9/5,  -9/5   0    0 >    921.1163135   ~231
1/5-cMT maj-6th       [ -8/5    3/5,   3/5   0    0 >    892.9612288   ~224

1/5-cMT min-6th       [ 19/5   -4/5,  -4/5   0    0 >    809.3850282   ~203
1/5-cMT aug-5th       [-28/5    8/5,   8/5   0    0 >    781.2299436   ~196
25:16 aug-5th         [ -4      0  ,   2     0    0 >    772.6274277   ~193 4/5

1/5-cMT dim-6th       [ 46/5  -11/5, -11/5   0    0 >    725.8088276   ~182

3:2 perfect-5th       [ -1      1  ,   0     0    0 >    701.9550009   ~176
12edo 5th             [  7/12   0  ,   0     0    0 >    700           ~175 4/7
1/6-cMT 5th           [ -1/3    1/3,   1/6   0    0 >    698.3706193   ~175 1/6
43edo 5th             [ 25/43   0  ,   0     0    0 >    697.6744186    175
1/5-cMT 5th           [ -1/5    1/5,   1/5   0    0 >    697.6537429   ~175
1/4-cMT 5th           [  0      0  ,   1/4   0    0 >    696.5784285   ~174 5/7
2/7-cMT 5th           [  1/7   -1/7,   2/7   0    0 >    695.8103467   ~174 1/2

10:7 tritone          [  1      0  ,   1    -1    0 >    617.4878074   ~154 8/9
1/5-cMT dim-5th       [ 26/5   -6/5,  -6/5   0    0 >    614.0775423   ~154
1/5-cMT aug-4th       [-21/5    6/5,   6/5   0    0 >    585.9224577   ~147
7:5 tritone           [  0      0  ,  -1     1    0 >    582.5121926   ~146 1/9

11:8                  [ -3      0  ,   0     0    1 >    551.3179424   ~138 2/7

1/5-cMT p-4th         [  6/5   -1/5,  -1/5   0    0 >    502.3462571   ~126
4:3 perfect 4th       [  2     -1  ,   0     0    0 >    498.0449991   ~125
1/5-cMT aug-3rd       [-41/5   11/5,  11/5   0    0 >    474.1911724   ~119

1/5-cMT dim-4th       [ 33/5   -8/5,  -8/5   0    0 >    418.7700564   ~105
43edo major-3rd       [ 14/43   0  ,   0     0    0 >    390.6976744    ~98
1/5-cMT major-3rd     [-14/5    4/5,   4/5   0    0 >    390.6149718    ~98
5:4 major-3rd         [ -2      0  ,   1     0    0 >    386.3137139    ~97

6:5 minor-3rd         [  1      1  ,  -1     0    0 >    315.641287     ~79 1/6
1/5-cMT min-3rd       [ 13/5   -3/5,  -3/5   0    0 >    307.0387712    ~77
1/5-cMT aug-2nd       [-34/5    9/5,   9/5   0    0 >    278.8836865    ~70
7:6 septimal 3rd      [ -1     -1  ,   0     1    0 >    266.8709056    ~67

8:7 septimal tone     [  3      0  ,   0    -1    0 >    231.1740935    ~58
1/5-cMT dim-3rd       [  8     -2  ,  -2     0    0 >    223.4625705    ~56     = 256:225 ratio
9:8 greater tone      [ -3      2  ,   0     0    0 >    203.9100017    ~51 1/7
43edo tone            [  7/43   0  ,   0     0    0 >    195.3488372     49
1/5-cMT tone          [ -7/5    2/5,   2/5   0    0 >    195.3074859    ~49
1/4-cMT tone          [  0      0  ,   1/2   0    0 >    193.1568569    ~48 4/9
10:9 lesser tone      [  1     -2  ,   1     0    0 >    182.4037121    ~45 3/4

1/5-cMT min-2nd       [  4     -1  ,  -1     0    0 >    111.7312853    ~28     =  16:15  ratio

1/5-cMT chr semitone  [-27/5    7/5,   7/5   0    0 >     83.57620062   ~21
JI chromatic semitone [ -3     -1  ,   2     0    0 >     70.67242686   ~17 5/7

43edo meride          [  1/43   0  ,   0     0    0 >     27.90697674     7
50edo degree          [  1/50   0  ,   0     0    0 >     24             ~6
Pythagorean comma     [-19     12  ,   0     0    0 >     23.46001038    ~5 8/9
53edo comma           [  1/53   0  ,   0     0    0 >     22.64150943    ~5 2/3
syntonic comma        [ -4      4  ,  -1     0    0 >     21.5062896     ~5 2/5
60edo degree          [  1/60   0  ,   0     0    0 >     20             ~5
75edo degree          [  1/75   0  ,   0     0    0 >     16             ~4
100edo degree         [  1/100  0  ,   0     0    0 >     12             ~3
kleisma               [ -6     -5  ,   6     0    0 >      8.107278862   ~2
152edo degree         [  1/152  0  ,   0     0    0 >      7.894736842   ~2
225:224 sept.kleisma  [ -5      2  ,   2    -1    0 >      7.711522991   ~2
skhisma               [-15      8  ,   1     0    0 >      1.953720788   ~1/2
```

Because 301-edo is a multiple of 43edo, which in turn is a very close approximation of 1/5-comma meantone, the system of heptamérides provides a system of integer interval-measurement for these meantones.

heptamérides also give nearly integer values for a fairly large central portion of the 7-limit lattice (as shown on the equal-temperament definition). Thus, they provide a very useful system of measurement where the interest is in comparing 7-limit JI with 43edo and its close relative 1/5-comma meantone, because most of the basic intervals in these tunings are nearly integer heptaméride values. For example, note that:

• 3:2 Pythagorean perfect-5th minus 43edo & 1/5-comma meantone 5th
= ~176.0737127 - 175 = ~1.073712717 heptaméride
and ~176.0737127 - ~174.9948139 = ~1.078898861 heptaméride
• 43edo & 1/5-comma meantone major-3rd minus 5:4 JI major-3rd
= 98 - ~96.90035656 = ~1.099643439 heptaméride
and ~97.97925542 - ~96.90035656 = ~1.078898861 heptaméride
• 43edo & 1/5-comma meantone augmented-6th minus 7:4 JI harmonic minor-7th
= 245 - ~243.0138315 = ~1.986168461 heptamérides
and ~244.9481386 - ~243.0138315 = ~1.934307017 heptamérides

So it is essentially correct to say that the meantone "5ths" are 1 heptaméride narrower than the "pure" 3:2, the meantone "major-3rds" are 1 heptaméride wider than the "pure" 5:4, and the meantone "augmented-6ths" are 2 heptamérides wider than the "pure" 7:4.

. . . . . . . . .

### heptamérides 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: = heptamérides

. . . . . . . . .
###### Reference

Ellis, Alexander. 1885.

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

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