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:
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.
Ellis, Alexander. 1885.
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