heptameride

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

A heptameride 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 heptameride is ~3.986710963 cents, or just under 4 cents.

The formula for calculating the heptameride-value of any ratio is:

heptamerides = log₁₀(ratio) * [301 / log₁₀(2)]

This interval therefore divides the "octave", which is assumed to have the ratio 2:1, into 301 equal parts. Thus a heptameride represents one degree in 301-EDO tuning.

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

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

The "savart" was originally identical to this, 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 heptameride 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 43edo, and some of the "5ths" of other meantones, with cents-values given for comparison:

                       ----- prime-factor vector -----
                         2      3       5      7    11     ~cents     heptamerides

8ve                     1                               1200            301 
1/5cMT dim-1me          7/5   -7/5   -7/5               1116.423799    ~280    
1/5-cMT maj-7th        -1      1      1                 1088.268715    ~273    = 15:8 ratio
1/5-cMT min-7th         2/5   -2/5   -2/5               1004.692514    ~252
     
43edo aug-6th          35/43                             976.744186     245    
1/5-cMT aug-6th        -2      2      2                  976.5374295   ~245    
7:4 harmonic 7th        0      0      0      1           968.8259065   ~243 
   
1/5-cMT dim-7th         9/5   -9/5   -9/5                921.1163135   ~231    
1/5-cMT maj-6th        -3/5    3/5    3/5                892.9612288   ~224
    
1/5-cMT min-6th         4/5   -4/5   -4/5                809.3850282   ~203    
1/5-cMT aug-5th        -8/5    8/5    8/5                781.2299436   ~196
25:16 aug-5th           0      2                         772.6274277   ~193 4/5

1/5-cMT dim-6th        11/5  -11/5  -11/5                725.8088276   ~182
    
3:2 perfect-5th         1                                701.9550009   ~176    
12edo 5th               7/12                             700           ~175 4/7
1/6-cMT 5th            -1/3    1/3    1/6                698.3706193   ~175 1/6
43edo 5th              25/43                             697.6744186    175    
1/5-cMT 5th            -1/5    1/5    1/5                697.6537429   ~175    
1/4-cMT 5th                           1/4                696.5784285   ~174 5/7
2/7-cMT 5th             1/7   -1/7    2/7                695.8103467   ~174 1/2

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

11:8                    0      0      0      0     1     551.3179424   ~138 2/7

1/5-cMT p-4th           1/5   -1/5   -1/5                502.3462571   ~126    
4:3 perfect 4th         2     -1                         498.0449991   ~125    
1/5-cMT aug-3rd       -11/5   11/5   11/5                474.1911724   ~119
   
1/5-cMT dim-4th         8/5   -8/5   -8/5                418.7700564   ~105    
43edo major-3rd        14/43                             390.6976744    ~98    
1/5-cMT major-3rd      -4/5    4/5    4/5                390.6149718    ~98    
5:4 major-3rd          -2      0      1                  386.3137139    ~97
    
6:5 minor-3rd           1      1     -1                  315.641287     ~79 1/6
1/5-cMT min-3rd         3/5   -3/5   -3/5                307.0387712    ~77    
1/5-cMT aug-2nd        -9/5    9/5    9/5                278.8836865    ~70    
7:6 septimal 3rd       -1     -1      0     1            266.8709056    ~67
    
8:7 septimal tone       0      0      0    -1            231.1740935    ~58    
1/5-cMT dim-3rd         2     -2     -2                  223.4625705    ~56    
9:8 greater tone        2                                203.9100017    ~51 1/7
43edo tone              7/43                             195.3488372     49    
1/5-cMT tone           -2/5    2/5    2/5                195.3074859    ~49    
1/4-cMT tone                          1/2                193.1568569    ~48 4/9
10:9 lesser tone       -2      1                         182.4037121    ~45 3/4

1/5-cMT min-2nd         1     -1     -1                  111.7312853    ~28    = 16:15 ratio

1/5-cMT chr semitone   -7/5    7/5    7/5                 83.57620062   ~21    
JI chromatic semitone         -1      2                   70.67242686   ~17 5/7

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

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

Heptamerides 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 heptameride values.

For example, note that:

• 3:2 Pythagorean perfect-5th minus 43edo & 1/5-comma meantone 5th
  = ~176.0737127 - 175 = ~1.073712717 heptameride
  and ~176.0737127 - ~174.9948139 = ~1.078898861 heptameride

• 43edo & 1/5-comma meantone major-3rd minus 5:4 JI major-3rd
  = 98 - ~96.90035656 = ~1.099643439 heptameride
  and ~97.97925542 - ~96.90035656 = ~1.078898861 heptameride

• 43edo & 1/5-comma meantone augmented-6th minus 7:4 JI harmonic minor-7th
  = 245 - ~243.0138315 = ~1.986168461 heptamerides
  and ~244.9481386 - ~243.0138315 = ~1.934307017 heptamerides

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

REFERENCES

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

[from Joe Monzo, JustMusic: A New Harmony]

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