In the following discussion I refer to octaves using the Helmholtz standard markings, where a’ is the same as the USA standard A4, the note A with a frequency of 440 cps. This is in order to use number suffixes as lattice identifiers (“names”) without confusion.

Taking A=440, a’, as the center, the generator, and the base of a stack of pure
fifths, name it A40. That is, assign it to a lattice identified with the suffix 40. The name given to the octave higher A=880, a”, becomes A50; the name for the octave lower, A=220, a, becomes A30.

The harmonic fifth above A=440 is in a higher Helmholtz octave, E=660, e”, but remains named in reference to the original A40, so is identified as E40. The fifth below A=440, D=293.33, d’, is identified as D40.

The preceding preliminary discussion is intended to clarify that the id numbers do not match octave numbers as in the USA standard.

The ’30’ and the ’40’ Pythagorean stacks of pure fifths, which form the backbones for two lattices identified as ’30’ and ’40’, are given below. Each note is given an approximate frequency and its Hemholtz octave marking. The center note of the 30 stack is A=220cps, an octave lower than the center note of the 40 stack (A=440cps). The 50 stack, not illustrated, is an octave higher than the 40 stack, and there are further octave transpositions above (60, 70, and so on) and below (20, 10, and so on, and ignoring for the moment the arithmetical problems the 00 stack might pose).

   E#30 etc             E#4  etc                
    A#30 = A#(3k+) =a#''''      A#4  = A#(7k+) = a#v
    D#30 = D#(2k+) =d#''''      D#4  = D#(5k+) = d#v
    G#30 = G#(1k+) =g#'''       G#4  = G#(3k+) = g#''''
    C#30 = C#(1k+) =c#'''       C#4  = C#(2k+) = c#''''
    F#30 = F#(74x) =f#''        F#4  = F#(1k+) = f#'''
    B 30 = B (495) =b'          B 4  = B (990) = b''
    E 30 = E (330) =e'          E 4  = E (660) = e''
--- A 30 = A (220) =a  ---  --- A 4  = A (440) = a' ---
    D 30 = D (293) =d           D 4  = D (293) = d'
    G 30 = G (196) =G           G 4  = G (196) = g
    C 30 = C (12x) =C           C 4  = C (12x) = c
    F 30 = F (08x) =F,          F 4  = F (08x) = F
    Bb30 = Bb(05x) =Bb,         Bb4  = Bb(05x) = Bb,
    Eb30 = Eb(03x) =Eb,,        Eb4  = Eb(03x) = Eb,
    Ab30 = Ab(02x) =Ab,,,       Ab4  = Ab(02x) = Ab,,
    Db30 etc                    Db4  etc          

The ’31’, ’41’ and ’51’ stacks of pure major thirds in relation to each of
the stack of fifths:

Fx31       fx'''' | Fx41       fxv    | Fx51       fxv'   
    D#30   d#'''' |     D#40   d#v    |     D#50   d#v'   
B#31       b#'''  | B#41       b#'''  | B#51       b#v       
    G#30   g#'''  |     G#40   g#'''  |     G#50   g#v       
E#31       e#'''  | E#41       e#'''  | E#51       e#v       
    C#30   c#'''  |     C#40   c#'''  |     C#50   c#v       
A#31       a#''   | A#41       a#'''  | A#51       a#''''   
    F#30   f#''   |     F#40   f#'''  |     F#50   f#''''   
D#31       d#''   | D#41       d#'''  | D#51       d#''''   
    B 30    b'    |     B 40    b''   |     B 50    b'''   
G#31       g#'    | G#41       g#''   | G#51       g#'''   
    E 30    e'    |     E 40    e''   |     E 50    e'''       
C#31       c#'    | C#41       c#''   | C#51       c#'''       
    A 30    a     |     A 40    a'    |     A 50    a''   
F#31       f#     | F#41       f#'    | F#51       f#''   
    D 30    d     |     D 40    d'    |     D 50    d''   
B 31        B     | B 41        b     | B 51        b'   
    G 30    G     |     G 40    g     |     G 50    g'   
E 31        E     | E 41        e     | E 51        e'   
    C 30    C     |     C 40    c     |     C 50    c'       
A 31        A,    | A 41        A     | A 51        a    
    F 30    F,    |     F 40    F     |     F 50    f    
D 31        D,    | D 41        D     | D 51        d    
    Bb30   Bb,,   |     Bb40   Bb,    |     Bb50   Bb     
G 31        G,,   | G 41        G,    | G 51        G     
    Eb30   Eb,,   |     Eb40   Eb,    |     Eb50   Eb     
C 31        C,,   | C 41        C,    | C 51        C       
    Ab30   Ab,,,  |     Ab40   Ab,,   |     Ab50   Ab,     
F 31        F,,,  | F 41        F,,   | F 51        F,     
    Db30   Db,,,  |     Db40   Db,,   |     Db50   Db,     
Bb31       Bb,,,, | Bb41       Bb,,,  | Bb51       Bb,,     
    Gb30   Gb,,,, |     Gb40   Gb,,,  |     Gb50   Gb,,     
Eb31       Eb,,,, | Eb41       Eb,,,  | Eb51       Eb,,    
    Cb30   Cb,,,, |     Cb40   Cb,,,  |     Cb50   Cb,,

To get from c to d, remembering the Hemholtz notation indicates the USA 3 octave, as a Pythagorean whole step, then, is not a simple two-step jump up the 40 stack: one must take into account the octave. One must go from C40 to D30, and the three shortest paths are C40-G40—D40-D30, C40-G40-G30-D30 and C40-C30-G30-D30. These paths have all have three steps, not two. Tracking the intervals by ratio for the first path, we have stepped 3/2 (up a fifth), 3/2 (up a fifth) and 1/2 (down an octave). Multiplying the ratios gives us (3*3*1) / (2*2*2) = 9/8, or a large, Pythagorean, whole step.

Further, the ‘quality’ of the steps within any one of the paths differ, though each path is equivalent. Each path requires one step of ‘octave quality’ and two steps of ‘fifth quality’.

If we scan the three partial lattices above for another d we find D51 to be apparently the same note. As we follow the new paths, however, we will find D51 to be a small whole step from C40, an interval of 10/9 ratio as opposed to the larger 9/8 ratio.

So to get from c to d as a small whole step, is a different set of paths: we must arrive at D51 from C40, and, remembering in these partial lattices the only interval steps are the octave, the fifth, and the major third, we find that two of the several equivalent paths are C40-F40-Bb40-D41-D51 and C40-C50-F50-Bb50-D51. Each path requires one step of ‘octave quality’, two steps of ‘fifth quality’ and one step of ‘major third’ quality. Tracking the intervals by ratio for the first path, we have stepped 2/3 (down a fifth) 2/3 (down a fifth) 5/4 (up a major third) and 2/1 (up an octave). Multiplying the ratios gives us (2*2*5*2) / (3*3*4*1) = 40 / 36 = 10/9, a small whole step.

And, naturally, when we add the next harmonic interval, the minor third of ratio 6/5, to the set of possible lattice steps, we can replace the combined one step of ‘fifth quality’ and one step of ‘major third quality’ with a single step of ‘minor third quality’. Without redrawing the lattices we can imagine the new path as C40-F40-D41-D51, and by ratios 2/3 (down a fifth) 5/6 (down a minor third) and 2/1 (up an octave). Multiplying the ratios gives us (2*5*2) / (3*6*1) = 20/18 = 10/9.

Lattice with pure thirds above and below:

    Edf+ Bdf+ Fb + Cb + Gb + Db + Ab + Eb + Bb + F  + C  + G  + D  + A  + E +  B
   /  \ /  \ /  \ /  \ /  \ /  \ /  \ /  \ /  \ /  \ /  \ /  \ /  \ /  \ /  \ /
* Cb * Gb * Db * Ab * Eb * Bb * F  * C  * G  * D  * A  * E  * B * F# * C# * G# *
 /  \ /  \ /  \ /  \ /  \ /  \ /  \ /  \ /  \ /  \ /  \ /  \ /  \ /  \ /  \ /  \
Ab - Eb - Bb - F  - C  - G  - D  - A  - E  - B - F# - C# - G# - D# - A# - E# - B#

Lattice with pure thirds above and below, twice removed:

(++)       ddf  adf  edf  bdf  fb   cb   gb   db   ab   eb   bb   f    c    g      
           / \ /  \ /  \ /  \ /  \ /  \ /  \ /  \ /  \ /  \ /  \ /  \ /  \ /  \ 
    Edf+ Bdf+ Fb + Cb + Gb + Db + Ab + Eb + Bb + F  + C  + G  + D  + A  + E +  B
   /  \ /  \ /  \ /  \ /  \ /  \ /  \ /  \ /  \ /  \ /  \ /  \ /  \ /  \ /  \ /
* Cb * Gb * Db * Ab * Eb * Bb * F  * C  * G  * D  * A  * E  * B * F# * C# * G# *
 /  \ /  \ /  \ /  \ /  \ /  \ /  \ /  \ /  \ /  \ /  \ /  \ /  \ /  \ /  \ /  \
Ab - Eb - Bb - F  - C  - G  - D  - A  - E  - B - F# - C# - G# - D# - A# - E# - B#
  \ / \  / \  / \  / \  / \  / \  / \  / \  / \  / \  / \  / \  / \  / \ / \
   c    g    d    a    e    b   f#   c#   g#   d#   a#   e#   b#   fx   cx   (--)

Same lattice as above, with approximate tuning in cents with respect to 12EDO
based on C:

(++)       ddf  adf  edf  bdf  fb   cb   gb   db   ab   eb   bb   f    c    g      
           +20  +22  +24 +26  +28  +30  +32  +34  +36  +38  +40  +42  +44 +48
           / \ /  \ /  \ /  \ /  \ /  \ /  \ /  \ /  \ /  \ /  \ /  \ /  \ /  \ 
    Edf+ Bdf+ Fb + Cb + Gb + Db + Ab + Eb + Bb + F  + C  + G  + D  + A  + E +  B
    +2   +4   +6   +8  +10  +12  +14  +16  +18  +20  +22  +24  +26  +28 +30
   /  \ /  \ /  \ /  \ /  \ /  \ /  \ /  \ /  \ /  \ /  \ /  \ /  \ /  \ /  \ /
* Cb * Gb * Db * Ab * Eb * Bb * F  * C  * G  * D  * A  * E  * B * F# * C# * G# *
 -14  -12  -10   -8  -6   -4   -2    0   +2   +4   +6   +8  +10  +12  +14  +16
 /  \ /  \ /  \ /  \ /  \ /  \ /  \ /  \ /  \ /  \ /  \ /  \ /  \ /  \ /  \ /  \
Ab - Eb - Bb - F  - C  - G  - D  - A  - E  - B - F# - C# - G# - D# - A# - E# - B#
    -28  -26 -24  -22  -20  -18  -16  -14  -12  -10   -8   -6   -4   -2   0
  \ / \  / \  / \  / \  / \  / \  / \  / \  / \  / \  / \  / \  / \  / \ / \
   c    g    d    a    e    b   f#   c#   g#   d#   a#   e#   b#   fx   cx   (--)
 -44  -42  -40  -38  -36  -34  -32  -30  -28  -26  -24  -22  -20  -18  -16