As you have seen, if a note has a frequency ν the ones an
octave 
and a
fifth upper have respectively frequency ν’=2ν and ν’=(3/2)·ν. More generally, the frequency ν’ is δ times ν (where ν’>ν) if δ is the ratio of the frequencies characteristic of the interval between the two notes (for example, δ=3/2 for a perfect fifth) . Obviously δ=ν’/ν: if you denote by δ(C-D) the ratio of the frequencies of C and D, by δ(D-E) the ratio of the frequencies of E and F (and so on), the second
intervals in the just intonation scale are characterized by the following ratios of frequencies:
- δ(C-D)=ν(D):ν(C)=(9/8):1=9/8
- δ(D-E)=ν(E):ν(D)=(5/4):(9/8)=(5/4)·(8/9)=10/9
- δ(E-F)=ν(F):ν(E)=(4/3):(5/4)=(4/3)·(4/5)=16/15
- δ(F-G)=ν(G):ν(F)=(3/2):(4/3)=(3/2)·(3/4)=9/8
- δ(G-A)=ν(A):ν(G)=(5/3):(3/2)=(5/3)·(2/3)=10/9
- δ(A-B)=ν(B):ν(A)=(15/8):(5/3)=(15/8)·(3/5)=9/8
- δ(B-C)=ν(C):ν(B)=2:(15/8)=2·(8/15)=16/15
Therefore there are three types of second in the just intonation scale:
| INTERVAL |
MAJOR TONE |
MINOR TONE |
SEMITONE |
| RATIO OF FREQUENCIES |
9/8 |
10/9 |
16/15 |
Not only the use of the just intonation scale implies two different types of tone (major and minor), but a semitone is not exactly half of a tone: in fact, since to ascend by one semitone you must multiply the frequency of the first note by that of the interval (16/15), if you ascend by two semitone from C1 (that has frequency 1) you obtain first 1·(16/15)=16/15 and later (16/15)·(16/15)=256/225, and not 9/8 or 10/9 as it would be ascending by a tone.