A finite sequence of non-negative integers is called a modal sequence ifany of its three consecutive terms of the form“aOb”is identified as a singleterm“a+b”,for example,13045 is identified as 175.For any two modal sequences α=a_1 a_2…a_h and β=b_1 b_2…b_k,if we canget the same sequence by replacing some(or none)even terms α_(2i)and some(or none)odd terms b_(2j+1)by smaller integers,then we say that a precedes β,denoted by α(?)β.It is easy to see that the relation(?)is an order relation.If α(?)βand β(?)α,then we say that α is equivalent to β,and denoted byEvidently we have infinitely many non-equivalent modal sequences.If weassume some new order relations,it may happen that in the result systemsonly a finite number of non-equivalent modal sequences can exist.Such sys-tems will be called finite modal systems.In any finite modal system thereare some properties which may be characterized by non-negative numbers(call-ed parameters).When the values of parameters are given,we may deducesome order relations in the system.Such order relations are called essentialrelations,which hold necessarily,in some sense,in every finite modal systems.For example,the relations n=n+1 and α=α~h+α~k are essential relations,whereα~h denoted αα…α(h in number).The number n is called the order of the sys-tem.The number pair(h,k)is called the type of the sequence α.Amongthe various types of all the sequences of the system,the strongest ones arecalled the types of the system.The present paper is to construct finite modal systems when the parame- ters alone are given,in other words,to construct the finite modal systems fromthe essential relations alone.Such systems constructed are called fundamentalfinite modal systems.The main results are as follows:We may construct the systems of order 0 and of order 1 readily.Theycontain 2 and 14 modal sequences respectively.When the type(h,2k)of the sequence 1 and the type(r_i,1)of the se-quence 21~(2i-1)(i=1,2,…,m,where m is the integral part of h/2)are given,we may construct the system of order two.The total number of the modalsequences in it may be computed out.We may construct the systems of the type(1,2),and the systems of order3 with type(2,2).In such construction,however,the sequences which havethe type(1,2)or(2,2)cannot be given arbitrarily.By means of zu-sequences(characterized by z(?)u,z~3=z~5,and zuz=zuz~3==z~3uz)we may construct infinitely many fundamental finite modal systems oforder three.Finally,we prove that(under a very broad assumption)if wecan construct all the fundamental finite modal systems of order three withtype(2,2),then we can construct all the fundamental finite modal systems.