Father–3 equivalence continuum/Godtone's approach: Difference between revisions
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* 128/125 is fundamental because it uniquely defines the relatively-very-accurate (strongly form-fitting) representation of the 2.5 subgroup in 3edo. | * 128/125 is fundamental because it uniquely defines the relatively-very-accurate (strongly form-fitting) representation of the 2.5 subgroup in 3edo. | ||
* 25/24 is fundamental because it gives the trivial way to relate ~5/4 = 1\3 to ~3/2 = 2\3 as 2 generators in 3edo. (By contrast, using 16/15 requires taking the octave-complement of one of the generators. There is also another stronger argument against using 16/15 detailed later in this list.) | * 25/24 is fundamental because it gives the trivial way to relate ~5/4 = 1\3 to ~3/2 = 2\3 as 2 generators in 3edo. (By contrast, using 16/15 requires taking the octave-complement of one of the generators. There is also another stronger argument against using 16/15 detailed later in this list.) | ||
* Using 25/24 is also useful because we then know how many intervals between ~6/5 and ~5/4 are guaranteed in a nontrivial tuning; because 25/24 is divided into ''n'' equal parts, the answer is ''n'' - 1. Meanwhile, if ''n'' = ''a''/''b'' is not an integer (meaning ''b'' > 1), then 25/24 is divided into ''a'' equal parts of ~(128/125)<sup>1/''b''</sup>, giving a clear meaning to the numerator and denominator (though more meanings are discussed later). | * Using 25/24 is also useful because we then know how many intervals between ~6/5 and ~5/4 are guaranteed in a nontrivial tuning; because 25/24 is divided into ''n'' equal parts, the answer is {{nowrap|''n'' - 1}}. Meanwhile, if {{nowrap|''n'' {{=}} ''a''/''b''}} is not an integer (meaning ''b'' > 1), then 25/24 is divided into ''a'' equal parts of {{nowrap|~(128/125)<sup>1/''b''</sup>}}, giving a clear meaning to the numerator and denominator (though more meanings are discussed later). | ||
* Because {{nowrap|25/24 {{=}} ([[25/16]])/([[3/2]])}}, this has the consequence of clearly relating the ''n'' in {{nowrap|(128/125)<sup>n</sup> {{=}} 25/24}} with how many 5/4's are used to reach 3/2 (when octave-reduced): | * Because {{nowrap|25/24 {{=}} ([[25/16]])/([[3/2]])}}, this has the consequence of clearly relating the ''n'' in {{nowrap|(128/125)<sup>n</sup> {{=}} 25/24}} with how many 5/4's are used to reach 3/2 (when octave-reduced): | ||
: * If {{nowrap|''n'' {{=}} 0}}, then it takes no 128/125's to reach 25/24, implying 25/24's size is 0 (so that it's tempered out), meaning that 3/2 is reached via (5/4)<sup>2</sup>. | : * If {{nowrap|''n'' {{=}} 0}}, then it takes no 128/125's to reach 25/24, implying 25/24's size is 0 (so that it's tempered out), meaning that 3/2 is reached via (5/4)<sup>2</sup>. | ||