## The boundaries between the Trojan symbols

Dave Keenan
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### The boundaries between the Trojan symbols

In Figure 10 of http://sagittal.org/sagittal.pdf we give a number-line marked in cents showing the approximate boundaries between the Trojan symbols, but we do not give precise numerical values. Here they are as fractions of a semitone plus or minus epsilon, where epsilon stands for an infinitesimal quantity, or the smallest change available in the number system being used.

Symbol	 Common	  Boundary
Boundary semitone  semitone
cents	 fraction  fraction
--------------------------------------
-3.999999999	-(1/25-ϵ)
no symbol, natural
3.999999999	1/25-ϵ
1/12
9.090909101	1/11+ϵ
1/10
11.99999999	3/25-ϵ
1/6
16.66666668	1/6+ϵ
1/5
22.22222223	2/9+ϵ
1/4
28.57142856	2/7-ϵ
1/3
33.33333334	1/3+ϵ
3/8
38.88888888	7/18-ϵ	See note below
2/5
42.85714285	3/7-ϵ
7/16
48.00000001	12/25+ϵ
1/2
51.99999999	13/25-ϵ
9/16
57.14285715	4/7+ϵ

Note: As first published, the boundary between and was as follows:
	2/5
40.00000001	2/5+ϵ	obsolete
5/12
But when working with 60-edo in 2012, Graham Breed made a case for the new 38.88888888 cent boundary, which George and I agreed was convincing.