Flattone

Flattone is an alternative extension to 5-limit meantone, the temperament that tempers out the syntonic comma (81/80). It is generated by a fifth that is typically flatter than that of meantone, and nine of those reach the pitch class of 8/7, so that 7/4 is a diminished seventh (C-B𝄫), 7/6 is a diminished third (C-E𝄫), and 7/5 is a doubly diminshed fifth (C-G𝄫).

11- and 13-limit extensions are fairly obvious, using the heavily tempered chromatic semitone for the undecimal quartertone of 33/32 and the tridecimal third tone of 27/26. 11/8 is an augmented fourth, and 13/8 is a minor sixth, conflated with 8/5.

Reasonable tunings lie between 19edo and 26edo. 19edo is the point where 7/4 and 12/7 are conflated. Any tuning whose fifth is sharper than 19edo's has the sizes of 7/4 and 12/7 inverted, so they can be more properly analysed as septimal meantone. Similarly, 26edo is the point where 7/5 and 10/7 are conflated. Any tuning whose fifth is flatter than 26edo's has the sizes of 7/5 and 10/7 inverted, so they can be more properly analysed as a flatter-of-flattone temperament.

See Meantone family #Flattone for technical data.

Interval chain

In the following table, odd harmonics 1–13 are in bold.

#	Cents*	Approximate Ratios
0	0.0	1/1
1	693.0	3/2
2	186.1	9/8, 10/9, 11/10
3	879.1	5/3
4	372.1	5/4, 16/13, 26/21
5	1065.1	11/6, 13/7, 15/8, 24/13
6	558.2	11/8, 18/13
7	51.2	25/24, 27/26, 33/32, 36/35, 55/54, 64/63
8	744.2	20/13, 32/21
9	237.3	8/7, 15/13
10	930.3	12/7, 22/13
11	423.3	9/7
12	1116.4	27/14, 40/21
13	609.4	10/7

* in 13-limit CTE tuning

Scales

• Flattone12 – 12-tone chromatic scale in 13-limit POTE tuning

Tunings

Tuning spectrum

Edo Generator	Eigenmonzo (Unchanged-interval)*	Generator (¢)	Comments
	64/63	689.609
	13/8	689.868
	11/6	689.873
19\33		690.909
	13/11	691.079
	21/16	691.152
	9/5	691.202	1/2 comma
53\92		691.304
	21/11	691.467
34\59		691.525
49\85		691.765
	11/8	691.886
	11/7	692.166
	13/12	692.285
15\26		692.308	Lower bound of 7-, 9-, 11-, and 13-odd-limit diamond monotone
	7/4	692.353
	21/13	692.437
	36/35	692.681
	49/48	692.858
41\71		692.958
	21/20	692.961
	13/10	693.223
	7/6	693.313
26\45		693.333
	7/5	693.653
37\64		693.750
48\83		693.976
	9/7	694.099
	15/13	694.193
	15/14	694.246
	13/7	694.340
11\19		694.737	Upper bound of 7-, 9-, 11-, 13-odd-limit diamond monotone
	5/3	694.786	1/3 comma
	25/24	695.810	2/7 comma
	5/4	696.578	1/4 comma, 5-odd-limit minimax
	15/8	697.654	1/5 comma
7\12		700.000
	3/2	701.955	Pythagorean tuning

* besides the octave