Meantone

Meantone equates four 3/2s to 5/1 and generates pentic (2L 3s) and mosdiatonic (5L 2s) scales.

Meantone, or rarely Syntonic or Didymus, is a widespread historical temperament that forms the basis of Western music theory, where the fifths are flattened to about 696c to produce a diatonic major third tuned to roughly 5/4, enabling the use of 5-limit harmony in the diatonic scale. When all the fifths are tuned the same, Meantone is a regular temperament, where the period is the octave, the generator is 3/2, and four generators stack to reach the 5th harmonic, meaning that the syntonic comma, which is the difference between Pythagorean intervals and nearby 5-limit intervals and has a ratio of 81/80, is tempered out.

As a monocot temperament, Meantone can be notated with standard diatonic notation, and in fact diatonic notation works the best for Meantone as the 5-limit 4:5:6 harmonic triad becomes simply C-E-G on C, and the chromatic semitone is usually smaller than the diatonic semitone. Meantone is a 7-form temperament, and is tuned well around the golden tuning of diatonic. Unsurprisingly, 7edo supports Meantone, and so does 12edo (which is the simplest ET to do so without exotempering the 5-limit), and so the best tunings of Meantone lie in between those two extremes. [7 & 12] is thus the edo join for meantone.

Other primes

If an unmapped (not equated to a stack of anything else, as 5/4 is in blackwood) prime 7 is introduced as a second generator, then the result can be called Didymus.7.

The edo join [7 & 12] results in an exotempered extension called dominant where 7/4 and 9/5 are equated, and is tuned best around Pythagorean tuning.

More accurate extensions of Meantone's diatonic structure to include other primes follow.

7/4 as the augmented sixth (12 & 19)

This is the primary extension of Meantone to the 7-limit, where 7/4 is the augmented sixth (C-A#, +10 fifths). It is best tuned with the generator around 696 cents. 5/4 is 384 cents, and 7/4 is 960 cents. It is notable for being the most accurate extension, as well as containing the golden tuning of the diatonic scale, and thus a melodically convenient chromatic and enharmonic scale. This means that the augmented diesis 128/125, is equated with the septimal quartertone 36/35, and the 5-limit supermajor and subminor intervals are equated with their septimal counterparts.

However, one drawback of this temperament is the large degree of complexity required to get to the 11th and 13th harmonics. In fact, there are two main options. In both cases, the tridecimal neutral third 16/13 is conflated with the undecimal neutral third 11/9, representing a characteristic tendency to make 11/9 the sharper of the two 11-limit neutral thirds. (As a result, one might find it useful to irregularly map 11/9.)

11-limit

The 11-limit form of 12 & 19 is an exotemperament called meanenneadecal, which tunes 11/8 very sharp (because both 12edo and 19edo do so). More accurate extensions are below.

11/8 as the double-augmented third (12 & 31)

This is best tuned around 697 cents, and places 11/9 as the double-augmented second (C-Dx, +16 fifths) and conflates 14/11 with 9/7 placed as the diminished fourth (C-Fb, -8 fifths). 13/8 is mapped to the double-diminished seventh (C-Bbb, -9 fifths).

11/8 as the double-diminished fifth (19 & 31)

This is best tuned around 696 cents, and places 11/9 as the double-diminished fourth (C-Fbb, -15 fifths). 13/8 is mapped to the double-augmented fifth (C-Gxx, +15 fifths).

7/4 as the diminished seventh (19 & 26)

This temperament, often called "Flattone", sets 7/4 equal to the diminished seventh, and is best tuned with the generator 3/2 around 693 cents, 5/4 at 372 cents, and 7/4 at 963 cents. It is a melodically intuitive extension, as it creates an equiheptatonic scale with a quartertone-sized chroma, and interval sizes tend to match with their corresponding interval categories. For example, it can be easily extended to map prime 11 to the augmented fourth (C-F#, +6 fifths) and 13 to the minor sixth (C-Ab, -4 fifths) tuned to around 558 and 828 cents respectively. 26edo is the most commonly used tuning, though it can be tuned more accurately with 45edo. It is a 7-cluster temperament, as indicated by the edo join (26 - 19 = 7).

Chords

Meantone's main feature is its conflation of the standard harmonic triad 4:5:6 with the diatonic major chord P1-M3-P5. This allows 5-limit harmony to be used in Meantone, and is the foundation of modern Western music theory, which treats P1-M3-P5 as stable and P1-m3-P5 as unstable (10:12:15). Meantone also contains an essentially tempered chord, where 1-9/8-3/2-5/3-2 contains steps of 9/8, 4/3, 9/8, and 6/5. Note that in just intonation, the top interval would be 27/16, not 5/3, or the two whole tones would be different sizes (resulting in a 40/27 wolf fifth).

Tunings

As essentially the only temperament that is both regular and attested outside xenharmony, Meantone has a number of historical tunings that today correspond to various extensions and approximate edos. Here, "comma" refers to the syntonic comma.

Quarter-comma Meantone

This tunes the fifth 1/4-comma flat, to a size of 696.57 cents. It has a just 5/4, and a step ratio of 1.65 (quasisoft). It approximates 31edo, and is often (rather insultingly to the rest of 31edo) seen as the latter's primary feature. It extends to 11-limit 19 & 31.

1/3-comma Meantone

This is the tuning of the fifth to 694.78 cents, which has a just 6/5 and is extremely close to 19edo, having a step ratio of 1.503 (~monosoft). As a result, it does not cleanly extend to the 11-limit, although as it is slightly sharp of 19edo it does technically extend to 7-limit 12 & 19.

2/5-comma Meantone

This is very close to the 45edo tuning of Meantone, tuning the fifth 693.35 cents and having a just 27/25 (note that 27/25 is tempered together with 16/15 in this system, resulting in a sharp minor second). As a result of the flat tuning, this extends to Flattone, rather than to Septimal Meantone. Its step ratio is 1.401 (parasoft).

1/2-comma Meantone

This is close to 33edo's diatonic tuning, which is not Meantone. As a result, it can be considered the lower bound of Meantone's tuning, where the tone is tuned to a just 10/9. It tunes the fifth to 691.2 cents. Its step ratio is 1.26 (ultrasoft).

2/7-comma Meantone

This is the tuning of Meantone situated roughly between 50edo's and 69edo's tunings, with a fifth of 695.81 cents and a step ratio of 1.584 (quasisoft). Consequently, it extends to Septimal Meantone, but with a rather poor approximation of 7/4. However, it tunes other septimal intervals like 9/7 and 7/6 more accurately. It tunes 25/24 purely, and can thus be considered a compromise between 1/4-comma's perfect 5/4 and 1/3-comma's perfect 6/5.

1/5-comma Meantone

This tuning of Meantone equalizes the error on 3/2 and 5/4, tuning the former to 697.65 cents and the latter to 390.61 cents. Equivalently, it tunes 16/15 purely. Its step ratio is 1.748 (minisoft), and consequently it is well approximated by 43edo.

1/11-comma Meantone

This tuning of Meantone is almost perfectly approximated by 12edo, having a fifth tuning of nearly exactly 700 cents and a step ratio of nearly exactly 2 (~basic). 12edo by definition lowers the fifth by 1/12 of a Pythagorean comma; setting 1/12 of a Pythagorean comma to 1/11 of a syntonic comma is done in edos such as 612edo.

List of patent vals

todo: format this better

5 (dominant)

12 (dominant, septimal meantone, meanenneadecal, huygens)

67

55

98

43 (septimal meantone, huygens)

117 (huygens)

74 (septimal meantone, huygens)

105 (septimal meantone, huygens)

31 (septimal meantone, huygens, meanpop)

81 (septimal meantone, meanpop)

50 (septimal meantone, meanpop)

69 (meanpop)

88

19 (flattone, septimal meantone, meanenneadecal, meanpop)

45 (flattone)

26 (flattone)

7 (dominant, flattone)