Insatinismic

Insatinismic is a rank-5 temperament for the add-29-19-limit that is a weak extension of insanismic and a detemperament of cassaschismic. It is not a member of any known temperament family or clan.

Data below is given because infobox regtemp doesn't work with rank-5 temperaments:

Temperament data

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 5767168/5767125, 11413376/11413325, 633556/633555

Mapping: [⟨ 1 0 0 0 5 8 -9 0 ], ⟨ 0 1 0 0 0 0 -2 1 ], ⟨ 0 0 1 0 0 -1 -2 1 ], ⟨ 0 0 0 1 1 0 0 2 ], ⟨ 0 0 0 0 -9 -3 8 -8 ]]

Mapping generators: ~2, ~3, ~5, ~7, ~608/385

Optimal tunings

WE: ~2 = 1199.999542¢, ~3/2 = 701.955847¢, ~5/4 = 386.315423¢, ~7/4 = 968.827501¢, ~385/608 = -791.051331¢

error map: -0.000458 0.000388 0.000793 0.000678 -0.000238 0.000173 -0.000847 -0.000598

CWE: ~2 = 1199.999542¢, ~3/2 = 701.955948¢, ~5/4 = 386.315507¢, ~7/4 = 968.827700¢, ~385/608 = -791.051640¢

error map: 0.000000 0.000947 0.001794 0.001794 0.00095 0.001912 0.000618 0.000724

Optimal ET sequence: 41g, 53, 94, 176g, 217, 270, 311, 487, 581, 1115g, 1385, 1696, 1966, 2597, 2814, 3084, 3178, 3395, 5144, 6573, 7958, 8269, 8539, 11934, 16808, 20203.

Badness (Sintel): 0.279

Subgroup: 2.3.5.7.11.13.17.19.29
Comma list: 5767168/5767125, 11413376/11413325, 633556/633555, 176176/176175

Mapping: [⟨ 1 0 0 0 5 8 -9 0 19 ], ⟨ 0 1 0 0 0 0 -2 1 -3 ], ⟨ 0 0 1 0 0 -1 -2 1 -3 ], ⟨ 0 0 0 1 1 0 0 2 3 ], ⟨ 0 0 0 0 -9 -3 8 -8 -21 ]]

Mapping generators: ~2, ~3, ~5, ~7, ~608/385

Optimal tunings

WE: ~2 = 1199.999454¢, ~3/2 = 701.955619¢, ~5/4 = 386.314591¢, ~7/4 = 968.825991¢, ~385/608 = -791.051281¢

error map: -0.000546 0.000618 0.000877 0.000084 -0.000591 -0.000463 0.00017 -0.001072 0.001239

Badness (Sintel): 0.354

Ploidacots, pergens, MOS scales and odd-limit detunings are unapplicable or too convoluted to be descriptive.

As a detemperament of cassaschismic, it does the following:

• It decouples the septimal comma from the chain of fifths (observing the garischisma),
• It observes 4096/4095 and 4225/4224 but tempers them together (observing the olympia and ibnsinma too), as seen in insanismic

As a weak extension of insanismic, it does the following:

• The fifth generator is trisected for primes 11, 13 and 29, and can be instead taken to be ~1540/1539, or a redundant pair of 4096/4095~4225/4224 and 10241/10240. 10241/10240 is used to reached 17/16 and 19/16.

Notation

Because this is a rank-5 temperament, there is no good way of visualizing intervals in a neatly compressed lattice, however, it can be somewhat easily notated with the following accidentals:

Minimal Insatinismic notation^{[idiosyncratic term]}

This notation is a direct mapping retrieved from Sintel's temperament evaluator here, shuffling the generators to have them be in decreasing order of size. Either way, it requires 5 pairs of accidentals to fully notate on a score.

• Sharps/Flats
• Pythagorean commas
• Garischismas
• Schismas
• Kevolismas

Through this notation, octave reduced harmonics can be described as follows:

• 2/1: octave
• 3/2: fifth
• 5/4: major third - pythagorean comma + schisma
• 7/4: minor seventh - pythagorean comma - garischisma
• 11/8: fourth + 2 pythagorean commas + 3 garischismas - 9 kevolismas
• 13/8: minor sixth + 2 pythagorean commas - schisma - 3 kevolismas
• 17/16: major second - 4 pythagorean commas + 2 schismas - 8 kevolismas
• 19/16: minor third + schisma - 2 garischismas + 8 kevolismas
• 29/16: major sixth + 5 pythagorean commas - 3 schismas - 3 garischismas + 21 kevolismas

The notation can be equivalently and naively described as using:

• Sharps/Flats
• Septimal commas
• Pythagorean commas
• Syntonic commas
• Kevolismas

Any of the above expressions can be rewritten by converting syntonic comma = pythagorean comma - schisma; septimal comma = pythagorean comma + garischisma.

A redundant accidental, the mercator comma, can be introduced to keep the amount of pythagorean commas needed less than two, since it is the difference between 4 pythagorean commas and one limma. 17/16 is a type of major second. Using a mercator comma allows it to be respelled as a minor second + pythagorean comma - mercator comma... etc. Same with 29/16, which instead of being spelt a major sixth can be spelt a minor seventh + pythagorean comma + mercator comma... etc.

Eufalesio's ULTRAOLYMPIC [sic] notation

Eufalesio had devised a somewhat accessible notation of differently sized accidentals can be devised by reusing the suit of accidentals that is used for 12e, cassandra, cassaschismic, and expanding upon them to accomodate the minuscule alterations. This has the benefit of being a natural representation of olympic, keeping the representations of primes 2-13,19 as simple as possible.

Buzzardmas are to ULTRAOLYMPIC notation what mercator commas are to minimal. They are the difference between 4 arrow-unison and a minor second, only that the arrow here is a septimal comma, not a pythagorean one.

Thus, as seen in the notation, octave reduced harmonics can be described as follows:

• 2/1: octave
• 3/2: fifth
• 5/4: major third - septimal comma + aberschisma
• 7/4: minor seventh - septimal comma
• 11/8: fourth + 2 septimal commas - 3 minas
• 13/8: minor sixth + 2 septimal commas - aberschisma - mina
• 17/16: minor second + septimal comma - buzzardma + aberschisma + schisma - 3 minas - tina
• 19/16: minor third + schisma + 3 minas + tina
• 29/16: minor seventh + septimal comma + buzzardma - aberschisma - 2 schismas - 7 minas OR minor seventh + septimal comma + buzzardma - 3 aberschismas + 10 minas - 3 tinas

And any interval between them can be found by subtracting each accidental independently, keeping in mind that an excess of septimal commas can be converted into buzzardmas.