Yahoo Tuning Groups Ultimate Backup tuning Re: What is a monzisma?

[Please let me explain that this post, like my article for Joseph
Pehrson, was completed on May 1, but only now am in a position to post
it, having regained access to my Internet account after some server
and network problems for my ISP. "About two weeks ago" means "around
the middle of April," as in the other article. While some of the
discussions about a week ago concerning the Monzonian lore celebrated
here might have made a post around May 1 especially timely, I hope
that what follows may still have interest for RI exponents and others,
and of course not least of them the Monz himself.

P.S. Catching up on some Digests, I'm delighted to see, Monz, that the
topic of "kleismatic" unison vectors or bridges came up again in TD 1283,
with a fascinating reference to Fokker -- maybe making this as
"timely" an occasion to post as any.]

Hello, there, everyone, and I'd like to propose a fitting name for an
interval defining a rather long kind of xenharmonic bridge: the
_monzisma_, naturally in honor of Joe Monzo, our "Monz."

As described in a post on neo-Gothic notations especially inspired by
some comments and questions from Joseph Pehrson, this story started
some months ago when the Monz reported using a 75:64 minor third in
place of a 7:6 in order to keep to the 5-prime-limit structure of a
piece entitled _3+4_.

Then, about two weeks ago, looking at the intervals of a 53-note
Pythagorean tuning in Manuel Op de Coul's Scala for MS-DOS, I came up
with the idea of a "tricomma tuning" with two 12-note Pythagorean
manuals tuned apart by three Pythagorean commas, the "tricomma" of
about 70.38 cents, or 150094635296999121:144115188075855872.

The main object was to obtain a 24-note tuning including the excellent
supraminor/submajor thirds of the "native" 53-note Pythagorean set
equal respectively to around 337.44 cents (32 fourths up minus 13
octaves) and 364.52 cents (33 fifths up minus 19 octaves). Since these
thirds have ratios of around 17:14 and 21:17, they are known in
neo-Gothic theory as "17-flavor" intervals.

As Scala also revealed, a tricomma tuning would yield rather complex
7-flavor intervals (most importantly 9:7, 7:6, 12:7, 7:4, and 8:7) at
about 7.42 cents from pure. Some of these interval sizes looked
familiar, and I soon realized that I had indeed found a pure
Pythagorean tuning not only combining a 7-flavor and a 17-flavor,
but with near-exact equivalent of the Monzonian 75:64 and related
"near-7" ratios (~32:25 for 9:7, ~225:128 for 7:4, etc.).

Both the Monz and I, months apart, had from one viewpoint faced
similar situations. We were aiming for 7-based ratios, but were
constrained by other intonational considerations to accept some
"inaccuracy," arriving at complex integer approximations

For the Monz, an overall 5-prime-limit structure fit 75:64 better than
a pure 7:6. For me, a somewhat inaccurate 7-flavor in the tricomma
tuning made it possible in a pure Pythagorean or 3-prime-limit tuning
of only 24 notes to combine this flavor with an ideal 17-flavor.

The small difference between the 7-flavor minor third of the tricomma
tuning (actually a Pythagorean 9:8 major second plus a tricomma) and
the Monz's 75:64 is equal to that between the 70.38-cent tricomma and
a pure 5-limit chromatic semitone at 25:24, or ~70.67 cents -- an
interval of about 0.29 cents.

More precisely, this interval has the precise and fairly large ratio
of 25:24 less 150094635296999121:144115188075855872, or

450359962737049600:450283905890997363 (~0.292396 cents)

Here I propose that this difference be termed a _monzisma_, if such a
choice would comply with the naming conventions, and assuming that
someone else has not already described this ratio (always a wise
caution in discussing "new" intervals and tunings).[1]

To explain the amount that the 7-flavor intervals of the tricomma
tuning differ from pure, about 7.42 cents, we might derive this
variation as the sum of the 3-7 schisma at about 3.80 cents and the
comma of Mercator at about 3.62 cents. This sum differs from the
225:224 comma or septimal kleisma of ~7.71 cents by a 0.29-cent
monzisma.

Let's set the 3-7 schisma and comma of Mercator in some musical
context. In a contiguous 24-note Pythagorean tuning formed by a chain
of 23 pure fifths, we approximate a 9:7 major third, for example, by
16 fifths up less the requisite octaves -- an approximation narrow of
9:7 by the 3-7 schisma of ~3.80 cents.

In the tricomma tuning, we approximate the same interval as a pure
fourth less a tricomma -- or 37 fifths down. The difference between
these two approximations of 9:7, 16 fifths up and 37 fifths down, is
equal to the 53-comma or "comma of Mercator": the difference between
53 pure fifths and 31 pure octaves. This is the small asymmetry of
about 3.62 cents encountered in a circulating 53-note Pythagorean
tuning, and equally distributed in 53-tone equal temperament
(53-tET).

As it happens, the effect of the 3-7 schisma and comma of Mercator is
cumulative, thus accounting for our 7.42-cent variation from pure
7-flavor ratios in the tricomma tuning.

Here is a table of the 7-flavor derivations in the tricomma tuning
which may make this pattern clear for the other types of 7-flavor
intervals also. Adding a tricomma to another interval is the
equivalent of moving 36 fifths up, while subtracting a tricomma is the
equivalent of moving 36 fourths up (or fifths down). Here I use "P" to
show a single Pythagorean comma, and "3P" to show a tricomma:

----------------------------------------------------------------------
Ratio Usual approximation Tricomma approximation
----------------------------------------------------------------------
9:7 M3 + P = 16 5ths up (-3.80) 4 - 3P = 37 4ths up (-7.42)
14:9 m6 - P = 16 4ths up (+3.80) 5 + 3P = 37 5ths up (+7.42)
......................................................................
7:6 m3 - P = 15 4ths up (+3.80) M2 + 3P = 38 5ths up (+7.42)
12:7 M6 + P = 15 5ths up (-3.80) m7 - 3P = 38 4ths up (-7.42)
......................................................................
7:4 m7 - P = 14 4ths up (+3.80) M6 + 3P = 39 5ths up (+7.42)
8:7 M2 + P = 14 5ths up (-3.80) m3 - 3P = 39 4ths up (-7.42)
----------------------------------------------------------------------

The monzisma also has interesting cross-cultural applications for the
17-flavor. While I've been describing these ratios for supraminor and
submajor thirds as ideally 14:17:21, one Scala file, PERSIAN.SCL
contributed by Dariush Anooshfar, documents a Persian Tar scale
including an interval of 243:200, or ~337.15 cents.

This is another complex 5-limit interval, most engagingly equal
precisely to a 6:5 minor third _plus_ a syntonic comma of 81:80 -- the
same syntonic comma by which a regular Pythagorean minor third at
32:27 (~294.13 cents) varies from 6:5 (~315.64 cents) in the _narrow_
direction.

This 243:200 or "reverse Pythagorean" minor third, identified in
Scala's INTNAM.PAR file as an "acute minor third," is almost exactly
approximated by the tricomma tuning -- where it is a monzisma larger
(~337.44 cents).

Likewise Scala's "grave major third" at 100:81 (~364.81 cents) might
be described as a "reverse Pythagorean" major third, differing from
5:4 (~386.31 cents) by the same 81:80 comma as the regular 81:64
(~407.82 cents), but again in the opposite direction. In our tricomma
tuning, this third at ~364.52 cents is a monzisma smaller.

Especially in a pure Pythagorean tuning such as the tricomma tuning (a
24-note subset of a 53-note cycle), this view of "17-flavor" thirds as
reverse Pythagorean thirds may have a certain aesthetic symmetry as
well as mathematical near-symmetry to commend it. In the tricomma
tuning, a "reverse Pythagorean" ratio of 200:243:300 is almost
perfectly realized.

Incidentally, 243:200 differs from 17:14, or 100:81 from 21:17, by
another "xenharmonic bridge": a superparticular ratio or epimore of
1701:1700 or ~1.02 cents. Tricomma versions differ from 17-based forms
by this small interval (in Scala nomenclature, some "septendecimal"
comma or schisma, I would guess) plus a monzisma, or ~1.31 cents.

Maybe the most fascinating aspect of this story is the way that
virtually the same interval could arise in two such different musical
settings: a 5-prime-limit composition where 75:64 fits the overall
context better than 7:6; and a 3-prime-limit tuning where a ratio very
close to 75:64 happens to fit a neo-Gothic scheme combining a 7-flavor
and a 17-flavor in the same 24-note tuning.

Yet maybe the common theme of artful compromise to achieve overall
balance unifies these two situations, so that the compromise itself
becomes a curious mark of perfection.

----
Note
----

1. A yet more cautious formulation would be: "assuming that no one _is
currently known_ to have described this ratio," a circumstance
sometimes subject to change with further historical research.

Most appreciatively,

Margo Schulter
mschulter@value.net

🔗monz <joemonz@yahoo.com>

--- In tuning@y..., "monz" <joemonz@y...> wrote:

/tuning/topicId_22165.html#22204

>
> --- In tuning@y..., mschulter <MSCHULTER@V...> wrote:
>
> /tuning/topicId_22165.html#22165
>
> > Hello, there, everyone, and I'd like to propose a fitting
> > name for an interval defining a rather long kind of
> > xenharmonic bridge: the _monzisma_, naturally in honor
> > of Joe Monzo, our "Monz."
> >
> > ...
> >
> > More precisely, this interval has the precise and fairly large
> > ratio of 25:24 less 150094635296999121:144115188075855872, or
> >
> > 450359962737049600:450283905890997363 (~0.292396 cents)
>
>
> Margo, thanks, I'm *truly* flattered!
>
> I wonder if this might be the smallest interval that's ever
> actually been given a name?
>
> ... aside from "unison", that is. ;-)

Well, no it's not. I guess I jumped too soon at that one.

A monzisma is nearly identical to one degree of 4104-EDO,
which would be ~0.292397661 cent.

There's an interval called a "jot", invented by someone named
de Morgan, which is one degree of 30103-EDO, and thus is only
~0.039863137 cent, quite a bit smaller than a "monzisma".
(Reference: Ellis's appendix XX in his translation of Helmholtz,
_On the Sensations of Tone_, p 437).

Note that Ellis states that John Curwen, who I've discussed
here briefly before, used jots for his measurements. A jot is
an extremely accurate way of calculating logarithmic values of
JI intervals, because it is simply log2 * 10000, rounded to the
nearest integer.

Even smaller is the interval I refer to as the "MIDI pitch-bend
unit", which is only ~0.024414062 cent and one degree of
49152-EDO. The MIDI spec divides a semitone into 4096 (= 2^12)
pitch-bend units. However, "MIDI pitch-bend unit" is not
exactly what I would call a "name"...

-monz
http://www.monz.org
"All roads lead to n^0"

🔗paul@stretch-music.com

🔗jpehrson@rcn.com

🔗Andreas Sparschuh <a_sparschuh@...>

--- In tuning@yahoogroups.com, paul@... wrote:
>
> Probably the smallest superparticular ratio that's been given a name
is Erv Wilson's 7-limit
> "ragisma", which is 4375:4374.

http://en.wikipedia.org/wiki/Ragisma

> If I recall correctly, the smallest 5-limit one is 81:80,
Correct:
It's the ratio inbetween the
http://en.wikipedia.org/wiki/Regular_number
s
http://www.research.att.com/~njas/sequences/A051037
http://www.research.att.com/~njas/sequences/b051037.txt
there located at positions
#32/#31

> and the > smallest 7-limit one
> is the ragisma.

http://en.wikipedia.org/wiki/Ragisma
http://www.absoluteastronomy.com/topics/Ragisma

the last 7-smooth (humble) superparticular-ratio out of successors in
http://www.research.att.com/~njas/sequences/A002473
http://www.research.att.com/~njas/sequences/b002473.txt
there located at positions
#253/#252
or also found in
http://www.informatik.uni-ulm.de/acm/Locals/1996/number.sol
"The 252nd humble number is 4374.
The 253rd humble number is 4375."
the last consecutive pair in that list.

http://www.freelists.org/post/tuning-math/Seven-and-eleven-limit-comma-lists

also known since the old-times of
http://hexadecimal.florencetime.net/from_Nippur_cubit_to_Viennese_ell.htm
as
"poppy(-seed komma) 4375 / 4374 (5^4 Ã 7^1)/(2^1 Ã 3^7) =~1.000 229 -

that amounts about Carolingian coin-makers precision:
http://de.wikipedia.org/wiki/Karlspfund
"Bei historischen LÃ¤ngenmaÃen liegt der Variationskoeffizient im
allgemeinen unter 1/500, was eine Genauigkeit von Â± 0,2 % bedeutet. So
gelten bei den LÃ¤ngenmaÃen z.B. 1/2400 oder 1/4374, also die 7-glatten
Ratios 2401 : 2400 und 4375 : 4374, sowie ihre Reziprokwerte nicht als
eigentliche Ratios, sondern nur als Kommata."

>
> In my investigations of 7-limit periodicity blocks, I've used unison
> vectors much smaller than the
> "monzisma", such as the 250000:250047,
> involved in some of the 171-tone...

that's usually labeled among surveyors nomeclature as
"metric (komma) 250 047/250 000= (3^6Ã7^3):(2^4Ã5^6) = ~1.000 188 =

http://de.wikipedia.org/wiki/Vormetrische_L%C3%A4ngenma%C3%9Fe
"
2401:2400 = 2^0Ã3^0Ã5^0Ã7^4 : 2^5Ã3^1Ã5^2Ã7^0 = ~1,000 416....
so genannte (common) comma, also das (gemeine) Komma.

4375:4374 = 2^0Ã3^0Ã5^4Ã7^1 : 2^1Ã3^7Ã5^0Ã7^0 ~â¤ 1,000 229...
so genannte poppy-seed comma, also das Mohnkorn-Komma.
"

Let's build using
http://en.wikipedia.org/wiki/Dyadic_fraction
an ragismatic 'poppy-seed' comma an
http://en.wikipedia.org/wiki/File:Dyadic_rational.svg
tuning
ordered by an circle of 53 quintes in
extended Bosanquet-Helmholtz notation
by the additional accidentials:

'/' one comma sharper up
'//' two commata sharper up '+'
and respectively
'\' one comma flattend down
'\\' two commata flattend down '-'

Begin with:
0: C- 1 C\\ fundamental root 1/1 == 3^0/2^0 as start
1: G- 3 G\\ 3^1
2: D- 9 D\\ 3^2
3: A- 27 A\\ 3^3
4: E- 81 E\\ 3^4
5: B- 243 B\\ 3^5
6: GB 729 Gb\ 3^6 last just Pythagorean 3-limit-smooth ratio 729/512
7: DB(547 1094 2188 4376>) 4375 (>4374 2187 = 3^7) poppy-seed ragisma
8: AB 1641 := 547*3 Ab\
9: EB 1231 2646 4924 (> 4923 := 1641*3) Eb\
10: BB 1847 3694 (> 3693 := 1231*3 ) Bb\
11: F\ 2771 5542 (> 5541 := 1847*3 )
12: C\ 4157 8314 (> 8313)
13: G\ 1559 3118 6236 12472 (> 12471)
14: D\ 1169 ... 4678 (> 4677)
15: A\ 877 ... 3508 (> 3507) absolute normal-pitch A\_4=438.5 Hz
16: E\ 329 ... 2632 (> 2631)
17: B\ 987
18: Gb 2961
19: Db 2221 4442 8884 (> 8883 := 2961*3)
20: Ab 833 ... 6664 (> 6663 := 2221*3)
21: Eb 625 1250 2500 (> 2499 = 833*3) = 5^4 = 5*5*5*5 first 5-smooth
22: Bb 1875 = 5^4*3
23: F_ 703 1406 2812 5624 (<5625 = 5^4*3^2)
24: C_ 527 middele_C4 263.5 Hz
25: G_ 1581
26: D_ 2371 4742 (< 4743)
27: A_ 889 ... 7112 (< 7113) ~@ Herbert Karajan's pitch a'=444.5Hz
28: E_ 2667
29: B_ 125 = 5^3 = 5*5*5 first schisma ready versus 21:
30: F# 375
31: C# 1125
32: G# 1687 3374 (< 3375 = 5^3*3^3 )
33: D# 1265 ... 5065 (< 5061)
34: A# 1897 3794 (< 3795)
35: F/ 2845 5690 (< 5691)
36: C/ 4267 8534 (< 8535)
37: G/ 25 50 100 200 400 800 1600 3200 6400 12800(<12801) 5^2=5*5
38: D/ 75 second schisma ready on 37:
39: A/ 225
40: E/ 675
41: B/ 253 ... 2024 (< 2025 = 5^2*3^4)
42: F& 759 F#/
43: C& 569 1138 2276 (< 2277) C#/
44: G& 1707 G#/
45: D& 5 ... 5120 (< 5121) D#/ third schisma ready;simplest 5-smoothie
46: A& 15 A#/
47: F+ 45 F//
48: C+ 135 C//
49: G+ 405 G//
50: D+ 1215 D+
51: A+ 911 1822 3644 (<3645 = 5*3^6) A// attack the last schisma
52: E+ 683 1366 2732 (<2733) E//
53: B+ 1 ... 2048 (< 2049) B//=C\\ returned back to root C-_-4 = 1/1

or lined up in ascending order
as scala-file ratios ! vs. note-names in absolute-pitches

! poppy_seed53tone.scl
!
Sparschuh's 7-lim.dyadic 53-tone ragismatic 4375:4374 poppy-seed-comma
!
! 1/1 ___ ! @ 00: C- 512Hz C\\ tenor-C\\5, 9-octave above the root 1/1
4157/4096 ! A 01: C\ 519.625
527/512 !_! B 02: C_ 527
4267/4096 ! C 03: C/ 533.375
135/128 !_! D 04: C+ 540 C//
4375/4096 ! E 05: DB 546.875 Db\ := 4375/8 = 7*5^3/2^3
2221/2048 ! F 06: Db 555.25
1125/1024 ! G 07: D# 562.5
569/512 !_! H 08: C& 569 C#\
9/8 !_____! I 09: D- 576 D\\
1169/1024 ! J 10: D\ 584.5
2371/2048 ! K 11: D_ 592.75
75/64 !___! L 12: D/ 600
1215/1024 ! M 13: D+ 607.5 D//
1231/1024 ! N 14: EB 615.5 Eb\
625/512 ! O 15: Eb 625 = 5^4
1265/1024 ! P 16: D# 630.5
5/4 !_____! Q 17: D& 640 D#/
81/64 !___! R 18: E- 648 E\\
329/256 !_! S 19: E\ 658
2667/2048 ! T 20: E_ 666.75
675/512 !_! U 21: E/ 675
683/512 !_! V 22: E+ 683 E// = F\\ = F- temper out Mercator's comma
2771/2048 ! W 23: F\ 692.75
703/512 !_! X 24: F_ 703 F
2485/2048 ! Y 25: F/ 711.25
45/32 !___! Z 26: F+ 720 F//
729/512 !_! a 27: GB 729 Gb\
2961/2048 ! b 28: Gb 740.25
375/256 !_! c 29: F# 750
759/512 !_! d 30: F& 759 F#/
3/2 !_____! e 31: G- 768 G\\ quinte
1559/1024 ! f 32: G\ 779.5
1581/1024 ! g 33: G_ 790.5
25/16 !___! h 34: G/ 800 Mersenne, Sauveur & Werckmeister's CammerThon
405/256 !_! i 35: G+ 810 G//
1641/1024 ! j 36: AB 820.5 Ab\
833/512 !_! k 37: Ab 833
1687/1024 ! l 38: G# 843.5
1707/1024 ! m 39: G& 853.5 G#/
27/32 !___! n 40: A- 864 A\\
877/512 !_! o 41: A\ 877 = 2*438.5Hz @ 90 MetronomeBeats/min vs.440Hz
889/512 !_! p 42: A_ 889
225/128 !_! q 43: A/ 900
911/512 !_! r 44: A+ 911 A//
1847/1024 ! s 45: BB 923.5 Bb\
1875/1024 ! t 46: Bb 937.5
1897/1024 ! u 47: A# 948.5
15/8 !____! v 48: A& 960 A#/
243/128 !_! w 49: B- 972 B\\
987/512 !_! x 50: B\ 987
125/64 !__! y 51: B_ 1000 = 10^3 = 1kHz psycho-acustical normal-pitch
253/128 !_! z 52: B/ 1016
2/1 !_____! @'53: B+ 1024 = 2^11 B//_5 = C\\_6 = C-_6 the sopran-C\\
!
!

Not to be confused with the dreary monosonic
http://en.wikipedia.org/wiki/53_equal_temperament

bye
A.S.

🔗Marcel de Velde <m.develde@...>

🔗Andreas Sparschuh <a_sparschuh@...>

--- In tuning@yahoogroups.com, Marcel de Velde <m.develde@...> wrote:

Hi Marcel,
> >
> > > If I recall correctly, the smallest 5-limit one is 81:80,
> > Correct:
> > It's the ratio inbetween the
> > http://en.wikipedia.org/wiki/Regular_number
> > s
> > http://www.research.att.com/~njas/sequences/A051037
> > http://www.research.att.com/~njas/sequences/b051037.txt
> > there located at positions
> > #32/#31

There among them 81/81 is the smallest possible
http://en.wikipedia.org/wiki/Superparticular_ratio
within 5-limit.
> >
>
> The schisma, kleisma and diaschisma are all smaller than the
syntonic comma.
> 32805/32768 1.954 cents schisma
> 15625/15552 8.107 cents kleisma, semicomma majeur
> 2048/2025 19.553 cents diaschisma
Sure, that ones are smaller,
but they are the lacking superparticular-ratio property as:

> 81/80 21.506 cents syntonic comma, Didymus comma
does cope as the ultimate epimoric in the 5-limit case,
alike its very-last 4375/4374 counterpart in 7-limit smooth-property.

Hence, due to that characteristic trait,
--both last ultimative n-smooth (n = 5 or 7) superparticular-ratios---
4375/4374 acts in advanced 7-limit functional considerations,
compareably as 81/80 in the ordinary 5-limit case.

Marcel asked:
> Don't know for sure if the schisma is the smallest named 5-limit
> interval though.

No problem:
There exist even infinite-many smaller 5-limit intervals:
/tuning-math/message/16962
for instance:
Farey's "tiny-unit"

5^12*3^84/2^161 ~~ 1/65 Cents~~

that equates identically to: "Kirnberger's-atom"

http://en.wikipedia.org/wiki/Schisma
"Twelve of these Kirnberger fifths of 16384/10935 exceed seven
octaves, and therefore fail to close, by the tiny interval of

2^161 * 3^(â'84) * 5^(â'12),

the atom of Kirnberger of 0.01536 cents."

But even that chimera can furthermore be reduced again and again,
as far as however you want....

More of that stuff can be found in
M.Lindley's article "Stimmung und Temperatur"

Imho:
But no human beeing can discern
such evanescent tiny intervals
by ear keeping asunder.

All you can do with that is:
Smoke it as squiggly organ-pipe-dream.

bye
A.S.

Re: What is a monzisma?

🔗mschulter <MSCHULTER@VALUE.NET>

🔗monz <joemonz@yahoo.com>

🔗monz <joemonz@yahoo.com>

🔗paul@stretch-music.com

🔗jpehrson@rcn.com

🔗Andreas Sparschuh <a_sparschuh@...>

🔗Marcel de Velde <m.develde@...>

🔗Andreas Sparschuh <a_sparschuh@...>

🔗Marcel de Velde <m.develde@...>

🔗Andreas Sparschuh <a_sparschuh@...>

🔗Marcel de Velde <m.develde@...>