19 Math

I'm investigating the mathematical set up of 19-tone temperment. If
there are any resources I'm not aware of, please hip me to them. I'm
interested in things like; Does 19 have a 3:2 fifth? Does 19 have a
5:4 third? et cetera...

fretlessjazz3000 wrote:
> I'm investigating the mathematical set up of 19-tone temperment. If > there are any resources I'm not aware of, please hip me to them. I'm > interested in things like; Does 19 have a 3:2 fifth? Does 19 have a > 5:4 third? et cetera...

Its fifth is 7.2 cents flat of 3:2, and the major third is about the same accuracy (7.4 cents flat), but the minor third is within 0.1 cent of just at 315.8 cents. This means that 19-ET is pretty close to 1/3 comma meantone (which has just minor thirds). It also has an interval near 9/7 (7 cents sharp, at 442.1 cents) but in other respects isn't very good for the 7-limit intervals.

The accuracy of the fifth and major third can be improved (at the expense of minor thirds and octaves) by stretching the octave to a little over 1202 cents. (One particular optimization of this, 5-limit TOP tuning, has an octave of 1202.281405 cents.)

Another thing to look at is which small intervals (commas) are tempered out (as 81/80 is tempered out in meantone temperament). This has an effect on the chord progressions that are possible. In addition to 81/80, some of the commas tempered out by 19-TET include 126/125 ("starling" temperament), 3125/3072 ("magic" temperament), 15625/15552 ("hanson" temperament), 16875/16384 ("negri" temperament), and 78732/78125 (the "semisixths" or "sensi" temperament).

In addition to the diatonic scale, there is a symmetrical 9- or 10-note scale which has useful harmonic and melodic resources, devised by Joe Negri (from which "negri" temperament gets its name). This scale is created by repeated intervals of 2 steps of 19-ET (i.e., skipping every other note).

--- In tuning@yahoogroups.com, Herman Miller wrote:
>
> fretlessjazz3000 wrote:
> > I'm investigating the mathematical set up of 19-tone temperment.
If there are any resources I'm not aware of, please hip me to them.
I'm interested in things like; Does 19 have a 3:2 fifth? Does 19
have a 5:4 third? et cetera...
>
> Its fifth is 7.2 cents flat of 3:2, and the major third is about
the same accuracy (7.4 cents flat), but the minor third is within 0.1
cent of just at 315.8 cents. This means that 19-ET is pretty close to
1/3 comma meantone (which has just minor thirds). It also has an
interval near 9/7 (7 cents sharp, at 442.1 cents) but in other
respects isn't very good for the 7-limit intervals.
>
> The accuracy of the fifth and major third can be improved (at the
expense of minor thirds and octaves) by stretching the octave to a
little over 1202 cents. (One particular optimization of this, 5-limit
TOP tuning, has an octave of 1202.281405 cents.)
>
> Another thing to look at is which small intervals (commas) are
tempered out (as 81/80 is tempered out in meantone temperament). This
has an effect on the chord progressions that are possible. In
addition to 81/80, some of the commas tempered out by 19-TET include
126/125 ("starling" temperament), 3125/3072 ("magic" temperament),
15625/15552 ("hanson" temperament), 16875/16384 ("negri"
temperament), and 78732/78125 (the "semisixths" or "sensi"
temperament).
>
> In addition to the diatonic scale, there is a symmetrical 9- or 10-
note scale which has useful harmonic and melodic resources, devised
by Joe Negri (from which "negri" temperament gets its name). This
scale is created by repeated intervals of 2 steps of 19-ET (i.e.,
skipping every other note).

Hi Herman!

That's a pretty thorough reply there. I'm interested in your last
paragraph, as you've had more experience than I with such tunings.
You say that a symmetrical 9-note scale can be made by skipping every
other step - that's easy enough to see; that it has useful harmonic
resources, by which I take it you mean it nicely approximates a few
interesting ratios; and that it has useful melodic properties.
Doesn't the lack of a leading tone make this a rather restless scale
melodically? And further, are there any really solid-sounding
harmonic progressions that are natural to a symmetric scale? On this
score, I mean without fudging it by using a few triads and then a
tetrad-with-seventh.

What I'm getting at, is just how useful you think the properties of
this scale are for composition: is it worth the trouble of tuning up
to experiment in, or would other scales in 19-EDO offer a richer
palette?

Regards,
Yahya

--- In tuning@yahoogroups.com, Herman Miller wrote:
>
> fretlessjazz3000 wrote:
> > I'm investigating the mathematical set up of 19-tone temperment.
If there are any resources I'm not aware of, please hip me to them.
I'm interested in things like; Does 19 have a 3:2 fifth? Does 19
have a 5:4 third? et cetera...
>
> Its fifth is 7.2 cents flat of 3:2, and the major third is about
the same accuracy (7.4 cents flat), but the minor third is within 0.1
cent of just at 315.8 cents. This means that 19-ET is pretty close to
1/3 comma meantone (which has just minor thirds). It also has an
interval near 9/7 (7 cents sharp, at 442.1 cents) but in other
respects isn't very good for the 7-limit intervals.
>
> The accuracy of the fifth and major third can be improved (at the
expense of minor thirds and octaves) by stretching the octave to a
little over 1202 cents. (One particular optimization of this, 5-limit
TOP tuning, has an octave of 1202.281405 cents.)
>
> Another thing to look at is which small intervals (commas) are
tempered out (as 81/80 is tempered out in meantone temperament). This
has an effect on the chord progressions that are possible. In
addition to 81/80, some of the commas tempered out by 19-TET include
126/125 ("starling" temperament), 3125/3072 ("magic" temperament),
15625/15552 ("hanson" temperament), 16875/16384 ("negri"
temperament), and 78732/78125 (the "semisixths" or "sensi"
temperament).
>
> In addition to the diatonic scale, there is a symmetrical 9- or 10-
note scale which has useful harmonic and melodic resources, devised
by Joe Negri (from which "negri" temperament gets its name). This
scale is created by repeated intervals of 2 steps of 19-ET (i.e.,
skipping every other note).

Hi Herman!

That's a pretty thorough reply there. I'm interested in your last
paragraph, as you've had more experience than I with such tunings.
You say that a symmetrical 9-note scale can be made by skipping every
other step - that's easy enough to see; that it has useful harmonic
resources, by which I take it you mean it nicely approximates a few
interesting ratios; and that it has useful melodic properties.
Doesn't the lack of a leading tone make this a rather restless scale
melodically? And further, are there any really solid-sounding
harmonic progressions that are natural to a symmetric scale? On this
score, I mean without fudging it by using a few triads and then a
tetrad-with-seventh.

What I'm getting at, is just how useful you think the properties of
this scale are for composition: is it worth the trouble of tuning up
to experiment in, or would other scales in 19-EDO offer a richer
palette?

Regards,
Yahya

--- In tuning@yahoogroups.com, "yahya_melb" <yahya@...> wrote:
>
> --- In tuning@yahoogroups.com, Herman Miller wrote:
> >
> >
> > In addition to the diatonic scale, there is a symmetrical 9- or
> > 10-note scale which has useful harmonic and melodic resources,
> > devised by Joe Negri (from which "negri" temperament gets its name).
> > This scale is created by repeated intervals of 2 steps of 19-ET
> > (i.e., skipping every other note).
>

> Doesn't the lack of a leading tone make this a rather restless scale
> melodically?

Without having heard the scale - I would guess it is not that
"restless". AFAIK, 2 steps are close to a semitone, good enough for a
leading tone - and if not, you can take the 10-note scale (9 intervals
of 2 steps and one of 1 step).

And "restlessness" is IMHO less connected to a small leading tone than
to the existence of non-trivial translation symmetries - such as the
whole-tone scale in 12edo. And this is property that no scale in 19edo
has (the scale above is not that "symmetric" in this aspect).

I once had the thought I might use 19edo or so for creating whole-tone
music without that infamous restlessness...
--
Hans Straub

Hi,

Yes, 19 does approximate 3:2 and 5:4. It doesn't do justice to a 7:4,
however. You might want to investigate 22 and 31 e.t.'s as well.

To check out a given e.t. to see how close you're going to get to a
given ratio, you need to use a calculator to find the 19th or whatever
root of 2. Multiply the answer times itself to get the next step, then
that answer times the (n-th) root again to get the next step, and so
on. It will help if you have a list of the ratios you're interested in
converted to decimal so you can check to see how close the e.t. is to
what you're after. (3:2 = 1.5, etc.)

You should also check out the links section of this list. There are
guitar players on the list who have experimented with all sorts of
e.t.'s. and other fretting systems.

You can also download (for free) Scala. In Scala, you can listen to
various e.t.'s and just-intoned scales and create your own.

Good luck,

Robin

--- In tuning@yahoogroups.com, "fretlessjazz3000" <bencole.miller@...>
wrote:
>
> I'm investigating the mathematical set up of 19-tone temperment. If
> there are any resources I'm not aware of, please hip me to them. I'm
> interested in things like; Does 19 have a 3:2 fifth? Does 19 have a
> 5:4 third? et cetera...
>

--- In tuning@yahoogroups.com, "yahya_melb" <yahya@...> wrote:

> What I'm getting at, is just how useful you think the properties of
> this scale are for composition: is it worth the trouble of tuning up
> to experiment in, or would other scales in 19-EDO offer a richer
> palette?

Aside from meantone/flattone, with an 11d19 fifth as generator,
logical things to try are magic/muggles, with a 6d19 major third as
generator, and keemun/hanson, with a 5d19 (nearly pure) minor third as
generator.
Meantone of course has nice MOS scales of size 7 and 12. For magic,
you could try 7, 10, or 13 notes, and for keemun, 7 or 11 notes.

Negri therefore is not the only alternative to meantone. For the
really brave, there is godzilla, with a 4d19 generator, which really
only makes sense if you want to emphasize the 7-limit (which 19
doesn't do that well.) A 9 or 14 note scale springs to mind.

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yahya_melb wrote:

> Hi Herman!
> > That's a pretty thorough reply there. I'm interested in your last > paragraph, as you've had more experience than I with such tunings. > You say that a symmetrical 9-note scale can be made by skipping every > other step - that's easy enough to see; that it has useful harmonic > resources, by which I take it you mean it nicely approximates a few > interesting ratios; and that it has useful melodic properties. > Doesn't the lack of a leading tone make this a rather restless scale > melodically? And further, are there any really solid-sounding > harmonic progressions that are natural to a symmetric scale? On this > score, I mean without fudging it by using a few triads and then a > tetrad-with-seventh. > > What I'm getting at, is just how useful you think the properties of > this scale are for composition: is it worth the trouble of tuning up > to experiment in, or would other scales in 19-EDO offer a richer > palette?

I don't think the 9-note scale by itself is anywhere near as useful as the diatonic scale, but it's one of the more notable features of 19-ET, and like the whole-tone scale in 12-ET, it could be used for extra flavor. I don't know whether you'd consider the scale as restless -- it depends on which note you use as the tonic. I think wide leading tones (126.3 cents in this case) can give a scale a more "relaxed" character, without the strong sense of resolving a dissonance that you get with scales that have narrower leading tones. One of the melodic features I find interesting is the ability to split the major third into three equal steps. Unfortunately it doesn't have many triads for a scale of its size (2 major + 2 minor + 2 augmented), and the available progressions are limited (e.g., Bb minor - A minor - F major - Gb major - Bb minor).

As Gene mentioned, hanson temperament has a scale with a chain of minor thirds, which I've found to be quite useful in the 11-note version. Although I haven't specifically tried it in 19-ET, the fact that 19-ET has such good minor thirds makes that a good option to try out. There are probably many other scales in 19-ET that could be of some use; here is a list of strictly proper 7-note scales (giving the number of 19-ET steps between each note of the scale)

1 3 3 3 3 3 3
1 4 2 3 3 2 4
2 2 3 3 3 3 3
2 3 2 3 3 2 4
2 3 2 3 3 3 3
2 3 2 4 2 3 3
2 3 3 2 3 3 3 (diatonic)

Each of these can have different modes, depending on which note you start with (compare the major 3 3 2 3 3 3 2 and minor 3 2 3 3 2 3 3 scales, which are both rotations of the diatonic scale structure). Then there are many 7-note scales that are proper but not strictly proper, as well as scales with different numbers of notes.

--- In tuning@yahoogroups.com, "Gene Ward Smith" wrote:
>
> --- In tuning@yahoogroups.com, "yahya_melb" wrote:
>
> > What I'm getting at, is just how useful you think the properties
of this scale are for composition: is it worth the trouble of tuning
up to experiment in, or would other scales in 19-EDO offer a richer
palette?
>
> Aside from meantone/flattone, with an 11d19 fifth as generator,
logical things to try are magic/muggles, with a 6d19 major third as
generator, and keemun/hanson, with a 5d19 (nearly pure) minor third
as generator.
> Meantone of course has nice MOS scales of size 7 and 12. For magic,
you could try 7, 10, or 13 notes, and for keemun, 7 or 11 notes.
>
> Negri therefore is not the only alternative to meantone. For the
really brave, there is godzilla, with a 4d19 generator, which really
only makes sense if you want to emphasize the 7-limit (which 19
doesn't do that well.) A 9 or 14 note scale springs to mind.

Hi Gene,

Thanks a lot for these suggestions. They should keep me busy for a
while ....

Regards,
Yahya

--- In tuning@yahoogroups.com, taylan wrote:
>
> hey, here is the list. some of the intervals are quite out-there,
but you'll see immediately which intervals are most closely
approximated. if you want to stick below a certain limit - say 13 -
just go to the nearest ratio. for instance, you could (if you wanted
to) read 52/29 as 13/7 (being 52/28)

Hey there, Taylan!

You could, but that wouldn't give very good approximations.
In case anyone is interested, I just spreadsheeted some
good approximate ratios for the notes of 19-EDO in Excel,
all the way from the 31-prime-limit down to the 7-prime-limit.

Folks, mail me offline if you want a copy of the spreadsheet. That
includes you, "fretlessjazz". It answers the questions about the 3/2
fifth, the thirds and so on.

Regards,
Yahya

>
> have fun, taylan.
>
> p.s. what i call "cents exactly", is the actual cents value of the
ratio
>
> 19-tone equal temperament
>
>
>
> cents 19t ratio cents exactly
>
>
>
> 0: 1/1 0
>
> 1: 63.158 28/27 62.961
>
> 2: 126.316 14/13 128.298
>
> 3: 189.474 29/26 189.050
>
> 4: 252.632 81/70 252.680
>
> 5: 315.789 6/5 315.641
>
> 6: 378.947 56/45 378.602
>
> 7: 442.105 40/31 441.278
>
> 8: 505.263 75/56 505.757
>
> 9: 568.421 25/18 568.717
>
> 10: 631.579 36/25 631.283
>
> 11: 694.737 121/81 694.816
>
> 12: 757.895 48/31 756.919
>
> 31/20 758.722
>
> 13: 821.053 45/28 821.398
>
> 14: 884.211 5/3 884.359
>
> 15: 947.368 19/11 946.195
>
> 16: 1010.526 52/29 1010.950
>
> 17: 1073.684 119/64 1073.781
>
> 18: 1136.842 27/14 1137.039
>
> 19: 2/1 1200
>
>
>
> 1 step = 252.6315
>
>
> On 11/28/06, fretlessjazz3000 wrote:
> >
> > I'm investigating the mathematical set up of 19-tone
temperment. If there are any resources I'm not aware of, please hip
me to them. I'm interested in things like; Does 19 have a 3:2 fifth?
Does 19 have a 5:4 third? et cetera...

--- In tuning@yahoogroups.com, Herman Miller wrote:
>
[snip]
> > What I'm getting at, is just how useful you think the properties
of this scale are for composition: is it worth the trouble of tuning
up to experiment in, or would other scales in 19-EDO offer a richer
palette?
>
> I don't think the 9-note scale by itself is anywhere near as useful
as the diatonic scale, but it's one of the more notable features of
19-ET, and like the whole-tone scale in 12-ET, it could be used for
extra flavor. I don't know whether you'd consider the scale as
restless -- it depends on which note you use as the tonic. I think
wide leading tones (126.3 cents in this case) can give a scale a
more "relaxed" character, without the strong sense of resolving a
dissonance that you get with scales that have narrower leading tones.
One of the melodic features I find interesting is the ability to
split the major third into three equal steps. Unfortunately it
doesn't have many triads for a scale of its size (2 major + 2 minor +
2 augmented), and the available progressions are limited (e.g., Bb
minor - A minor - F major - Gb major - Bb minor).
>
> As Gene mentioned, hanson temperament has a scale with a chain of
minor thirds, which I've found to be quite useful in the 11-note
version.
> Although I haven't specifically tried it in 19-ET, the fact that 19-
ET has such good minor thirds makes that a good option to try out.
There are probably many other scales in 19-ET that could be of some
use; here is a list of strictly proper 7-note scales (giving the
number of 19-ET steps between each note of the scale)
>
> 1 3 3 3 3 3 3
> 1 4 2 3 3 2 4
> 2 2 3 3 3 3 3
> 2 3 2 3 3 2 4
> 2 3 2 3 3 3 3
> 2 3 2 4 2 3 3
> 2 3 3 2 3 3 3 (diatonic)
>
> Each of these can have different modes, depending on which note you
start with (compare the major 3 3 2 3 3 3 2 and minor 3 2 3 3 2 3 3
scales, which are both rotations of the diatonic scale structure).
Then there are many 7-note scales that are proper but not strictly
proper, as well as scales with different numbers of notes.

Thank you for these ideas, Herman.

Two of the "strictly proper" scales you listed have three step sizes,
and one (the second) has four - quite an unusual quantity to handle
in an EDO. It's "angular" enough to probably have a distinctive
flavour, so I'll probably give it a go.

IIRC, "strictly proper" means that no interval comprising a given
number of scale steps is smaller than one that comprises more scale
steps; eg no "fourth" is smaller than any "third". A very useful
property when writing stretches of parallel multi-part harmony, as
you don't get gaps widening and contracting uncontrollably.

I note you chose only 7-note scales; I will open a fresh discussion
on that.

Regards,
Yahya

--- In tuning@yahoogroups.com, "fretlessjazz3000" <bencole.miller@...>
wrote:
>
> investigating the mathematical set up of 19-tone temperment. If
> there are any resources I'm not aware of, please hip me to them. Doe
> 19 have a 3:2 fifth? Does 19 have a
> 5:4 third? et cetera...
>
here comes an 19-division with above demanded/asked properties:

A (55, 110, 440)111, 222, 444 Hz absolute start
E 165:= 55*3
B 247, 494(495):= 165*3
F# 185, 370 740(741:= 247*3)
C# 69, 138, 276, 552(555:= 185*3)
G# 103, 206(207:= 69*3)
D# 77, 154, 308(309:= 103*3)
A# (115, 230)231:= 77*3
E#=Fb 43, 86, 172, 344(345:= 115*3)
Cb (127)128(129:= 43*3)
Gb (95, 190, 380)381:= 127*3
Db (71, 142, 284)285:= 95*3
Ab (53, 106, 212)213:= 71*3
Eb (79, 158)159:= 53*3
Bb (59, 118, 236)237:= 79*3
F (11,22,44,88,176)177:= 59*3
C 33:= 11*3
G 99
D (37, 74 148, 296)297:= 99*3
A 111:= 37*3 cycle closed, easily tuned within a few minutes

order all that frequencies in ascending sequence

264 C 1/1 middle C with 264 Hz
276 C#
285 Db
297 D
308 D#
318 Eb
330 E 5/4
344 E#=Fb
354 F (4/3)*(176/177)
370 F#
381 Gb
396 G 3/2
416 G#
426 Ab
444 A 5/3*(111/110) that's 4Hz above Normal-Pitch 440Hz
462 A# 7/4
474 Bb
512 Cb'
528 C' 2/1

it contains even an just C:E:G:A#:D==4:5:6:7:9 pure 9th-chord in
extended C-major. Sounds nice on:
http://www.denzilwraight.com/roman.htm#cimbalocromatico
http://www.denzilwraight.com/cs19phs3.jpg
but works there also well in all other 18 keys too,
but increasing less brilliant, the more one modulates away from C.

have a lot of fun and joy with that
A.S.

yahya_melb wrote:

> I note you chose only 7-note scales; I will open a fresh discussion > on that.

I happened to have a list of those already in a convenient location.

Here's some 8-note scale patterns that are also strictly proper:

1 3 2 3 2 3 2 3
2 2 2 3 2 3 2 3
2 2 3 2 2 3 2 3

There are also two 9-note strictly proper scale patterns:

1 3 2 2 2 2 2 2 3
2 2 2 2 2 2 2 2 3

And a whole bunch of smaller scales.

--- In tuning@yahoogroups.com, "fretlessjazz3000" <bencole.miller@...>
wrote:
>
> I'm investigating the mathematical set up of 19-tone temperment. If
> there are any resources I'm not aware of, please hip me to them. I'm
> interested in things like; Does 19 have a 3:2 fifth? Does 19 have a
> 5:4 third? et cetera...

Here's my webpage showing a diagram of a 19-edo guitar
fretboard, showing the degree of 19-edo and the Semitone
value (i.e., cents divided by 100, to 2 decimal places)
for each fret.

http://sonic-arts.org/monzo/fretboards/eq-fbs.htm

-monz
http://tonalsoft.com
Tonescape microtonal music software

--- In tuning@yahoogroups.com, "fretlessjazz3000" <bencole.miller@...>
wrote:
>
> I'm investigating the mathematical set up of 19-tone temperment.
> If there are any resources I'm not aware of, please hip me to
> them. I'm interested in things like; Does 19 have a 3:2 fifth?
> Does 19 have a 5:4 third? et cetera...

There's also my Tonalsoft Encyclopedia page about 19-edo,
which goes into pretty good detail, and also explores its
relationship to 1/3-comma meantone (which was pointed out
by Herman Miller in the first response to your post):

http://tonalsoft.com/enc/number/19edo.aspx

-monz
http://tonalsoft.com
Tonescape microtonal music software

hi taylan,

--- In tuning@yahoogroups.com, taylan <taylan.susam@...> wrote:

> p.s. what i call "cents exactly", is the actual cents value
> of the ratio
>
> 19-tone equal temperament
>
>
>
> cents 19t ratio cents exactly
>
>
>
> 0: 1/1 0
> 1: 63.158 28/27 62.961
> 2: 126.316 14/13 128.298
>
> <etc., snip>

Except that your "cents exactly" are still only an
approximation to 3 decimal places.

For 19-edo, you have to use fractions to get the
exact values, and here they are:

... 0 ...... 0
... 1 ..... 63 + 3/19
... 2 .... 126 + 6/19
... 3 .... 189 + 9/19
... 4 .... 252 + 12/19
... 5 .... 315 + 15/19
... 6 .... 378 + 18/19
... 7 .... 442 + 2/19
... 8 .... 505 + 5/19
... 9 .... 568 + 8/19
.. 10 .... 631 + 11/19
.. 11 .... 694 + 14/19
.. 12 .... 757 + 17/19
.. 13 .... 821 + 1/19
.. 14 .... 884 + 4/19
.. 15 .... 947 + 7/19
.. 16 ... 1010 + 10/19
.. 17 ... 1073 + 13/19
.. 18 ... 1136 + 16/19
.. 19 ... 1200 ( = 0 )

-monz
http://tonalsoft.com
Tonescape microtonal music software

hi monz,

> For 19-edo, you have to use fractions to get the
> exact values, and here they are:

yes, of course - thanks for pointing it out! i forgot that i was
posting a list created for my own use... the tolerance i use is by the
way actually much larger. what are your thoughts concerning this?

best,

taylan

hi taylan,

--- In tuning@yahoogroups.com, taylan <taylan.susam@...> wrote:

> > For 19-edo, you have to use fractions to get the
> > exact values, and here they are:
>
> yes, of course - thanks for pointing it out! i forgot that
> i was posting a list created for my own use... the tolerance
> i use is by the way actually much larger. what are your
> thoughts concerning this?

My own personal preference is to use exact values
wherever possible, so as to avoid rounding errors
in calculations.

Thus, for EDOs i like to use the fractional cents values.

But even beyond that, i prefer to express musical pitches
in terms of a "monzo" (Gene's name for the prime-factor
exponent vector of a ratio) or its equivalent (for a
frequency which can be factored exactly by some non-prime
base).

By utilizing an exponent vector, the math is generally
integer math, which is simpler than floating-point,
and accumulation of errors is also avoided.

(Of course, cents can also be viewed as an exponent of 2.)

If you're new to this terminology and don't know what
i'm talking about, here are some of my webpages:

http://tonalsoft.com/enc/m/monzo.aspx

http://tonalsoft.com/monzo/article/article.htm

-monz
http://tonalsoft.com
Tonescape microtonal music software

--- In tuning@yahoogroups.com, "monz" <monz@...> wrote:

> By utilizing an exponent vector, the math is generally
> integer math, which is simpler than floating-point,
> and accumulation of errors is also avoided.

When it isn't integer math, it is often rational number math, which
has the same characteristics.