Yahoo Tuning Groups Ultimate Backup tuning The use of degrees-minutes-seconds in pitch measurement

I'm finding myself doing this: expressing multiples and fractions of a natural interval not as a decimal number, but as sexagesimal (base-60).

The obvious application would be to treat an octave as complete circle, so that a degree is 1/360 of an octave, a minute is 1/60 of a degree, and a second is 1/60 of a minute. (What's smaller than second in the same pattern?) This can be done, obviously, to analyze generators in relation to periods. The fifth would be expressed as 210ï¿½ 35' 11" in untempered Pythagorean, if the octave is 360ï¿½.

For my own personal practice, I use deg-min-sec notation differently: a degree for me is a Pythagorean comma (so we're not talking 360-degree circles anymore). So when I want to play some JI interval on my fretless bass, I look at the 53-tone guide taped on the side of the neck and place the finger appropriately, based on which comma to play and what fraction to go sharp or flat. I'd like to memorize all the commas up to 31-limit, but I don't see myself going beyond 11/13-limit much right now.

I prefer a sexagesimal understanding because unlike base-10, 60 is divisible by 3 and 6, so it's easier to think in thirds. The Sumerians and Babylonians, and also the Arabs during the Umayyad era, used base-60 mathemathics, because 60 is the smallest number divisible by 2, 3, 4, 5 and 6.

My minute is about 1/3069 of an octave, and my second is about 1/184143 of an octave.

~Danny~

🔗Danny Wier <dawiertx@sbcglobal.net>

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Danny Wier <dawiertx@sbcglobal.net>

From: "Gene Ward Smith"

>> For my own personal practice, I use deg-min-sec notation differently: a
>> degree for me is a Pythagorean comma (so we're not talking 360-degree
>> circles anymore).
>
> Double that to 720 to a Pythagorean comma and you have the tuning
> unit. Then my usual complaint applies: why not multiply by 5/4, so
> now you have (5/2)(360)=900 parts in a Pythagorean comma, which makes
> things work out so you actually do have an equal division of the
> octave, namely, 46032. Since the octave is an important musical
> interval, this seems like a good idea to me.

Actually the Pythagorean comma in my system isn't divided by 360, but by 60, so you're multiplying by 12 instead of doubling. A tuning unit then would be 5 seconds, and one 900th of a comma is 4 seconds.

And I failed to post some examples of how DMS would be applied to JI:

81/80 ~ 0ï¿½ 55' 00", or 11/12 of a P-comma.
128/125 ~ 1ï¿½ 45' 01", or 1 3/4 of a P-comma.
64/33 ~ 1ï¿½ 09' 44", just under 1 1/6 of a P-comma.
33/32 ~ 2ï¿½ 16' 15", a little more than 2 1/4 P-commas.

I still need to calculate all the values of the Sagittal intervals.

~Danny~

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Danny Wier <dawiertx@sbcglobal.net>

🔗monz <monz@tonalsoft.com>

hi Danny,

--- In tuning@yahoogroups.com,
"Danny Wier" <dawiertx@s...>
wrote:

> I'm finding myself doing
> this: expressing multiples
> and fractions of a natural
> interval not as a decimal
> number, but as sexagesimal
> (base-60).

you remark about the Babylonians
and Sumerians ... i wonder if you
know my webpages about this?

http://tonalsoft.com/monzo/sumerian/sumerian-tuning.htm

http://tonalsoft.com/monzo/sumerian/simplified-sumerian-tuning.htm

> The obvious application
> would be to treat an octave
> as complete circle, so that
> a degree is 1/360 of an octave,
> a minute is 1/60 of a degree,
> and a second is 1/60 of a
> minute.

> (What's smaller than second in the same pattern?)

that would be the Latin word for "third":
Tertius, Tertia, Tertium -- depending on the case.

here's a table of data for all these units:

unit .... fraction of octave .. --------- cents ------ ...... edo
................................ ~decimal .. fraction

degree . 1 / 360 .............. 3.333333333 . 3 & 1/3 .......... 360
minute . 1 / ( 360 * (60^1) ) . 0.055555556 . 1 / 18 ........ 21,600
second . 1 / ( 360 * (60^2) ) . 0.000925926 . 1 / 1080 ... 1,296,000
tertia . 1 / ( 360 * (60^3) ) . 0.000015432 . 1 / 64800 . 77,760,000

"degree" is the first, main, largest part, then "minute"
means the smaller part. when the ancient Latin mathematicians
admitted the next smaller part, it was the "second" small part.
by "part" here, i mean the 2-digit sexagesimal place.

> This can be done, obviously, to analyze generators in relation
> to periods. The fifth would be expressed as 210° 35' 11" in
> untempered Pythagorean, if the octave is 360°.

i have a better suggestion: keep things simple and have *all*
the places be sexagesimal: let the whole octave be 1, unity,
which also equals 60. i propose the new term "sexagesa" for
this unit -- note that a sexagesa is in the size range for
the strict definition of "comma". here's a table:

unit ........ --------- cents ------ ....... edo
............ ~decimal ..... fraction

sexagesa .. 20 ................................. 60
minute ..... 0.333333333 .. 1 / 3 ........... 3,600
second ..... 0.005555556 .. 1 / 180 ....... 216,000
tertia ..... 0.000092593 .. 1 / 10800 .. 12,960,000

but i'd prefer to coin new names instead of "minute,
second, tertia" since minute and second are already
well-established as fractions of a degree.

i think 216,000-edo is quite good enough for most measurements
(it's smaller than a 14mu), let alone 12,960,000-edo.

> For my own personal practice, I use deg-min-sec notation
> differently: a degree for me is a Pythagorean comma (so
> we're not talking 360-degree circles anymore). So when
> I want to play some JI interval on my fretless bass,
> I look at the 53-tone guide taped on the side of the neck
> and place the finger appropriately, based on which comma
> to play and what fraction to go sharp or flat. I'd like
> to memorize all the commas up to 31-limit, but I
> don't see myself going beyond 11/13-limit much right now.
>
> I prefer a sexagesimal understanding because unlike
> base-10, 60 is divisible by 3 and 6, so it's easier to
> think in thirds. The Sumerians and Babylonians, and also
> the Arabs during the Umayyad era, used base-60 mathemathics,
> because 60 is the smallest number divisible by 2, 3, 4, 5
> and 6.

indeed, base-60 math is wonderful. it only takes a little
practice to get used to it, and it's *much* more accurate
than decimal, as well as much easier to manipulate mentally
because it's so easily divisible.

> My minute is about 1/3069 of an octave, and my second
> is about 1/184143 of an octave.

unit ........ ---------- cents ---------- ......... edo
............. ~decimal ........ ~fraction

degree .... 23.46001038 ................................ 51
minute ..... 0.391000173 .... 300459 / 768437 ....... 3,069
second ..... 0.00651667 ....... 5234 / 803171 ..... 184,143
tertia ..... 0.000108611 ........ 70 / 644501 .. 11,048,588

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

🔗Danny Wier <dawiertx@sbcglobal.net>

From: "monz"

> you remark about the Babylonians
> and Sumerians ... i wonder if you
> know my webpages about this?
>
> http://tonalsoft.com/monzo/sumerian/sumerian-tuning.htm
>
> http://tonalsoft.com/monzo/sumerian/simplified-sumerian-tuning.htm

I remember you telling me about these tunings a while ago. Did I talk about base-60 some weeks ago, because I'm having some dï¿½jï¿½ vu about it. I forget things easily.

> that would be the Latin word for "third":
> Tertius, Tertia, Tertium -- depending on the case.

I was going to suggest "terce", which is related, and reflects a post-Latin convention (French specifically). And most Latinate words in English come from French. You're probably familiar with the use of the word "terce" to refer to a third interval.

(Or is "tertia" already accepted terminology?)

> i have a better suggestion: keep things simple and have *all*
> the places be sexagesimal: let the whole octave be 1, unity,
> which also equals 60. i propose the new term "sexagesa" for
> this unit -- note that a sexagesa is in the size range for
> the strict definition of "comma". here's a table:

60-TET is not a bad temperament either; I just haven't heard of it being used much. Of course we're now dealing with tenth-tones, 20 cents each.

> but i'd prefer to coin new names instead of "minute,
> second, tertia" since minute and second are already
> well-established as fractions of a degree.

You're probably right; I'm just using the terms degree-minute-second as analogy. I've referred to steps in 53-tone as "degrees" for some time, since I think of "step" as being a whole step, that is, 9 degrees of 53-tone. The other two seem to fall into place.

(I wonder if anyone besides me has proposed an alternative metric system using base-60 instead of base-10, and Planck's units rather than the more arbitrary/geocentric SI standards. I have no idea what to call the units or what analogizes to all the base-10 prefixes.)

> tertia ..... 0.000108611 ........ 70 / 644501 .. 11,048,588

What kind of super-advanced alien civilization would have the need to worry about an 11 millionth of an octave....

~Danny~

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Ozan Yarman <ozanyarman@superonline.com>

Danny, it is an ingenious suggestion to use arc measurement values as a basic for interval calculations. I might, with your approval, implement this suggestion with the pitch-bender wheels of my Ultratonal Piano (tm) project.

Cordially,
Ozan
----- Original Message -----
From: Danny Wier
To: tuning@yahoogroups.com
Sent: 30 Nisan 2005 Cumartesi 9:02
Subject: Re: [tuning] Re: The use of degrees-minutes-seconds in pitch measurement

I just remembered that DMS could also be used as a sort of Eitz-like
notation. A couple examples:

The fifth of 1/4-comma meantone in the key of C could be written as: G - 0°
13' 45", meaning G is lowered by 13 minutes and 45 seconds.

The 13/8 neutral sixth from C would be Ab + 2° 03' 39", meaning Pythagorean
A-flat (128/81) is raised by 1053/1024, which measures to 2 P-commas, 3
minutes and 39 seconds. 4 minutes is 1/15 of a Pythagorean comma (and since
I'm interested in primes higher than 11, I decided to abandon the practice
of simply using degress of 612-EDO, which only divides P-commas into
twelfths).

A fifteenth of a comma is a very small interval indeed, so practically, if I
want 13/8, I'll just play a minor sixth two commas higher and not worry
about the fraction. 11/8 is different since it's 2 1/4 commas sharp of 4/3.

🔗monz <monz@tonalsoft.com>

hi Ozan and Danny,

--- In tuning@yahoogroups.com,
"Ozan Yarman" <ozanyarman@s...>
wrote:

> Danny, it is an ingenious
> suggestion to use arc measurement
> values as a basic for interval
> calculations. I might, with your
> approval, implement this
> suggestion with the pitch-bender
> wheels of my Ultratonal Piano (tm)
> project.

the first mention i know of for arc measurement is
from W. S. B. Woolhouse:

http://tonalsoft.com/monzo/woolhouse/essay.htm

look under the section "730-EDO as a basic unit":

>> [p 20] Woolhouse mentions likening the octave to a circle,
>> and using degrees, minutes, and seconds to measure the
>> intervals. This amounts to 1296000-EDO (yes, that's over
>> a million degrees: 360 * 60 * 60), which Woolhouse
>> dismisses as 'of no advantage in musical computations'.

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

🔗Ozan Yarman <ozanyarman@superonline.com>

Ouch. I have to pay him royalty then.

----- Original Message -----
From: monz
To: tuning@yahoogroups.com
Sent: 01 Mayıs 2005 Pazar 2:09
Subject: [tuning] Re: The use of degrees-minutes-seconds in pitch measurement

hi Ozan and Danny,

--- In tuning@yahoogroups.com,
"Ozan Yarman" <ozanyarman@s...>
wrote:

the first mention i know of for arc measurement is
from W. S. B. Woolhouse:

http://tonalsoft.com/monzo/woolhouse/essay.htm

look under the section "730-EDO as a basic unit":

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