Sharing the juicy details of modified 79-tone extra-ordinaire

>
> Oz, the number of (apparently) exact brats (both major and minor) is
> quite remarkable. How did you arrive at these numbers?
>
> --George
>

Why, it was no easy task. Please scrutinize:
http://www.ozanyarman.com/misc/79tone.xls

The first step is to remember the pattern of the cycle of fifths derived
from the logarithmic division of 4:3 into 33 equal parts + extending the
`comma` to the 79th step + completing the 79th degree to the octave +
transposing the scale over to the 33rd key.

Notice the pattern that goes AB AB AB AAB, etc...

The original approach involving the division of 4:3 into 33 equal parts
yields a scale practically equivalent not only to 79 MOS 159-tET, but also
temperament by 19/53 of a syntonic comma as specified in the excel file that
comprises all the juicy details.

Now flip to the second sheet of this excel file. Observe how I truncated the
beat rates of the appreciable fifths. Nonetheless, the other beat
frequencies are far from pleasing me. So what I did was round off the
decimal values of the frequencies in column F to their quarts as seen in
column R (I could not automate this procedure in Excel. Do you know a way?).

Amazingly enough, the scale is intact by an highest absolute difference of
about a cent compared with the original version. Better yet, the meantone
fifths are now 695 cents in average and they resonate perfectly with the
modified beat frequencies. Moreover, the chords yield excellent brats, the
Maqamat is preserved over every key, and the notation is correct to the
utmost detail.

Now, I would like you to create a Sagittal template for 80 MOS 159-tET (an
extension of the modified extra-ordinaire 79-tone version, with the addition
of a meantone fifth at 391.5 hz), so that I might notate all pitches
according to a super-pythagorean cycle.

Is this an impressive thing or what?

Also, can you tell me what is the easiest way to fill the cycle of intervals
columns in the second sheet?

Cordially,
Oz.

Uh, I came up with this tuning:

0: 1/1 C Dbb unison, perfect prime
1: 93.255 cents C# Db
2: 196.552 cents D Ebb
3: 302.375 cents D# Eb
4: 391.592 cents E Fb
5: 498.871 cents F Gbb
6: 590.517 cents F# Gb
7: 701.955 cents G Abb
8: 792.077 cents G# Ab
9: 897.524 cents A Bbb
10: 1004.330 cents A# Bb
11: 1094.425 cents B Cb
12: 1200.000 cents C Dbb

It has the following beat rates:

Base frequency : 262.0000 Hertz
Beat frequencies of 3/2 5/4 6/5
0: 0.000: 0.0000 4.0000 -12.0000
1: 93.255: -1.5000 15.5000 -16.5000
2: 196.552: -0.5000 6.5000 -13.5000
3: 302.375: 0.0000 12.0000 -29.5000
4: 391.592: 0.5000 13.5000 -6.0000
5: 498.871: -0.5000 12.5000 -27.0000
6: 590.517: 0.5000 29.5000 -11.0000
7: 701.955: -5.0000 7.0000 -18.0000
8: 792.077: 6.0000 26.0000 -19.0000
9: 897.524: -6.0000 12.0000 -20.0000
10: 1004.330: -6.0000 8.0000 -43.0000
11: 1094.425: -5.0000 31.0000 -23.0000
12: 1200.000: 0.0000 8.0000 -24.0000
Total abs. beats : 31.5000 177.5000 238.5000
Average abs. beats: 2.6250 14.7917 19.8750
Highest abs. beats: 6.0000 31.0000 43.0000

Is it extra-ordinaire too?

Cordially,
Oz.

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
>
> Uh, I came up with this tuning:
>
> 0: 1/1 C Dbb unison, perfect prime
> 1: 93.255 cents C# Db
> 2: 196.552 cents D Ebb
> 3: 302.375 cents D# Eb
> 4: 391.592 cents E Fb
> 5: 498.871 cents F Gbb
> 6: 590.517 cents F# Gb
> 7: 701.955 cents G Abb
> 8: 792.077 cents G# Ab
> 9: 897.524 cents A Bbb
> 10: 1004.330 cents A# Bb
> 11: 1094.425 cents B Cb
> 12: 1200.000 cents C Dbb
>
> It has the following beat rates:
>
> Base frequency : 262.0000 Hertz
> Beat frequencies of 3/2 5/4 6/5
> 0: 0.000: 0.0000 4.0000 -12.0000
> 1: 93.255: -1.5000 15.5000 -16.5000
> 2: 196.552: -0.5000 6.5000 -13.5000
> 3: 302.375: 0.0000 12.0000 -29.5000
> 4: 391.592: 0.5000 13.5000 -6.0000
> 5: 498.871: -0.5000 12.5000 -27.0000
> 6: 590.517: 0.5000 29.5000 -11.0000
> 7: 701.955: -5.0000 7.0000 -18.0000
> 8: 792.077: 6.0000 26.0000 -19.0000
> 9: 897.524: -6.0000 12.0000 -20.0000
> 10: 1004.330: -6.0000 8.0000 -43.0000
> 11: 1094.425: -5.0000 31.0000 -23.0000
> 12: 1200.000: 0.0000 8.0000 -24.0000
> Total abs. beats : 31.5000 177.5000 238.5000
> Average abs. beats: 2.6250 14.7917 19.8750
> Highest abs. beats: 6.0000 31.0000 43.0000
>
> Is it extra-ordinaire too?
>
> Cordially,
> Oz.

Um, I wouldn't be inclined to think of this as very _extraordinaire_,
because it's the *ratios* between the beat rates (or brats, which are
not affected by the base frequency) that count, not the beat rates
themselves (which change if the base frequency is changed). To see
these in Scala, enter
show/relative beats 5/4 3/2
for the major triads, and
show/relative beats 6/5 3/2
for the minor triads.

The only reasonably simple brats are on C, Eb, A, F, and E major, and
on C, Eb, A, C#, and G minor, which is less than half of the triads.

The questions in your other message will not be as quick for me to
answer; I will need more time to study your "79 MOS 159tET Ratios
based on simple frequencies" and accompanying spreadsheet. It would
help me if I had the actual ratios for each tone rather than just
cents. (If you don't have these, then perhaps Scala has a command to
convert cents to ratios?)

Best,

--George

Dear George, the SCALA command for a quick conversion from cents to ADO is a
Farey approximation (CTRL+SHIFT+F) where the maximum denominator should be
specified as 1048. This gives you the ratios for the modified 79-tone
extraordinaire by a common numerary nexus of 1048.

You know that I cannot extract a closed 12-tone cycle without employing a
super-pythagorean fifth somewhere along the chain. Unfortunately, this may
also be the reason why the maximum error of thirds far-exceeds the
academical standard of piano-tuning as you mentioned (18.5 cents?). Is there
any way to improve triad brats?

Cordially,
Oz.

SNIP

Um, I wouldn't be inclined to think of this as very _extraordinaire_,
because it's the *ratios* between the beat rates (or brats, which are
not affected by the base frequency) that count, not the beat rates
themselves (which change if the base frequency is changed). To see
these in Scala, enter
show/relative beats 5/4 3/2
for the major triads, and
show/relative beats 6/5 3/2
for the minor triads.

The only reasonably simple brats are on C, Eb, A, F, and E major, and
on C, Eb, A, C#, and G minor, which is less than half of the triads.

The questions in your other message will not be as quick for me to
answer; I will need more time to study your "79 MOS 159tET Ratios
based on simple frequencies" and accompanying spreadsheet. It would
help me if I had the actual ratios for each tone rather than just
cents. (If you don't have these, then perhaps Scala has a command to
convert cents to ratios?)

Best,

--George