Meantones with stretched octaves

I'm looking for meantone scales in the Scala archive with the octave
stretched so that the tempered fifth is closer to the just fifth. I found
two.

One was proposed by Dave Keenan to this list in December 1999, the file name
is 'meansixthso.scl'. The tempered octave is 1/6 comma (3.58458 cents) sharp
of the just octave and the tempered fifth is 1/6 comma flat of the just
fifth. The major third is exactly 5/4 and the augmented fifth is exactly
25/16.

The other is called 'Top 5&7 limit meantone', file named
'smithgw_meantop.scl' (so I assume it's Gene Ward Smith). The octave is
stretched by 1.69852 cents, about 1/8 comma. The fifth is flat by 4.39016
cents.

I know how the first scale was generated, but how was the second? And what
other similar scales are there?

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

> The other is called 'Top 5&7 limit meantone', file named
> 'smithgw_meantop.scl' (so I assume it's Gene Ward Smith). The octave is
> stretched by 1.69852 cents, about 1/8 comma. The fifth is flat by
4.39016
> cents.
>
> I know how the first scale was generated, but how was the second?
And what
> other similar scales are there?

The second is an example of TOP tuning, a huge thread you may have
missed. TOP tuning can be applied to regular temperaments in general,
so you can TOP tune 12-equal and so forth.

Danny Wier wrote:

> I know how the first scale was generated, but how was the second? And what
> other similar scales are there?

pormswe

octave sharp by 1.4 cents

fifth flat by 4.9 cents

see

/tuning-math/message/10228

graham

>> I know how the first scale was generated, but how was
>> the second? And what other similar scales are there?
>
>pormswe
>
>octave sharp by 1.4 cents
>
>fifth flat by 4.9 cents
>
>see
>
>/tuning-math/message/10228

Wonderful post. I'm very interested in this approach.

-Carl

>I'm looking for meantone scales in the Scala archive with the octave
>stretched so that the tempered fifth is closer to the just fifth.
//
>And what other similar scales are there?

Don't forget the 7th-root-of-3/2 tuning, which actually has just
fifths!

-Carl

--- In tuning@yahoogroups.com, "Danny Wier" <dawiertx@s...> wrote:
> I'm looking for meantone scales in the Scala archive with the octave
> stretched so that the tempered fifth is closer to the just fifth. I
found
> two.
>
> One was proposed by Dave Keenan to this list in December 1999, the
file name
> is 'meansixthso.scl'. The tempered octave is 1/6 comma (3.58458
cents) sharp
> of the just octave and the tempered fifth is 1/6 comma flat of the
just
> fifth. The major third is exactly 5/4 and the augmented fifth is
exactly
> 25/16.
>
> The other is called 'Top 5&7 limit meantone', file named
> 'smithgw_meantop.scl' (so I assume it's Gene Ward Smith). The
octave is
> stretched by 1.69852 cents, about 1/8 comma. The fifth is flat by
4.39016
> cents.
>
> I know how the first scale was generated, but how was the second?

Hi Danny.

To review, TOP meantone tuning is defined as follows:

Prime 2 gets tempered to

cents(2/1) + cents(81/80)*log(2/1)/log(81*80) = ~1201.7 cents

Prime 3 gets tempered to

cents(3/1) - cents(81/80)*log(3/1)/log(81*80) = ~1899.3 cents

Prime 5 then necessarily gets tempered to

cents(5/1) + cents(81/80)*log(5/1)/log(81*80) = ~2790.3 cents

If you look at all JI ratios and see how they're approximated in
meantone, this form of meantone minimizes the maximum of
(|mistuning|/complexity), where complexity is the Tenney Harmonic
Distance, HD=log2(n*d), of the ratio n/d. Confused? Look at this
table:

Ratio......Tmprmnt...Mistuning..HD=log2(n*d)..|Mistuning|/HD
2/1........1201.70....1.70...........1..............1.70
3/1........1899.26....2.69..........1.58............1.70
4/1........2403.40....3.40...........2..............1.70
5/1........2790.26....3.94..........2.32............1.70
3/2.........697.56....4.39..........2.58............1.70
6/1........3100.96....0.99..........2.58............0.38
8/1........3605.10....5.10...........3..............1.70
9/1........3798.53....5.38..........3.17............1.70
10/1.......3991.96....5.64..........3.32............1.70
4/3.........504.13....6.09..........3.58............1.70
12/1.......4302.66....0.70..........3.58............0.20
5/3.........890.99....6.64..........3.91............1.70
15/1.......4689.52....1.25..........3.91............0.32
16/1.......4806.79....6.79...........4..............1.70
9/2........2596.83....7.08..........4.17............1.70
18/1.......5000.22....3.69..........4.17............0.88
5/4.........386.86....0.55..........4.32............0.13
20/1.......5193.65....7.34..........4.32............1.70
8/3........1705.83....7.79..........4.58............1.70
24/1.......5504.36....2.40..........4.58............0.52
25/1.......5580.52....7.89..........4.64............1.70
6/5.........310.70....4.94..........4.91............1.01
10/3.......2092.69....8.33..........4.91............1.70
30/1.......5891.22....2.95..........4.91............0.60
32/1.......6008.49....8.49...........5..............1.70
36/1.......6201.92....1.99..........5.17............0.38
8/5.........814.84....1.15..........5.32............0.22
40/1.......6395.35....9.04..........5.32............1.70
9/5........1008.27....9.33..........5.49............1.70
45/1.......6588.78....1.44..........5.49............0.26
16/3.......2907.53....9.49..........5.58............1.70
48/1.......6706.06....4.10..........5.58............0.73
25/2.......4378.82....6.19..........5.64............1.10
50/1.......6782.21....9.59..........5.64............1.70
27/2.......4496.09....9.77..........5.75............1.70
54/1.......6899.49....6.38..........5.75............1.11
12/5.......1512.40....3.24..........5.91............0.55
15/4.......2286.12....2.15..........5.91............0.36
20/3.......3294.39...10.03..........5.91............1.70
60/1.......7092.92....4.65..........5.91............0.79
Etc.

As shown in the last column, the maximum value of
(|mistuning|/complexity) for any ratio is 1.70. Any other form of
meantone tuning would have a higher maximum. So TOP meantone is
optimal in at least this one sense.

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >I'm looking for meantone scales in the Scala archive with the
octave
> >stretched so that the tempered fifth is closer to the just fifth.
> //
> >And what other similar scales are there?
>
> Don't forget the 7th-root-of-3/2 tuning, which actually has just
> fifths!
>
> -Carl

That seems far more similar to 12-equal than to meantone.

>> >I'm looking for meantone scales in the Scala archive with the
>> >octave stretched so that the tempered fifth is closer to the
>> >just fifth.
>> //
>> >And what other similar scales are there?
>>
>> Don't forget the 7th-root-of-3/2 tuning, which actually has just
>> fifths!
>
>That seems far more similar to 12-equal than to meantone.

Maybe so; I was reading "meantone" in the most general sense.

-Carl

From: "wallyesterpaulrus" <paul@...>

> Prime 2 gets tempered to
>
> cents(2/1) + cents(81/80)*log(2/1)/log(81*80) = ~1201.7 cents

That can't be right, I got exactly 2400 cents. Or am I misreading the
'cents' function?

--- In tuning@yahoogroups.com, "Danny Wier" <dawiertx@s...> wrote:
> From: "wallyesterpaulrus" <paul@>
>
> > Prime 2 gets tempered to
> >
> > cents(2/1) + cents(81/80)*log(2/1)/log(81*80) = ~1201.7 cents
>
> That can't be right, I got exactly 2400 cents. Or am I misreading
the
> 'cents' function?

Someone else got 2400 cents too. They misread "81*80" as "81/80".
Make sure you're careful about that.