New quasi-meantone tuning - another way of playing East African folk tunes on modern keyboards

Hi all.

In 2004, I made two quasi-meantone F..A# scales which were meant to be tuned
by beat rates. Now I've made a new one. Similarly to the previous two
scales, Its chain of fifths also goes from F to A# and the tuning comes from
beat rates as well. Here is how to tune the scale (by "higher" or "lower" I
mean compared to JI):

1. A4 = 439Hz
2. E4 = Perfect 4th downwards (slightly lower to get -6 bps)
3. D4 = Perfect 5th downwards from A4 (slightly higher to get 4 bps)
4. C4 = Major 3rd downwards from E4 (slightly lower to get -1 bps)
5. G4 = Perfect 5th upwards from C4 (slightly lower to get -1 bps)
6. F4 = Major 3rd downwards from A4 (slightly lower to get -2 bps)
7. B4 = Major 3rd upwards from G4 (slightly higher to get 0.5 bps)
8. F#4 = Perfect 4th downwards from B4 (slightly lower to get -3 bps)
9. C#4 = Minor 3rd downwards from E4 (slightly higher to get 9 bps)
10. G#4 = Minor 3rd downwards from B4 (slightly higher to get 7 bps)
11. D#4 = Minor 6th downwards from B4 (slightly higher to get 1 bps)
12. A#4 = Augmented 6th from C4 (slightly lower than 7/4 to get 1 bps)

As you can see, the preferred chords in this tuning are C major and G major.
Also E major and B major sound quite OK, though not so smoothly. A very
strong sonority appears in the C-E-G-A# tetrad as many of the beat rates
there are equal. This is interesting because playing G-A#-C-D-E-G comes out
close to 6:7:8:9:10:12, which is one of the most popular scales in East
Africa. So it may actually be a good idea to play East African folk melodies
in this tuning (these were promoted mainly by the Master Musicians of
Tanzania and their lead singer Hukwe Zawose at the end of the 1980s).

Now the actual scale follows.

! qmeb3.scl
!
Equal beating quasi-meantone tuning no. 3 - F...A# (1/1 = 262Hz)
12
!
2197/2096
147/131
9819/8384
1311/1048
877/655
5877/4192
785/524
3281/2096
439/262
1833/1048
1963/1048
2/1

Petr

I was incorrect when I said that C4-A#4 did 1 bps. I meant -1 bps, of
course.

On Sun, 8 Jan 2006, Petr Pa��zek wrote:
> In 2004, I made two quasi-meantone F..A# scales which were meant to be
tuned
> by beat rates. Now I've made a new one. Similarly to the previous two
> scales, Its chain of fifths also goes from F to A# and the tuning comes
from
> beat rates as well. Here is how to tune the scale (by "higher" or "lower"
I
> mean compared to JI):
>
> 1. A4 = 439Hz
> 2. E4 = Perfect 4th downwards (slightly lower to get -6 bps)
> 3. D4 = Perfect 5th downwards from A4 (slightly higher to get 4 bps)
> 4. C4 = Major 3rd downwards from E4 (slightly lower to get -1 bps)
> 5. G4 = Perfect 5th upwards from C4 (slightly lower to get -1 bps)
> 6. F4 = Major 3rd downwards from A4 (slightly lower to get -2 bps)
> 7. B4 = Major 3rd upwards from G4 (slightly higher to get 0.5 bps)
> 8. F#4 = Perfect 4th downwards from B4 (slightly lower to get -3 bps)
> 9. C#4 = Minor 3rd downwards from E4 (slightly higher to get 9 bps)
> 10. G#4 = Minor 3rd downwards from B4 (slightly higher to get 7 bps)
> 11. D#4 = Minor 6th downwards from B4 (slightly higher to get 1 bps)
> 12. A#4 = Augmented 6th from C4 (slightly lower than 7/4 to get 1 bps)
>
> As you can see, the preferred chords in this tuning are C major and G
major.
> Also E major and B major sound quite OK, though not so smoothly. A very
> strong sonority appears in the C-E-G-A# tetrad as many of the beat rates
> there are equal. This is interesting because playing G-A#-C-D-E-G comes
out
> close to 6:7:8:9:10:12, which is one of the most popular scales in East
> Africa. So it may actually be a good idea to play East African folk
melodies
> in this tuning (these were promoted mainly by the Master Musicians of
> Tanzania and their lead singer Hukwe Zawose at the end of the 1980s).

Hi Petr,

This is interesting! I played the G-A#-C-D-E-G
scale, improvising a bit, in 12-EDO, using an
accordion patch. The sound I heard was classic
Cajun ...!

The only interval in this I find awkward in 12-EDO
is the tritone (or diminished fourth) A#-E. It
would probably sound better as a 7:10 ratio, but
I haven't done much seven-limit listening of late.

> Now the actual scale follows.

I've calculated the cents equivalents of each note
and the step intervals between them:

1/1 .................... 0.00000 c .........
> 2197/2096 .. 81.47538 c ....... 81.47538 c
> 147/131 ......... 199.49921 c ..... 118.02383 c
> 9819/8384 .. 273.52486 c ... 74.02565 c
> 1311/1048 .... 387.63476 c ... 114.1099 c
> 877/655 ........ 505.29832 c ... 117.66356 c
> 5877/4192 .. 584.92942 c ... 79.6311 c
> 785/524 ........ 699.75101 c .... 114.82159 c
> 3281/2096 .. 775.79625 c ... 76.04524 c
> 439/262 ....... 893.58495 c .... 117.78870 c
> 1833/1048 .. 967.88168 c ..... 74.29673 c
> 1963/1048 .. 1086.50575 c .. 118.62407 c
> 2/1 ................... 1200.00000c .... 113.49425 c

The steps are of two general sizes:
small - from ~74 to ~81.5 c ----- averaging ~ 77 c
large - from ~113.5 to ~118.6 c - averaging ~ 116 c

How different would a piece of slow music
sound, tuned in -
a) Your scale
b) An approximation of it using 77 & 116 c
steps?

Would the answer be the same for fast
music?

Do you have any music samples using this
scale?

Regards,
Yahya

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Hi Yahya.
You wrote:

> This is interesting! I played the G-A#-C-D-E-G
> scale, improvising a bit, in 12-EDO, using an
> accordion patch. The sound I heard was classic
> Cajun ...!

Unfortunately, I'm pretty unaware what Cajun means.

> The only interval in this I find awkward in 12-EDO
> is the tritone (or diminished fourth) A#-E. It
> would probably sound better as a 7:10 ratio, but
> I haven't done much seven-limit listening of late.

BTW: you probably wanted to say "diminished fifth".
Yes, I agree. And not only do I agree. I can prove the point by recommending
you some recordings by the Master Musicians of Tanzania or by Hukwe Zawose
alone. Almost Everything he played and sung was in the simple scale of
6:7:8:9:10:12.

> How different would a piece of slow music
> sound, tuned in -
> a) Your scale
> b) An approximation of it using 77 & 116 c
> steps?

First of all, if you want to find averages, you don't have to start with
5-fifth (i.e. diatonic) or 7-fifth (i.e. chromatic) steps. The best way is
to take the whole chain of fifths and make the average using that. For a
12-tone system, we have to count 11 fifths, which comes out as an augmented
third + 6 octaves. In case of this scale, the augmented third is the F-A#
with a ratio of 9165/7016. If you widen that by 6 octaves, you get
73320/877. Convert this to cents, divide the result by 11, and you get an
average fifth of ~696.598487 cents. This means that if you make a regular
meantone temperament using a fifth of this size, the interval between the
first and the last tone of the chain will be just the same. You can try it
yourself in Scala by typing "Equal 11 73320/877", then "Normalize" and then
"Key 7". In the end, you get something similar to quarter-comma meantone,
which means that:
1. Major thirds are detuned just slightly, while fifths are detuned much
more.
2. This is true for all keys and therefore the "sweet clearness" of the C
major and G major chords is lost.

> Would the answer be the same for fast
> music?

Probably not. The scale behaves somewhat similarly to a meantone temperament
so the melodic properties are hardly distinguishable.

> Do you have any music samples using this
> scale?

Not yet. But I can make some improvisations and send them to you, if you
like. Let me just ask you two brief questions:
1. Can you decode Ogg/Vorbis?
2. Can you decode Rar?

Petr

--- In tuning@yahoogroups.com, Petr Paøízek <p.parizek@c...> wrote:
>
> I was incorrect when I said that C4-A#4 did 1 bps. I meant -1 bps, of
> course.

What practical difference does it make? How could one hear the
distinction?

Hi Paul.
You wrote:

> What practical difference does it make? How could one hear the
> distinction?

It was not meant to show a difference in how it's heard. It was originally
meant just to go along with the other instructions on tuning the scale. If I
say that A# should be tuned slightly lower than a 7/4 from C, it's obvious
that the resulting beating comes out as -1 bps in this particular case. I
just wanted to avoid confusion for the case someone decided to tune this
scale on an acoustic instrument, which may more or less require the beat
rates to match the ones I've given. In such a situation, one needs to know
if they want to get positive or negative beats (or in other words, if the
tone they're tuning should be higher or lower than for a JI interval). It
has nothing to do with actual music performance after the instrument has
been tuned.

Petr

Hi Petr,

On Mon, 9 Jan 2006 Petr Par�zek wrote:
> > This is interesting! I played the G-A#-C-D-E-G
> > scale, improvising a bit, in 12-EDO, using an
> > accordion patch. The sound I heard was classic
> > Cajun ...!
>
> Unfortunately, I'm pretty unaware what Cajun means.

See eg:
http://en.wikipedia.org/wiki/Cajun#Music.2C_including_Zydeco
for a good backgrounder. Then try listening to
some zydeco - it's real party music! ;-)

> > The only interval in this I find awkward in 12-EDO
> > is the tritone (or diminished fourth) A#-E. It
> > would probably sound better as a 7:10 ratio, but
> > I haven't done much seven-limit listening of late.
>
> BTW: you probably wanted to say "diminished fifth".

Yup.

> Yes, I agree. And not only do I agree. I can prove the point by
recommending
> you some recordings by the Master Musicians of Tanzania or by Hukwe Zawose
> alone. Almost Everything he played and sung was in the simple scale of
> 6:7:8:9:10:12.

Do you have a link? Tanzanian music has hardly
ever been heard in Australia; I remember hearing
one song on 3PBS (www.pbsfm.org.au) a couple of
years back.

> > How different would a piece of slow music
> > sound, tuned in -
> > a) Your scale
> > b) An approximation of it using 77 & 116 c
> > steps?
>
> First of all, if you want to find averages, you don't have to start with
> 5-fifth (i.e. diatonic) or 7-fifth (i.e. chromatic) steps. The best way is
> to take the whole chain of fifths and make the average using that. For a
> 12-tone system, we have to count 11 fifths, which comes out as an
augmented
> third + 6 octaves. In case of this scale, the augmented third is the F-A#
> with a ratio of 9165/7016. If you widen that by 6 octaves, you get
> 73320/877. Convert this to cents, divide the result by 11, and you get an
> average fifth of ~696.598487 cents. This means that if you make a regular
> meantone temperament using a fifth of this size, the interval between the
> first and the last tone of the chain will be just the same. You can try it
> yourself in Scala by typing "Equal 11 73320/877", then "Normalize" and
then
> "Key 7". In the end, you get something similar to quarter-comma meantone,
> which means that:
> 1. Major thirds are detuned just slightly, while fifths are detuned much
> more.
> 2. This is true for all keys and therefore the "sweet clearness" of the C
> major and G major chords is lost.

Hmm, I think I understand this procedure. And
presumably one could apply the same procedure
to any tuning that makes an "augmented third" a
specific interval distinct from a "fourth", to arrive
at the average size of a fifth in that tuning.

But that would only give a useful indication of the
nature of the tuning if the fifths do fall very close
to that average, wouldn't it? In particular, for
some of the equal-beating temperaments people
have been discussing in the last month or so, small
deviations from that average would produce
markedly different effects.

> > Would the answer be the same for fast
> > music?
>
> Probably not. The scale behaves somewhat similarly to a meantone
temperament
> so the melodic properties are hardly distinguishable.

Fair enough.

> > Do you have any music samples using this
> > scale?
>
> Not yet. But I can make some improvisations and send them to you, if you
> like. Let me just ask you two brief questions:
> 1. Can you decode Ogg/Vorbis?
> 2. Can you decode Rar?

Yes, and yes! WinAmp plays Ogg Vorbis, and i have WinRAR.

Regards,
Yahya

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Hi Yahya.
You wrote:

> But that would only give a useful indication of the
> nature of the tuning if the fifths do fall very close
> to that average, wouldn't it? In particular, for
> some of the equal-beating temperaments people
> have been discussing in the last month or so, small
> deviations from that average would produce
> markedly different effects.

Sure, I definitely agree with this. And it's a very similar matter if you do
averages of the one-step intervals, isn't it?

> WinAmp plays Ogg Vorbis, and i have WinRAR.

Great. So be ready to find a couple of examples during tomorrow.

Petr