Yahoo Tuning Groups Ultimate Backup tuning A recording of "fifthless" music

Carl wrote:

> Is this the 3-less temperament you discovered last year,
> which is well-found in 87-ET?

Probably not the one you mean because both the 3-less temperaments I've used before are 2D tunings, not 3D. One of them (which I was noticing for the first time, I think, as early as 2006) used the 9th root of 32/5 as the generator. The other one (which I found almost by chance in March 2008) uses the 13th root of 130 as the generator.

> How many tones did the scale you use have, and how was
> it mapped to the keyboard?

To be honest, I didn't care too much about what keyboard mapping came out as long as it could be converted to the Yamaha XG tuning format (i.e. every tone is no more than 63 cents away from 12-equal). So I used a 12-tone version of the temperament, which means that the tempered 13/8 was used only once in the 3D chain and the tempered 13/10 was used 5 times, and I chose the 12 keys in simple ascending order. And it worked.

Petr

🔗Carl Lumma <carl@...>

Hi Petr,

> > Is this the 3-less temperament you discovered last year,
> > which is well-found in 87-ET?
>
> Probably not the one you mean because both the 3-less
> temperaments I've used before are 2D tunings, not 3D.

Though 87-ET, like all ETs, is a 1D temperament, you can
use it as a tuning for 2- and 3D temperaments (much like
31-ET is a tuning of meantone).

> One of them (which I was noticing for the first time, I think,
> as early as 2006) used the 9th root of 32/5 as the generator.
> The other one (which I found almost by chance in March 2008)
> uses the 13th root of 130 as the generator.

How do you know they're not the same temperament?

The latter generator is 648.22 cents, which is ~ 16/11.
Usually we give the smallest period-inversion of the
generator, so we'd report this generator as 11/8. 87-ET
has an interval of 551.72 cents, which is even closer to
what you gave than 11/8. March 2008 sounds right.

The former generator is 357.08 cents, and 87-ET has 358.62,
which is less than a cent away (the expected error for an
arbitrary interval in 87-ET is about 4.6 cents).

When telling us about your music, I'd love it if you
reported the mapping as well as the generator size(s).
The March '08 fifthless (and 7-less) temperament deserves
a name in my opinion. The 1D version has
val <87 202 301 322], and came up as one of the very best
temperaments when I surveyed nontraditional JI limits.

You can add fifths to it without too much trouble, as
87-ET has excellent fifths. And it looks like you
get 37-ET if you add 7s instead. If we try to make it
a complete 13-limit temperament, perhaps its the 37&87
2D temperament. A quick trip to Graham's site...

http://x31eq.com/cgi-bin/temperament.cgi?et1=37&et2=87&limit=13

does give the 551.8 cent generator. And the following
mapping

2 3 5 7 11 13
<1 14 6 12 3 6] 2/1
<0 -27 -8 -20 1 -5] 11/8

Anybody care to wedge this for me?

> > How many tones did the scale you use have, and how was
> > it mapped to the keyboard?
>
> To be honest, I didn't care too much about what keyboard
> mapping came out as long as it could be converted to the
> Yamaha XG tuning format (i.e. every tone is no more than
> 63 cents away from 12-equal). So I used a 12-tone version
> of the temperament, which means that the tempered 13/8 was
> used only once in the 3D chain and the tempered 13/10 was
> used 5 times, and I chose the 12 keys in simple ascending
> order. And it worked.

Are most of your keyboard mappings 12-tone ones? You have
an uncanny knack for finding your way around extended
temperaments on the keyboard. Tell us your secret! Do
you mark keys with tape, work out fingerings for chords
on paper, do it all in your head? Or do you just
improvise without trying to find particular chords?

Thanks,

-Carl

🔗Herman Miller <hmiller@...>

🔗Carl Lumma <carl@...>

🔗Petr Pařízek <p.parizek@...>

Carl wrote:

> Though 87-ET, like all ETs, is a 1D temperament, you can
> use it as a tuning for 2- and 3D temperaments (much like
> 31-ET is a tuning of meantone).

Sure.

> How do you know they're not the same temperament?

I'm not sure if I know what you mean. If one uses a semi-augmented fourth (or a semi-diminished fifth) and the other uses a neutral third as a generator, then the only possible situation where they could both temper out the same intervals migh be when the actual generator in one temperament splits the generator of the other into an integer number of parts (like othra or squares which both contain meantone or like superpelog which contains mavila). For one thing, the 2006 tuning doesn't do 11s while the 2008 tuning doesn't do 7s. For another thing, 5s are mapped to -8 generators in the one from 2008 (if I use a generator of a tempered 11/8 instead of 16/11) and -9 generators in the one from 2006.

> When telling us about your music, I'd love it if you
> reported the mapping as well as the generator size(s).

Okay, even though I seem to only rarely come up with something new to the TL members (which I realized after rediscovering semisixths, hanson, hedgehog, tetracot, orwell, and rodan), this is what I can say about the three temperaments we're discussing, in order of used prime limits, mappings, and selected generator sizes:
<<<<<<
2.5.7.13:
[(1, 0), (5, -9), (4, -4), (4, -1)]
357.076254
>>>>>>
<<<<<<
2.5.11.13:
[(1, 0), (6, -8), (3, 1), (6, -5)]
551.781433
>>>>>>
<<<<<<
2.5.11.13:
[(1, 0, 0), (3, -1, -1), (3, -1, 2), (4, -1, 0)]
358.999861, 455.158902
>>>>>>
Although I prefer to describe the 2008 temperament using a tempered 16/11 rather than 11/8 because then you need a smaller number of periods for the important approximations.

> 87-ET has excellent fifths. And it looks like you
> get 37-ET if you add 7s instead.

In my mailbox on my laptop, there's a message entitled "A case of 37-EDO" which I was apparently posting to the TL on the May 23rd, 2006. There I was talking about exactly these interesting properties of 37-equal.

> Are most of your keyboard mappings 12-tone ones?

Yes, most of them are, just because I want to have a chance to play the scales live, to have them at my finger tips. The only cases when I can't play them "live" are scales of more than 12 tones, which means I have to tune different MIDI channels to different parts of the scale. But I usually try, in situations like these, to get as much as I can on a single MIDI channel (like making a 16-tone scale by using 12 tones on one channel and 4 tones on another instead of, let's say, 8 and 8), which allows me to record a substantial portion of the chords on one track and then only add a few more notes on another. For example, the pieces I called "Run Down The Whistle 3" and "Still Differently" are both in 5-limit semisixths. But when I wanted to include a comma pump there, I realized I had to use at least a 16-tone set to do that. And I can tell you, it was a pretty difficult thing to play, in both cases. But because I knew the exact pitches I was looking for, at first, I was singing or whistling the tones which weren't in the 12-tone set to fill in the "gaps" in my mind, and after recording a few measures of the passage, I added the second track where I used only 4 tones so it wasn't so difficult for me to find them. In this particular case, what I did was take a 12-tone set of semisixths and tuned the second MIDI channel to the inversion of the first one. You can imagine something similar by, for example, getting a meantone chain from Bbb to D on one channel and from D to F## on another, with the difference that this was semisixths and not meantone but the method is the same. So there was one tpitch in common on both channels. Although this allowed me to use as much as 23 tones in the scale, I used only 16. And the one tone which had the same pitch on both channels helped me not only to find the necessary pitches while recording the second track, but also to appropriately tune the scale even with the limitations of the +/-63-cent maximum; I'll tell you something, I was using pitch-bends to have the same pitch on the "central" key while allowing different tuning offsets for the key on each channel, which meant, of course, that I couldn't use pitch-bends for melodic ornamentation. But when I was experimenting with this in 2002, I solved the problem by setting the channel aftertouch parameter to change the pitch.

> You have an uncanny knack for finding your way around extended
> temperaments on the keyboard. Tell us your secret! Do
> you mark keys with tape, work out fingerings for chords
> on paper, do it all in your head? Or do you just
> improvise without trying to find particular chords?

Wow, thanks: :-) -- Well, for many temperaments, I carefully read the interval sizes when making them in Scala, which helps me a lot to know which key maps to which interval. And then, what follows is just about 5 to 10 minutes of practicing, and I get it.

Petr

🔗Carl Lumma <carl@...>

Petr wrote:

> Although I prefer to describe the 2008 temperament using a
> tempered 16/11 rather than 11/8 because then you need a
> smaller number of periods for the important approximations.

Can't we just say we will remember to apply the 11/8 generator
downward from the unison?

Care to have a stab at naming this temperament?

> > Are most of your keyboard mappings 12-tone ones?
>
> Yes, most of them are, just because I want to have a chance
> to play the scales live, to have them at my finger tips.

Some folks map > 12 tones to the keyboard and just play
chords with two hands. This approach has always been
intimidating to me, but others seem to do OK.

> The only cases when I can't play them "live" are scales of
> more than 12 tones, which means I have to tune different
> MIDI channels to different parts of the scale.

What synth are you using? Many synths support full 128-note
retuning (per channel).

> For example, the pieces I called "Run Down The
> Whistle 3" and "Still Differently" are both in 5-limit
> semisixths. But when I wanted to include a comma pump there,
> I realized I had to use at least a 16-tone set to do that.
> And I can tell you, it was a pretty difficult thing to play,
> in both cases. But because I knew the exact pitches I was
> looking for, at first, I was singing or whistling the tones
> which weren't in the 12-tone set to fill in the "gaps" in my
> mind, and after recording a few measures of the passage,
> I added the second track where I used only 4 tones so it
> wasn't so difficult for me to find them. In this particular
> case, what I did was take a 12-tone set of semisixths and
> tuned the second MIDI channel to the inversion of the
> first one. You can imagine something similar by, for example,
> getting a meantone chain from Bbb to D on one channel and
> from D to F## on another, with the difference that this was
> semisixths and not meantone but the method is the same.
> So there was one tpitch in common on both channels. Although
> this allowed me to use as much as 23 tones in the scale,
> I used only 16. And the one tone which had the same pitch
> on both channels helped me not only to find the necessary
> pitches while recording the second track, but also to
> appropriately tune the scale even with the limitations of
> the +/-63-cent maximum; I'll tell you something, I was using
> pitch-bends to have the same pitch on the "central" key while
> allowing different tuning offsets for the key on each channel,
> which meant, of course, that I couldn't use pitch-bends for
> melodic ornamentation. But when I was experimenting with this
> in 2002, I solved the problem by setting the channel
> aftertouch parameter to change the pitch.

Thanks for sharing your process! Having notes in common
when switching between keyboards (or retuning a single
keyboard) certainly does make things easier.

Have you ever considered making a page for comma pumps?
I think exotic comma pumps are one of the best ways to
illustrate the point of exotic temperaments. They're also
directly connected to the number theory behind "good"
temperaments. I would love to see a page from you giving
a brief explanation of your pump-finding algorithm, with a
permanent home for some audio examples your favorite pumps.
It would be ideal to get simple 3-times through progressions
to illustrate the pump in each case... if you think it's
a good idea let me know if I can help (web hosting, etc).

-Carl

🔗Petr Parízek <p.parizek@...>

Carl wrote:

> Can't we just say we will remember to apply the 11/8 generator
> downward from the unison?

That wasn't my point. I meant that if the generator is a tempered 11/8, then the 5/1 is mapped to "(6, -8)", while with a tempered 16/11, it's "(-2, 8)". Similarly, with a tempered 11/8, 13/1 maps to "(7, -5)", while with a tempered 16/11, it's "(1, 5)". This is, among others, one of the reasons why I prefer to say that the meantone generator is a tempered fifth rather than a fourth, because then you don't need any periods at all for mapping the 5/1.

> Care to have a stab at naming this temperament?

I was thinking about something like "emka", which is a "hidden" description of the approximated primes in terms of falling generator numbers -- the highest number of generators is needed for mapping 5 ("e" is the 5th letter in the alphabet), then less generators are needed for 13 ("m" is the 13th letter), only one generator makes 11 ("k" is the 11th), and the "a" (or the #1) just says that it's my first 2D tuning which intentionally approximates the primes of 5 and 11.

> Some folks map > 12 tones to the keyboard and just play
> chords with two hands. This approach has always been
> intimidating to me, but others seem to do OK.

Unfortunately, the Yamaha XG instruments don't allow this because you can only retune 12 pitches and it affects all the octaves so you can't say "C4 - 50 cents, C5 + 40 cents".

> What synth are you using? Many synths support full 128-note
> retuning (per channel).

It's a Yamaha QS-300. I'm afraid that the MTS is a newer invention than all of the XG-compatible instruments I've ever played, so I'm not surprised that they don't support MTS.

> Have you ever considered making a page for comma pumps?
> I think exotic comma pumps are one of the best ways to
> illustrate the point of exotic temperaments. They're also
> directly connected to the number theory behind "good"
> temperaments. I would love to see a page from you giving
> a brief explanation of your pump-finding algorithm, with a
> permanent home for some audio examples your favorite pumps.

I'm working on something like that so I can let you know when I have it ready.

> It would be ideal to get simple 3-times through progressions
> to illustrate the pump in each case...

Not sure what you mean.

> if you think it's
> a good idea let me know if I can help (web hosting, etc).

It certainly is. And I'll definitely do something about it.

Petr

🔗Daniel Forró <dan.for@...>

🔗Carl Lumma <carl@...>

--- In tuning@yahoogroups.com, Petr Parízek <p.parizek@...> wrote:

> > What synth are you using? Many synths support full 128-note
> > retuning (per channel).
>
> It's a Yamaha QS-300. I'm afraid that the MTS is a newer
> invention than all of the XG-compatible instruments I've ever
> played, so I'm not surprised that they don't support MTS.

Not possible, since MTS predates the XG spec itself.
But point taken. Have you considered using your QS-300
as a MIDI controller for softsynths running on your PC?
Then you could have > 12 notes/octave. You could also
do things like splitting the keyboard and putting two
slightly different 12-tone scales in the same register
on each half.

> > Have you ever considered making a page for comma pumps?
> > I think exotic comma pumps are one of the best ways to
> > illustrate the point of exotic temperaments. They're also
> > directly connected to the number theory behind "good"
> > temperaments. I would love to see a page from you giving
> > a brief explanation of your pump-finding algorithm, with a
> > permanent home for some audio examples your favorite pumps.
>
> I'm working on something like that so I can let you know when
> I have it ready.

Fantastic! Probably you have your own plans, but just to
share what I was thinking, is to have a bare progression for
each -- possibly a more musical version also, but always at
least a plain version -- so that other musicians could hear
the raw 'impossibility', quickly go and try to play the same
chords on the piano and see the difference, and to compare
pumps without extra melodic material in the picture.

> > It would be ideal to get simple 3-times through progressions
> > to illustrate the pump in each case...
>
> Not sure what you mean.

Playing each pump three times through, so that examples
are consistent with one another.

-Carl