Canton

This one is strictly proper, but that's Ohio for you.

! canton.scl
A 2.3.11/7.13/7 subgroup scale
12
!
14/13
9/8
13/11
14/11
4/3
39/28
3/2
11/7
22/13
16/9
13/7
2/1

! cantonpenta.scl
Canton scale in 13-limit pentacircle (351/350 and 364/363) temperament, 271et
12
!
128.41328
208.11808
287.82288
416.23616
495.94096
575.64576
704.05904
783.76384
912.17712
991.88192
1071.58672
1200.00000

Gene,

At some point I'm just going to push all of these tunings you've been
producing up to my website.
I think we could use a repository for composers to peruse and find ones they
like.

If you have any suggestions on how that should be set up I'd like to hear
them. Only caveat is that I'm not going to make zips since it makes it too
much of a problem for me to update - at least right now with the rate you
are producing tunings.

Perhaps in the future I'll do that.

Chris

On Thu, Mar 3, 2011 at 5:23 PM, genewardsmith
<genewardsmith@...>wrote:

>
>
> This one is strictly proper, but that's Ohio for you.
>
> ! canton.scl
> A 2.3.11/7.13/7 subgroup scale
> 12
> !
> 14/13
> 9/8
> 13/11
> 14/11
> 4/3
> 39/28
> 3/2
> 11/7
> 22/13
> 16/9
> 13/7
> 2/1
>
> ! cantonpenta.scl
> Canton scale in 13-limit pentacircle (351/350 and 364/363) temperament,
> 271et
> 12
> !
> 128.41328
> 208.11808
> 287.82288
> 416.23616
> 495.94096
> 575.64576
> 704.05904
> 783.76384
> 912.17712
> 991.88192
> 1071.58672
> 1200.00000
>
>
>

--- In tuning@yahoogroups.com, Chris Vaisvil <chrisvaisvil@...> wrote:
>
> Gene,
>
> At some point I'm just going to push all of these tunings you've been
> producing up to my website.

Sounds good, however I've been sticking them on the "Gallery of 12-tone Just Intonation" on the Xenwiki, so they have not been wandering without a home. Andrew Heathwaite suggested I add to this gallery which he had started and that has led to this orgy of 12-note JI scale creation. Since I'd already started my own orgy of 12-note tempered scale creation, it fit right in.

I wonder if your site and the Xenwiki could be brought into closer coordination somehow.

Here's a likely-to-be-dumb question.

Why give a version of this scale in 271et? Each step is 4.43 cents,
which is (nearly?) undetectable. 271 seems arbitrary to me (though I
may just not understand the reason for choosing it). You're

On 3/3/11, genewardsmith <genewardsmith@...> wrote:
> This one is strictly proper, but that's Ohio for you.
>
> ! canton.scl
> A 2.3.11/7.13/7 subgroup scale
> 12
> !
> 14/13
> 9/8
> 13/11
> 14/11
> 4/3
> 39/28
> 3/2
> 11/7
> 22/13
> 16/9
> 13/7
> 2/1
>
> ! cantonpenta.scl
> Canton scale in 13-limit pentacircle (351/350 and 364/363) temperament,
> 271et
> 12
> !
> 128.41328
> 208.11808
> 287.82288
> 416.23616
> 495.94096
> 575.64576
> 704.05904
> 783.76384
> 912.17712
> 991.88192
> 1071.58672
> 1200.00000
>
>
>
>
>
>
> ------------------------------------
>
> You can configure your subscription by sending an empty email to one
> of these addresses (from the address at which you receive the list):
> tuning-subscribe@yahoogroups.com - join the tuning group.
> tuning-unsubscribe@yahoogroups.com - leave the group.
> tuning-nomail@yahoogroups.com - turn off mail from the group.
> tuning-digest@yahoogroups.com - set group to send daily digests.
> tuning-normal@yahoogroups.com - set group to send individual emails.
> tuning-help@yahoogroups.com - receive general help information.
> Yahoo! Groups Links
>
>
>
>

Sorry, slipped on the "send" button.

Gene,

Here's a likely-to-be-dumb question.

Why give a version of this scale as an extract from 271 ET?

Each step is 4.43 cents, which is (nearly?) undetectable. 271 seems
arbitrary to me (though I may just not understand the reason for
choosing it). You're still giving very precise numbers in cents, so
you're not saving effort by reducing precision.

What is 271 ET tempering, and why choose that temperament instead of
the almost-equivalent-sounding 272 ET, say? Or the more precise 542
ET?

I'm just trying to understand what I'm seeing.

Thanks,
Jake

On 3/3/11, Jake Freivald <jdfreivald@...> wrote:
> Here's a likely-to-be-dumb question.
>
> Why give a version of this scale in 271et? Each step is 4.43 cents,
> which is (nearly?) undetectable. 271 seems arbitrary to me (though I
> may just not understand the reason for choosing it). You're
>
> On 3/3/11, genewardsmith <genewardsmith@...> wrote:
>> This one is strictly proper, but that's Ohio for you.
>>
>> ! canton.scl
>> A 2.3.11/7.13/7 subgroup scale
>> 12
>> !
>> 14/13
>> 9/8
>> 13/11
>> 14/11
>> 4/3
>> 39/28
>> 3/2
>> 11/7
>> 22/13
>> 16/9
>> 13/7
>> 2/1
>>
>> ! cantonpenta.scl
>> Canton scale in 13-limit pentacircle (351/350 and 364/363) temperament,
>> 271et
>> 12
>> !
>> 128.41328
>> 208.11808
>> 287.82288
>> 416.23616
>> 495.94096
>> 575.64576
>> 704.05904
>> 783.76384
>> 912.17712
>> 991.88192
>> 1071.58672
>> 1200.00000
>>
>>
>>
>>
>>
>>
>> ------------------------------------
>>
>> You can configure your subscription by sending an empty email to one
>> of these addresses (from the address at which you receive the list):
>> tuning-subscribe@yahoogroups.com - join the tuning group.
>> tuning-unsubscribe@yahoogroups.com - leave the group.
>> tuning-nomail@yahoogroups.com - turn off mail from the group.
>> tuning-digest@yahoogroups.com - set group to send daily digests.
>> tuning-normal@yahoogroups.com - set group to send individual emails.
>> tuning-help@yahoogroups.com - receive general help information.
>> Yahoo! Groups Links
>>
>>
>>
>>
>

--- In tuning@yahoogroups.com, Jake Freivald <jdfreivald@...> wrote:

> What is 271 ET tempering, and why choose that temperament instead of
> the almost-equivalent-sounding 272 ET, say? Or the more precise 542
> ET?

First off, I want to temper out the commas 351/350 and 364/383, and 271 does this. But it's not the only choice; I could have used, say, 208 instead. However, I'm also working in the just intonation subgroup generated by 2, 3, 11/7 and 13/7, and I want to temper those ratios specifically. If for some integer N I round off to the nearest integer the quantities N*log2(3), N*log2(11/7) and N*log2(13/7) I get a mapping, and if I require 351/350 and 364/363 to be tempered out, I get a finite number of possible mappings. Choosing the best one by the "Tenney-Euclidean" error measure picks out 271 as the optimal choice. It may not make much difference, but it gives me a quick way of finding a good equal temperament tuning. I like those because they are in some circumstances convenient--say, for instance, I wanted to reconstruct an exact specification for the scale, or I want to find related temperaments, or I am working on a score in this temperament, etc.

Well, having just looked at the wiki you have loads of great information up
there. However, I've been collecting more than just your 12 note tunings,
and more than just yours. (Often, say when that first cup of coffee is
kicking in) I don't read much and just cut n paste tunings to my folder.
However, just a slew of .scl files isn't nearly as valuable as the work you
have been putting in the xenwiki. If any of the following files are of
interest please tell me. Otherwise unless you can think of something my site
can add my best contribution is probably coming up with example pieces.

Here is the list with the obvious stuff removed (Lucy tuning and the scala
archive)

Folder PATH listing for volume RAID 1
Volume serial number is 0024F72C F0AC:E39E
E:.
³ 12edo.scl
³ 12root618phi.scl
³ 12throotofphi12.scl
³ 12throotofphi168.scl
³ 12to30harm12.scl
³ 12to30harmonic18.scl
³ 12to30subharm12.scl
³ 13edo.scl
³ 14edo.scl
³ 15edo.scl
³ 17.gly
³ 17et.scl
³ 1800cents-in12.scl
³ 18ET.scl
³ 18rat.scl
³ 19ET.scl
³ 1overphi.scl
³ 1overphi2fix.tun
³ 1overphiv2-fixed.scl
³ 1overphiv2.scl
³ 1oversilver.scl
³ 20100513scales.txt
³ 22et.mid
³ 22ET.scl
³ 23ET.scl
³ 27edo.scl
³ 27et-.mid
³ 36edo.scl
³ 36hairs2.scl
³ 3octaves-in-12.scl
³ 41near-EDO.scl
³ 6oct88.scl
³ 7th-heaven-john-os.scl
³ 98harmonics-5skip1.scl
³ AbsynthLucyTunings.zip
³ allscala.txt
³ alpha-cjv.scl
³ archytas12.scl
³ bad-john.scl
³ bad.scl
³ bali-balaeb_14.scl
³ bamm20.scl
³ bamm20b.scl
³ bamm24b.scl
³ beatles17.scl
³ blackwood-major.scl
³ blackwood-minor.scl
³ blue-ji-improv.mid
³ blue-ji.scl
³ Blue-temper.scl
³ bohpier25.scl
³ bp-in25.scl
³ BP13.scl
³ cal46-mod.scl
³ caleb46.scl
³ CENTAUR A 7-CAP TUNING.scl
³ centmarv.scl
³ chris-zither-1.scl
³ chris-zither-2.scl
³ cjv-and-other-scales.rar
³ cjv-harm-01.scl
³ ComplexityPlot.xls
³ cps-to-23.scl
³ dimensionsquared.scl
³ double-gamma.scl
³ duodenelike-ji.scl
³ dwarf17marveq.scl
³ ErlangenMonochord.scl
³ fts-blue-temp.scl
³ fts-blue-temp.tun
³ genes-decatonic.scl
³ golden new2.scl
³ goldenfixed.scl
³ harm16-41-skip2.scl
³ harm3-26-skip2.mid
³ harm3-26skip2reduce2oct.scl
³ harm3-27skip2reduce2oct.scl
³ harm30-54by2.scl
³ harm39-127in1.scl
³ harmonic-4to16.scl
³ harmonic-scale.scl
³ harmonic16-28.scl
³ hemifamity27.scl
³ hexany357more.scl
³ john20110212.scl
³ jove41.scl
³ jubilismic10.scl
³ julius22.scl
³ julius24.scl
³ justchromatic.scl
³ kalle-aho-12-rwt.scl
³ love-in-your-heart.mid
³ Lucy4Cam5000.rar
³ LucyMicrotuningsForLogic.zip
³ magic16septimage.scl
³ magic16terzbirat.scl
³ mavila[7] in 30-ET for C Major keyboard mapping.scl
³ mean25rat.scl
³ mike-cameron-tuning.scl
³ mir10.scl
³ mir11.scl
³ mir12-a.mid
³ mir12.scl
³ mir9.scl
³ myna15br25.scl
³ myna15br3.scl
³ New Text Document.txt
³ newts.scl
³ O3-bamm24b_E.scl
³ O3-wilsonistic_pivot_C-ji.scl
³ O3-wilsonistic_pivot_C.scl
³ octacot27.scl
³ octasquare.mid
³ octasquare25.scl
³ orwell12_po.scl
³ orwell13eb.scl
³ orwell13eb.tun
³ otherworld-1.scl
³ oz-ultimate.scl
³ oz-uwt-scala.scl
³ pan-temperament-maybe.scl
³ PentatonicMOS.txt
³ petr-golden.scl
³ Phi-JI.scl
³ Play along.wav
³ pluto17.scl
³ portent46.scl
³ prelude_in_shur-ji.scl
³ quart.scl
³ quarter-mod-12.scl
³ ra1.scl
³ ra2.scl
³ README.rtf
³ real-other-world.scl
³ SCALA Scales.lsc
³ scala.ini
³ ScalaVista-1r0x2.exe
³ Secor5_23TX.scl
³ secor_19p3.scl
³ semimarvelousdrawf.mid
³ semimavdrawf2.mid
³ sensi19br1.scl
³ sethares-neutral10.scl
³ sethares-neutral10in12.scl
³ soria17_plus2a.scl
³ soria17_plus2b.scl
³ sorog9.scl
³ spectral20090105.scl
³ squares8.scl
³ TayyipErdogan.scl
³ thrush15.scl
³ Turko-Arabic_Hijaz-Humayun-Zirgule_on_D.scl
³ ugly-by-john.scl
³ UWT.scl
³ valamute.scl
³ wilson_chromatic10.scl
³ wilson_chromatic8.scl
³ zeus22.scl
³ zeus24.scl
³
ÃÄÄÄ24of41.wav
³ 24of41.mid
³ 24of41.wav
³
³
ÃÄÄÄGene
³ ³ canton.scl
³ ³ cantonpenta.scl
³ ³ dock0.mid
³ ³ dock3.mid
³ ³ dock6.mid
³ ³ dock9.mid
³ ³ EC2lesfip.scl
³ ³ EClesfip.scl
³ ³ hanson11.scl
³ ³ hemifamity27.scl
³ ³ jbf0.mid
³ ³ jbf3.mid
³ ³ jbf6.mid
³ ³ jbf9.mid
³ ³ locomotive.scl
³ ³ magni132.mid
³ ³ magni133.mid
³ ³ nakika12.scl
³ ³ New Text Document.txt
³ ³ omaha.scl
³ ³ parizekmic14.scl
³ ³ parizekmic9.scl
³ ³ portsmouth.scl
³ ³ schoe19a-bihexany.mid
³ ³ schoe19a.mid
³ ³ semafip.scl
³ ³ tamwind.mid
³ ³ unimajor.scl
³ ³ unimajorpenta.scl
³ ³ unimarv19.scl
³ ³
³ ÃÄÄÄconvex
³ ³ diaconv1029.scl
³ ³ diaconv225.scl
³ ³ diaconv2401.scl
³ ³
³ ÃÄÄÄdwarfs
³ ³ dwarf11marv.scl
³ ³ dwarf12marv.scl
³ ³ dwarf13marv.scl
³ ³ dwarf14marv.scl
³ ³ dwarf15marv.scl
³ ³ dwarf15_7_1.jpg
³ ³ dwarf15_9_2.jpg
³ ³ dwarf16marv.scl
³ ³ dwarf17marv.scl
³ ³ dwarf18marv.scl
³ ³ dwarf19marv.scl
³ ³ dwarf20marv.scl
³ ³ dwarf21marv.scl
³ ³ dwarf22marv.scl
³ ³ dwarf25marv.scl
³ ³ dwarflist.txt
³ ³
³ ÃÄÄÄfokker10
³ ³ blackchrome1.scl
³ ³ blackchrome2.scl
³ ³ diablack.scl
³ ³ diachrome1.scl
³ ³ diachrome2.scl
³ ³ diachrome3.scl
³ ³ diachrome4.scl
³ ³ diachrome5.scl
³ ³
³ ÃÄÄÄfokker12
³ ³ centr.scl
³ ³ classr.scl
³ ³ diadie1.scl
³ ³ diadie2.scl
³ ³ diadieorw1.scl
³ ³ diadieorw2.scl
³ ³ diamist123.scl
³ ³ diamisty1.scl
³ ³ diamisty2.scl
³ ³ duodene.scl
³ ³ duodene_skew.scl
³ ³ duowell.scl
³ ³ hahnmaxr.scl
³ ³ indian_12.scl
³ ³ majraj1.scl
³ ³ majraj2.scl
³ ³ majraj3.scl
³ ³ majsyn1.scl
³ ³ majsyn2.scl
³ ³ majsyn3.scl
³ ³ mdar.scl
³ ³ mistyschism1.scl
³ ³ mistyschism2.scl
³ ³ mistyschism3.scl
³ ³ mistyschism4.scl
³ ³ pel1.scl
³ ³ pel1v.scl
³ ³ pel2.scl
³ ³ pel2v.scl
³ ³ pel3.scl
³ ³ pel3v.scl
³ ³ prismr.scl
³ ³ ragisyn1.scl
³ ³ ragisyn10.scl
³ ³ ragisyn11.scl
³ ³ ragisyn12.scl
³ ³ ragisyn2.scl
³ ³ ragisyn3.scl
³ ³ ragisyn4.scl
³ ³ ragisyn5.scl
³ ³ ragisyn6.scl
³ ³ ragisyn7.scl
³ ³ ragisyn8.scl
³ ³ ragisyn9.scl
³ ³ ramis.scl
³ ³ schisdia1.scl
³ ³ schisdia2.scl
³ ³ schisdia3.scl
³ ³ schisdia4.scl
³ ³ schisdia5.scl
³ ³ schisdia6.scl
³ ³ syndia1.scl
³ ³ syndia2.scl
³ ³ syndia3.scl
³ ³ syndia4.scl
³ ³ syndia5.scl
³ ³ syndia6.scl
³ ³ syndie1.scl
³ ³ syndie2.scl
³ ³ syndie3.scl
³ ³ syndie4.scl
³ ³ syndwell1.scl
³ ³ syndwell2.scl
³ ³ syndwell3.scl
³ ³ syndwell4.scl
³ ³ tamil_vi.scl
³ ³ tertiadia1.scl
³ ³ tertiadia2.scl
³ ³ tertiadia3.scl
³ ³ tertiadia4.scl
³ ³ tertiadia5.scl
³ ³ tertiadia6.scl
³ ³ tertiadie1.scl
³ ³ tertiadie2.scl
³ ³ tertiadie3.scl
³ ³ tertiadie4.scl
³ ³ thirds.scl
³ ³
³ ÃÄÄÄfokker14
³ ³ ddimlim1.scl
³ ³ ddimlim10.scl
³ ³ ddimlim11.scl
³ ³ ddimlim12.scl
³ ³ ddimlim13.scl
³ ³ ddimlim14.scl
³ ³ ddimlim2.scl
³ ³ ddimlim3.scl
³ ³ ddimlim4.scl
³ ³ ddimlim5.scl
³ ³ ddimlim6.scl
³ ³ ddimlim7.scl
³ ³ ddimlim8.scl
³ ³ ddimlim9.scl
³ ³ diaddim1.scl
³ ³ diaddim2.scl
³ ³ diaddim3.scl
³ ³ diaddim4.scl
³ ³ diaddim5.scl
³ ³ diaddim6.scl
³ ³ diaddim7.scl
³ ³ dialim1.scl
³ ³ dialim2.scl
³ ³ dialim3.scl
³ ³ dialim4.scl
³ ³ dialim5.scl
³ ³ dialim6.scl
³ ³ dialim7.scl
³ ³
³ ÃÄÄÄfokker5
³ ³ semilim1.scl
³ ³ semilim2.scl
³ ³ semilim3.scl
³ ³ semilim4.scl
³ ³ semilim5.scl
³ ³ semisyn1.scl
³ ³ semisyn2.scl
³ ³ semisyn3.scl
³ ³ semisyn4.scl
³ ³ semisyn5.scl
³ ³ synlim1.scl
³ ³ synlim2.scl
³ ³ synlim3.scl
³ ³ synlim4.scl
³ ³ synlim5.scl
³ ³
³ ÃÄÄÄfokker7
³ ³ mavchrome1.scl
³ ³ mavchrome2.scl
³ ³ mavchrome3.scl
³ ³ mavchrome4.scl
³ ³ mavchrome5.scl
³ ³ mavchrome6.scl
³ ³ mavchrome7.scl
³ ³ porchrome1.scl
³ ³ porchrome2.scl
³ ³ porchrome3.scl
³ ³ porchrome4.scl
³ ³ porchrome5.scl
³ ³ porchrome6.scl
³ ³ porchrome7.scl
³ ³ synchrome1.scl
³ ³ synchrome2.scl
³ ³ synchrome3.scl
³ ³ synchrome4.scl
³ ³ synchrome5.scl
³ ³ synchrome6.scl
³ ³ synchrome7.scl
³ ³ synmav1.scl
³ ³ synmav2.scl
³ ³ synmav3.scl
³ ³ synmav4.scl
³ ³ synmav5.scl
³ ³ synmav6.scl
³ ³ synmav7.scl
³ ³ synpor1.scl
³ ³ synpor2.scl
³ ³ synpor3.scl
³ ³ synpor4.scl
³ ³ synpor5.scl
³ ³ synpor6.scl
³ ³ synpor7.scl
³ ³
³ ÃÄÄÄfokker8
³ ³ semimaj1.scl
³ ³ semimaj2.scl
³ ³ semipor1.scl
³ ³ semipor2.scl
³ ³ semipor3.scl
³ ³ semipor4.scl
³ ³ semipor5.scl
³ ³ semipor6.scl
³ ³ semipor7.scl
³ ³ semipor8.scl
³ ³
³ ÃÄÄÄfokker9
³ ³ dielim1.scl
³ ³ dielim2.scl
³ ³ dielim3.scl
³ ³ mavdie1.scl
³ ³ mavdie2.scl
³ ³ mavdie3.scl
³ ³ mavlim1.scl
³ ³ mavlim2.scl
³ ³ mavlim3.scl
³ ³ mavlim4.scl
³ ³ mavlim5.scl
³ ³ mavlim6.scl
³ ³ mavlim7.scl
³ ³ mavlim8.scl
³ ³ mavlim9.scl
³ ³
³ ÃÄÄÄhobbits
³ ³ akea46_13.scl
³ ³ archytas12.scl
³ ³ archytas12sync.scl
³ ³ ares12.scl
³ ³ ares12opt.scl
³ ³ didymus19sync.scl
³ ³ indra31.scl
³ ³ jove41.scl
³ ³ jubilismic10.scl
³ ³ madagascar19.scl
³ ³ minvera12.scl
³ ³ nakika12.scl
³ ³ New Text Document.txt
³ ³ pele29.scl
³ ³ portent26.scl
³ ³ prodigy12.scl
³ ³ thrush12.scl
³ ³ thrush15.scl
³ ³ unimarv22.scl
³ ³
³ ÃÄÄÄmarvell
³ ³ marvell13.scl
³ ³ marvell15.scl
³ ³ marvell19 - Copy.scl
³ ³ marvell19.scl
³ ³ marvell21.scl
³ ³ marvell31.scl
³ ³ marvell9.scl
³ ³
³ ÀÄÄÄproper-pumps
³ prop12_8.scl
³ prop19_10.scl
³ prop19_9a.scl
³ prop19_9b.scl
³ propw12_8a.scl
³ propw12_8b.scl
³ propw12_8d.scl
³ pump1.scl
³ pump10.scl
³ pump2.scl
³ pump3.scl
³ pump4.scl
³ pump5.scl
³ pump6.scl
³ pump7.scl
³ pump8.scl
³ pump9.scl
³
ÃÄÄÄJacques
³ balasept-above.scl
³ balasept-under.scl
³ bala_ribbon.scl
³ bala_ribbon19.scl
³ bala_ribbon24.scl
³ bala_semifo.scl
³ neutr_pent2.scl
³ New Text Document.txt
³ quadraparizekmic39.scl
³

³
ÃÄÄÄmargo
³ bamm24b-pegasus12_D.scl
³ bamm24b-pegasus16_D.scl
³ bamm24b-pegasus20-efg3x59_D.scl
³
ÃÄÄÄmike
³ harmonicious scale.scl
³ harmonicious14scale.scl
³ PHITER.scl
³ phiter3v2.scl
³ phiter3v2scala.scl
³ PHITER_revised.scl
³ SILVERRATIO.scl
³
ÃÄÄÄMikeB
³ 11-of-17.scl
³ driftwood_30.scl
³
³
ÀÄÄÄsecor
AKJ-GDS-RWT.scl
pelog11i.scl
pelog9i.scl
Secor-VRWT.scl
secor17htt1.scl
secor17htt2.scl
secor17htt3.scl
secor17htt4.scl
secor17wt.scl
secor19p3.scl
secor19wt.scl
Secor1_4TX.scl
Secor1_5TX.scl
Secor1_5WT.scl
Secor1_7WT.scl
secor22ji29C.scl
secor22ji29F.scl
secor29htt.scl
Secor2_11WT.scl
secor34wt.scl
secor41htt.scl
Secor5_23TX.scl
Secor5_23WT.scl
SecorVRWT-24e.scl
SecorWTPB-24a.scl
SecorWTPB-24b.scl
SecorWTPB-24f.scl

Gene,

Thanks for your response. I'm trying to understand your it. If these questions get to be too much, please don't feel pressured to answer them.

Me:
>> What is 271 ET tempering, and why choose that temperament instead of
>> the almost-equivalent-sounding 272 ET, say? Or the more precise 542
>> ET?

Gene:
> First off, I want to temper out the commas 351/350 and 364/383, and
> 271 does this.

Okay.

As I understand it, when people temper commas, it's to eliminate the difference between two notes. For instance, tempering 81/80 removes the difference between a major and a minor whole tone; similarly, it removes the difference between (4 perfect 5ths) and (2 octaves + 1 Major 3rd).

Why do you want to temper out 351/350 or 364/383? What groups of intervals are you making equivalent, and why (if there's a reason)?

> But it's not the only choice; I could have used, say, 208 instead.

I don't know how you know this, but I understand how it's possible and I'm willing to push the "I believe" button on this point. :)

> However, I'm also working in the just intonation subgroup generated by
> 2, 3, 11/7 and 13/7, and I want to temper those ratios specifically.

If I understand this correctly, it means you're trying to work with a group of intervals that can be reached by arbitrary multiplication and division of these four numbers.

It seems to me that by using 11/7 and 13/7, you are enforcing the fact that you will always have two larger primes if you have any at all (i.e., if you have 7 you have 13 or 11; if you have 13 you have 7 or 11; if you have 11 you have 7 or 13). I'm not sure why that's useful or harmonically / melodically interesting, but you wouldn't have that constraint if you had 2, 3, 7, 11, 13 as your generators. Is that your goal, or just an artifact of some other goal? If an artifact, I suspect it's related to the mapping (below): Am I right? If a goal, why have that goal?

Although, given that you aren't using all possible intervals generated by the generators, why do you need to specifically force this limitation? In other words, it seems that you could have other intervals in here that you haven't included; for instance, between the 13/11 and the 14/11, you could have 33/26, but you don't. Is this interval eliminated by some other constraint? And if you don't have to have a constraint that prevents 33/26, why have a constraint that prevents, say, 7/6? Why not just arbitrarily select intervals?

> If for some integer N I round off to the nearest integer the quantities
> N*log2(3), N*log2(11/7) and N*log2(13/7) I get a mapping,

This makes sense to me, I think; you're rounding off to the nearest integer to determine the corresponding step number in the ET you're using, yes?

> and if I require 351/350 and 364/363 to be tempered out, I get a finite number of
> possible mappings.

...because not all ETs temper those commas. That makes sense, except I still don't know why you don't want to temper 351/350 and 364/363.

> Choosing the best one by the "Tenney-Euclidean" error measure picks out 271 as
> the optimal choice.

I had to look this up, and it involves a lot of math I don't understand yet. But it sounds like you're using some formula to determine which of the "finite number of possible mappings" has the smallest error for some given level of complexity. And, to answer my own question, you could have gone with 542 ET, but that would have added to the complexity without signficantly reducing the error. (I think.)

> It may not make much difference, but it gives me a quick way of finding a good
> equal temperament tuning. I like those because they are in some circumstances
> convenient--say, for instance, I wanted to reconstruct an exact specification
> for the scale, or I want to find related temperaments, or I am working on a
> score in this temperament, etc.

Okay. My actual ability to make microtonal music is pretty limited at the moment, so I'll take this on faith as well.

Thanks,
Jake

By the way, it's a really cool scale. I used it to modify a silly test piece I'm using right now.

Original:
http://www.freivald.org/~jake/documents/ji_bells.mp3

Canton:
http://www.freivald.org/~jake/documents/canton_bells.mp3

You may find this music repetitive, but I don't really understand how to change chords microtonally other than mimicking typical 12TET progressions -- so I don't. Not here, anyway. For the most part, I just keep a single chord and try to make the notes inside it change. That said, the Canton scale adds a lot more variety (even though it uses fewer notes than my original), and when I use the 22/13 chord instead of the harmonic 7 chord that I used in the original, it's a pretty significant difference.

So thanks for the lead.

Thanks,
Jake

--- In tuning@yahoogroups.com, Jake Freivald <jdfreivald@...> wrote:

> > First off, I want to temper out the commas 351/350 and 364/383, and
> > 271 does this.

Sorry, I butchered this. Should have been 352/351 and 364/363.

> As I understand it, when people temper commas, it's to eliminate the
> difference between two notes. For instance, tempering 81/80 removes the
> difference between a major and a minor whole tone; similarly, it removes
> the difference between (4 perfect 5ths) and (2 octaves + 1 Major 3rd).

In this case, 352/351 is the difference between 13/11 and 32/27: (32/27)/(13/11) = 352/351. Also, 364/363 is the difference between (13/11)*(14/11) = 182/121 and 3/2: (182/121)/(3/2) = 364/363. These relationships are all over the place in the just intonation "Canton" scale. Also, if you put 352/351, 364/363 and 147/143 together, you get the mapping <12 19 8 11| of the 2.3.13/11.14/11 subgroup corresponding to 12edo.

> It seems to me that by using 11/7 and 13/7, you are enforcing the fact
> that you will always have two larger primes if you have any at all
> (i.e., if you have 7 you have 13 or 11; if you have 13 you have 7 or 11;
> if you have 11 you have 7 or 13). I'm not sure why that's useful or
> harmonically / melodically interesting, but you wouldn't have that
> constraint if you had 2, 3, 7, 11, 13 as your generators.

2.3.11/7.13/7 is a large enough group to have both 1-13/11-3/2 minor and 1-14/11-3/2 major triads, which seems like enough of a reason to me. Isn't your question a bit like asking why restrict yourself to 2, 3 and 5; or 2, 3, 5 and 7?

Is that your
> goal, or just an artifact of some other goal? If an artifact, I suspect
> it's related to the mapping (below): Am I right? If a goal, why have
> that goal?
>
> Although, given that you aren't using all possible intervals generated
> by the generators, why do you need to specifically force this
> limitation? In other words, it seems that you could have other intervals
> in here that you haven't included; for instance, between the 13/11 and
> the 14/11, you could have 33/26, but you don't.

It's not there in the untempered scale; 33/26 is less by 364/363 from 14/11 so they occupy the same four steps out of twelve slot. In a functional sense, though not in terms of pitches, it is "there" in the tempered version since 364/363 is tempered out.

Is this interval
> eliminated by some other constraint? And if you don't have to have a
> constraint that prevents 33/26, why have a constraint that prevents,
> say, 7/6? Why not just arbitrarily select intervals?

I didn't arbitrarily select intervals because this is a systematically constructed scale for a particular subgroup, but of course nothing prevents you or anyone from selecting whatever intervals you like. In case it isn't clear, 7/6 is not in the subgroup I am using.

> Okay. My actual ability to make microtonal music is pretty limited at
> the moment, so I'll take this on faith as well.

Have you thought about taking the microtonal plunge and giving it a try? If you have a system where you can retune the notes of a keyboard, twelve note scales are an obvious way to start.

--- In tuning@yahoogroups.com, Jake Freivald <jdfreivald@...> wrote:
>
> By the way, it's a really cool scale. I used it to modify a silly test
> piece I'm using right now.
>
> Original:
> http://www.freivald.org/~jake/documents/ji_bells.mp3
>
> Canton:
> http://www.freivald.org/~jake/documents/canton_bells.mp3

There you go--you are now officially a microtonal composer! You up to writing something more extensive as a demo piece for canton, or for the tempered cantonpenta scale?

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:

>Also, if you put 352/351, 364/363 and 147/143 together, you get the mapping <12 19 8 11| of the 2.3.13/11.14/11 subgroup corresponding to 12edo.

This should be the 2.3.11/7.13/7, as that is what the numbers 12, 19, 8 and 11 correspond to.

On Thu, Mar 3, 2011 at 11:43 PM, Jake Freivald <jdfreivald@...> wrote:
>
> As I understand it, when people temper commas, it's to eliminate the
> difference between two notes. For instance, tempering 81/80 removes the
> difference between a major and a minor whole tone; similarly, it removes
> the difference between (4 perfect 5ths) and (2 octaves + 1 Major 3rd).
>
> Why do you want to temper out 351/350 or 364/383? What groups of
> intervals are you making equivalent, and why (if there's a reason)?

Gene says he was talking about 352/351 and 364/363 now. 352/351
equates 16/13 and 27/22, among other things. 364/363 equates 14/11 and
33/26, among other things.

Let me give you a bit of historical context here, most of which
admittedly precedes my own involvement here. Tuning theory has gone
far beyond just equating similarly sized intervals and . This
relatively simple concept has now turned into an extremely deep
group-theory based approach to temperament (and making music) and once
you start learning about it, you will actually never stop. There are
so many nuances that I'm still way behind catching up on most of it.

Frankly, the quality of the work done was so strong that I don't think
that anyone here really doubts that it's only a matter of time before
academia catches onto it and interest in this field explodes. Until
then, you're now on the bleeding edge, so congrats and good luck
figuring it all out. Go post some basic questions about it on
tuning-math and you'll pretty much watch your brain enlarge before
your very eyes. Gene is a mathematician and he was involved in the
collaboration over on tuning-math, along with a lot of other people
you don't know, and that's where this all came from. And that's why
you're getting such esoteric mathematical answers.

So the reason why 352/351 and 364/363 are chosen is, I assume, that
Gene loaded up a Maple program to search for interesting commas of the
aforementioned subgroups without even thinking about what specific
intervals those might temper. The goal is usually to find what they
call rank 2 temperaments around here, which consist of a single
generator and a period (or two generators, if you're hip. So something
like quarter-comma meantone is a rank 2 temperament, with the
generator a slightly flat fifth and the period an octave).

If you're in the 2.3.11/7.13/7 subgroup, then you're in rank 4,
because every interval in the system can be represented as a
combination of four generators - 2, 3, 11/7, and 13/7. So by tempering
two commas you reduce the dimensionality of the system by two,
bringing you down to rank 2. Once you have a generator and a period,
you can search for MOS's of that tuning, find a "diatonic"-ish sized
one, and a "chromatic"-ish sized one, or scales sized in between, or,
if you're Gene, you're happy with MOS's that are much larger.

The whole thing builds somewhat off of the work of Adrian Fokker, so
perhaps reading about Fokker periodicity blocks will help get you
started.

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> Gene says he was talking about 352/351 and 364/363 now. 352/351
> equates 16/13 and 27/22, among other things. 364/363 equates 14/11 and
> 33/26, among other things.

Also, 364/363 and 352/351 are commas of 2.3.11/7/13/7, and 351/350 isn't. It's not an interval of that subgroup.
h esoteric mathematical answers.
>
> So the reason why 352/351 and 364/363 are chosen is, I assume, that
> Gene loaded up a Maple program to search for interesting commas of the
> aforementioned subgroups without even thinking about what specific
> intervals those might temper.

Actually, I loaded up a Maple program which told me the 12et comma basis in the subgroup was 147/143, 352/351 and 364/363. That drew attention to 352/351 and 364/363 right off the bat. Then I looked at the approximations in the scale, and found it was loaded with approximations off by 352/351, 364/363 and 896/891. And 896/891 = 352/351 * 364/363. At that point it's a no-brainer to temper by 352/351 and 364/363, though if you prefer you could just use the approximations.

This doesn't need to happen. Portsmouth seems fine as it is. Omaha could use 243/242 tempering; 243/242 being a comma of 2.3.11. But the 2.3.7.11 subgroup of Portsmouth has 12et commas 64/63, 99/98 and 896/891, which I don't find being used.

> If you're in the 2.3.11/7.13/7 subgroup, then you're in rank 4,
> because every interval in the system can be represented as a
> combination of four generators - 2, 3, 11/7, and 13/7. So by tempering
> two commas you reduce the dimensionality of the system by two,
> bringing you down to rank 2.

In this case, you get [<1 0 7 12|, <0 1 -4 -7|], with a POTE generator of ~3/2 = 704.052 cents. This gives MOS of sizes 5, 7, 12, 17, 29, 46 etc., and so the 12-note MOS, basically Leapday[12], suggests itself.

> The whole thing builds somewhat off of the work of Adrian Fokker, so
> perhaps reading about Fokker periodicity blocks will help get you
> started.

I should finish the Wikipedia article on Fokker blocks. :( But I don't think it's necessary to get into them.

Me:
>> As I understand it, when people temper commas, it's to eliminate the
>> difference between two notes. For instance, tempering 81/80 removes the
>> difference between a major and a minor whole tone; similarly, it removes
>> the difference between (4 perfect 5ths) and (2 octaves + 1 Major 3rd).

Gene:
> In this case, 352/351 is the difference between 13/11 and 32/27:
> (32/27)/(13/11) = 352/351. Also, 364/363 is the difference between
> (13/11)*(14/11) = 182/121 and 3/2: (182/121)/(3/2) = 364/363.

*Light bulb*

I guess I should already have known how to see this, since I knew what the purpose was supposed to be, but reading the explanation for how you did it really makes a difference. Having / learning Maple would probably help, too, I guess....

I've been putting scales into a spreadsheet that highlights important "normal" intervals so I can see how each note in the scale can be harmonized. (When I made it, "normal" was basically 7-limit, so the spreadsheet has its limitations, too.) I see how the temperament gives me a bunch of 704-cent 5ths instead of a mix of 702-cent and 707-cent ones. And, just like in 12-TET, a stack of m3/M3 or M3/m3 gives me a fifth.

And yes, I see the similar way that 32/27s (294 cents) are mixed in with the 13/11s (289 cents).

And I see how it's all done by shifting pitches by only two or three cents at a time, which is imperceptible to me.

> These relationships are all over the place in the just intonation "Canton"
> scale. Also, if you put 352/351, 364/363 and 147/143 together, you get the
> mapping <12 19 8 11| of the 2.3.13/11.14/11 subgroup corresponding to 12edo.

Yeah, I'm still trying to figure out what all this means. :)

And, in a similar vein, I think I'm misunderstanding something about subgroups. You said:

> I didn't arbitrarily select intervals because this is a systematically
> constructed scale for a particular subgroup, but of course nothing prevents
> you or anyone from selecting whatever intervals you like. In case it isn't
> clear, 7/6 is not in the subgroup I am using.

It's clear that 7/6 isn't in the subgroup, because it can't be derived through multiplications and divisions of 2, 3, 11/7, and 13/7.

What's not clear is why 33/26 isn't in the subgroup. The xenharmonic wiki says, "By a just intonation subgroup is meant a group generated by a finite set of positive rational numbers via arbitrary multiplications and divisions." 33/26 = 3 * (11/7) * (13/7)^-1 * 2^-1, so I thought it could be in the subgroup, and you *chose* not to use it.

It sounds like you're saying there's something deterministic that I'm missing, and that 33/26 can't be in the subgroup. If that's the case, then

a) your explanation here:

> 2.3.11/7.13/7 is a large enough group to have both 1-13/11-3/2 minor
> and 1-14/11-3/2 major triads, which seems like enough of a reason to me.

...makes all the sense in the world, and

b) I have to understand subgroups better.

>> Okay. My actual ability to make microtonal music is pretty limited at
>> the moment, so I'll take this on faith as well.
>
> Have you thought about taking the microtonal plunge and giving it a try?
> If you have a system where you can retune the notes of a keyboard,
> twelve note scales are an obvious way to start.

[and, from another email]

> There you go--you are now officially a microtonal composer! You up to
> writing something more extensive as a demo piece for canton, or for the
> tempered cantonpenta scale?

I'm absolutely *not* a microtonal composer -- I have two little non-twelve ditties about three minutes long total that don't totally annoy me -- but thanks for the encouragement. :)

I don't have a retunable keyboard, and when I mentioned to my wife that Chris V. talked about spending $250 to get a fully-functional software synth setup, she basically laughed at me. So I'm using free things at the moment.

I have Lilypad and Scala, which enables to me do annoyingly beepy-sounding retuned MIDI files pretty easily, and I have Csound and a not-horrific soundfont, which enables me to do artificial but not-quite-nails-on-a-chalkboard-sounding renderings with some significant difficulty. I can't get Csound to render my MIDI files, and I have downloaded a MIDI-to-Csound app or two that I haven't really worked on enough to get working.

But it's a start. So yes, the Canton tempered scale seems like something worth playing with. I've also just started messing with Blackwood[10] based on Mike Battaglia's repeated recommendations, and a 13-limit 12-note scale that's heavy in sevens:

! C:\Program Files (x86)\Scala22\septimal-12-note.scl
!
Septimal Scale
12
!
15/14
8/7
7/6
9/7
4/3
10/7
3/2
11/7
12/7
7/4
13/7
2/1

While playing with Scala I discovered that I really like the sound of the septimal m3 paired with a P4 or P5 (I had to look it up: Those are 6:7:8 and 6:7:9 chords), so I wanted to make music that contains them. The rest of the notes are just spread out enough over the octave to make them useful. There are some weird things about it, though, and I might think about how I can temper that, based on what you've done here, too. Maybe more on that in another post.

Thanks very much again for your patience and explanations. Also thanks to Mike Battaglia for his commentary, which was very helpful but about which I have less to say. And to Igs, who previously told me about tuning spaces, which made the rank 4 --> tempering two commas --> rank 2 description make sense.

Regards,
Jake

--- In tuning@yahoogroups.com, Jake Freivald <jdfreivald@...> wrote:

> I guess I should already have known how to see this, since I knew what
> the purpose was supposed to be, but reading the explanation for how you
> did it really makes a difference. Having / learning Maple would probably
> help, too, I guess....

> It's clear that 7/6 isn't in the subgroup, because it can't be derived
> through multiplications and divisions of 2, 3, 11/7, and 13/7.
>
> What's not clear is why 33/26 isn't in the subgroup. The xenharmonic
> wiki says, "By a just intonation subgroup is meant a group generated by
> a finite set of positive rational numbers via arbitrary multiplications
> and divisions." 33/26 = 3 * (11/7) * (13/7)^-1 * 2^-1, so I thought it
> could be in the subgroup, and you *chose* not to use it.

I didn't say it wasn't in the subgroup, I said it was occupying the same slot as 14/11. Hence, it's a choice between 33/26 or 14/11; you can use either, but not both, if you want the scale to correspond to the mapping ("epimorphic" property.) On the other hand, if you temper out 364/363, the question goes away.

> I have Lilypad and Scala, which enables to me do annoyingly
> beepy-sounding retuned MIDI files pretty easily, and I have Csound and a
> not-horrific soundfont, which enables me to do artificial but
> not-quite-nails-on-a-chalkboard-sounding renderings with some
> significant difficulty. I can't get Csound to render my MIDI files, and
> I have downloaded a MIDI-to-Csound app or two that I haven't really
> worked on enough to get working.

I suggest you download TiMidity++. It can not only render a midi file, it can render a midi file which uses the midi tuning standard, an unusual ability in software.

http://sourceforge.net/projects/timidity/

> ! C:\Program Files (x86)\Scala22\septimal-12-note.scl

I'll check it out.

On Sat, Mar 5, 2011 at 11:43 PM, genewardsmith
<genewardsmith@...> wrote:
>
> I suggest you download TiMidity++. It can not only render a midi file, it can render a midi file which uses the midi tuning standard, an unusual ability in software.

In other news, if I were to try to re-render classic GWS compositions
using the latest in 21st century cutting edge technology, technology
which up until now has been unattainable for the masses and generally
requires lots of esoteric and arcane knowledge to operate, I'd need
MIDI files in which something like note 60 is C and then note 61 is
the next note up in the tuning. No MTS or pitch bends, but some kind
of stretched tuning like that. Can we make this happen? But later, but
still at some point?

-Mikexcited to have Kontakt working finally

--- In tuning@yahoogroups.com, Jake Freivald <jdfreivald@...> wrote:

> ! C:\Program Files (x86)\Scala22\septimal-12-note.scl
> !
> Septimal Scale
> 12

According to Scala, this is a transposition of "Terry Riley, tuning for Cactus Rosary (1993)". Thanks for pointing this scale out. There are various possibilities for tempering it, such as putting it all in 31 equal.

I used to be in your shoes

you can get a nanokorg usb keyboard for $42 from Amazon

http://www.amazon.com/Korg-nanoKEY-Controller-Keyboard-Black/dp/B001J8INY2/ref=sr_1_2?s=musical-instruments&ie=UTF8&qid=1299387424&sr=1-2

and then any freeware microtonal software - such as open modplug
tracker or probably some linux stuff would get you going.

Chris

On Sat, Mar 5, 2011 at 10:46 PM, Jake Freivald <jdfreivald@...> wrote:

>
> I don't have a retunable keyboard, and when I mentioned to my wife that
> Chris V. talked about spending $250 to get a fully-functional software
> synth setup, she basically laughed at me. So I'm using free things at
> the moment.
>

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> In other news, if I were to try to re-render classic GWS compositions
> using the latest in 21st century cutting edge technology, technology
> which up until now has been unattainable for the masses and generally
> requires lots of esoteric and arcane knowledge to operate, I'd need
> MIDI files in which something like note 60 is C and then note 61 is
> the next note up in the tuning. No MTS or pitch bends, but some kind
> of stretched tuning like that. Can we make this happen? But later, but
> still at some point?

Could be done!

I just found this in my "draft" folder, and it looks like Gmail says I didn't send it. Sorry if this is a resend.

----

Thanks to Chris and Gene for the leads on the keyboard and Timidity++.

While I was asking questions about Canton and why he chose certain ratios and not others to be in it, Gene said:

> I didn't say it [33/26] wasn't in the [2.3.11/7.13/7] subgroup, I said it was occupying the same slot as 14/11.

I stand corrected, and I think I understand.

Then, later, we were talking about a septimal scale that I'd been playing around with:
1: 15/14 119.443 major diatonic semitone
2: 8/7 231.174 septimal whole tone
3: 7/6 266.871 septimal minor third
4: 9/7 435.084 septimal major third, BP third
5: 4/3 498.045 perfect fourth
6: 10/7 617.488 Euler's tritone
7: 3/2 701.955 perfect fifth
8: 11/7 782.492 undecimal augmented fifth
9: 12/7 933.129 septimal major sixth
10: 7/4 968.826 harmonic seventh
11: 13/7 1071.702 16/3-tone
12: 2/1 1200.000 octave

Gene said:
> According to Scala, this is a transposition of "Terry Riley, tuning for Cactus Rosary
> (1993)".

I knew that at one point, but didn't know if it had any historical significance.

> Thanks for pointing this scale out. There are various possibilities for tempering it,
> such as putting it all in 31 equal.

I guess I have to decide what I want out of it before I do that, though. I ultimately figured out that I think I need to reduce the prime limit and possibly temper it in 41 equal, which provides very close approximations to this scale -- though I'm not sure it's the "mathematically correct" thing to do.

Fair warning: I'm writing the rest of this because writing it all down helps me figure things out. Maybe someone will find it interesting, but nobody needs to pay attention; I won't get offended. :) I've actually started writing this three times with different approaches, and each time I've realized that my approach had a problem -- but I didn't notice the new problem until I started writing down the "solution".

This septimal scale is one of the first scales I put together to play around with, and I wasn't paying attention to primes, only ratios. So it has 8/7, 9/7, 10/7, 11/7, 12/7, and 13/7 in it. But my real intent was to get the 0-267-498 and 0-267-702 chords (6:7:8 and 6:7:9) that I like so much. I also want the harmonic 7, which sounds beautiful over those triads (12:14:16:21 and 12:14:18:21). So the base of the scale is 0-267-498-702-969-1200 cents. The rest is filling in the gaps in a quasi-diatonic way, and filling in more gaps in a quasi-chromatic way (apropos of the "chromatic pairs" discussion going on in another thread).

Now that I'm considering prime limits, I'm approaching this in a different way.

The original scale is heinously complicated (10 step sizes ranging from 49/48 (36 cents) to 54/49 (168 cents), which -- I think -- is at the root of it sounding "sloppy" in the little song I made with it. Based on what Igs told me about "tuning spaces", that's in part because it's a selection from subgroup 2,3,5,7,11,13; too many primes lining up in too many ways will make for too many incommensurable ratios. So now I'm thinking that if I'm looking for good septimal minor seconds, fourths, fifths, and harmonic 7ths, I should eliminate the 11 and 13 and focus on 7-limit ratios that give me that instead.

That leads me to looking at the 4/3 step of the scale as a root of a chord. The 11/7 gave me a 284-cent minor third over the 4/3. If I change that to 14/9, I get a "proper" (i.e., one that was like other modes of the scale) 267-cent septimal minor third. Now I could play traditional i-iv-v chords if I wanted to.

Oh, except the 5th up from the 3/2 step (the V mode of the scale) is way out of whack: The 8/7 septimal whole tone drives that way up to 729 cents. I kinda like the 8/7, though, so I'm not sure I want to get rid of it. Maybe I just won't rely on the fifth for the V chord.

Finally, eliminating the 13/7 still requires some kind of major-seventh-ish tone. Options include 50/27 (1067 cents), 28/15 (1081 cents), or 15/8 (1088 cents). Since I'm trying to use septimal tones, 28/15 seems like a better option here than the others.

So this looks like what I'm dealing with now.

15/14 119.443 major diatonic semitone
8/7 231.174 septimal whole tone
7/6 266.871 septimal minor third
9/7 435.084 septimal major third, BP third
4/3 498.045 perfect fourth
10/7 617.488 Euler's tritone
3/2 701.955 perfect fifth
14/9 764.916
12/7 933.129 septimal major sixth
7/4 968.826 harmonic seventh
28/15 1080.557
2/1 1200.000 octave

Things are still a little strange. There are these little 49/48 steps (36 cents) between the 8/7 & 7/6 and the 12/7 & 7/4, but that's the price you pay for having both the 8/7 and the 7/6.

Here's something new I learned: If you tell Scala Analyze / Show Interval Differences, you get lists of commas and their prime factors. So, for Canton (before tempering), you get:
10648/10647, 0.1626 cents 11.11.11/3.3.7.13.13
364/363, 4.7627 cents 7.13/3.11.11
352/351, 4.9253 cents 11/3.3.3.13
1701/1664, 38.0732 cents 3.3.3.3.3.7/13
3969/3872, 42.8359 cents 3.3.3.3.7.7/11.11
693/676, 42.9985 cents 3.3.7.11/13.13
147/143, 47.7612 cents 3.7.7/11.13

(Unfortunately, there's a bug or something that's preventing prime 2 from showing up.)

The 364/363, 352/351, and 147/143 are all commas that Gene told me about in his discussion of tempering Canton. He also noted that there were approximations in the scale that were 364/363 or 352/351, as well as others that were off by the product of those two commas.

So I tried the same thing with this new septimal scale I got:

225/224, 7.7115 cents 3.3.5.5/2.2.2.2.7
81/80, 21.5063 cents 3.3.3.3/2.2.2.2.5
64/63, 27.2641 cents 2.2.2.2.2.2/3.3.7
50/49, 34.9756 cents 2.5.5/7.7
36/35, 48.7704 cents 2.2.3.3/5.7
405/392, 56.4819 cents 3.3.3.3.5/2.2.2.7.7

(I added in the 2s that Scala left out.)

I started playing around with factorizations of these, but quickly realized why that wouldn't help.

So it looks like I could temper 225/224. And Scala helps you do that, too, with Modify / Temper. When I do that, I get this Wedgie: <<1 2 -2 -5|| -- I only sort of know what that is conceptually -- and this scale:

0: 1/1 0.000 unison, perfect prime
1: 114.568 cents 114.568
2: 229.136 cents 229.136
3: 270.060 cents 270.060
4: 430.744 cents 430.744
5: 499.196 cents 499.196
6: 613.764 cents 613.764
7: 700.804 cents 700.804
8: 769.256 cents 769.256
9: 929.940 cents 929.940
10: 970.864 cents 970.864
11: 1085.432 cents 1085.432
12: 2/1 1200.000 octave

The biggest difference I can see immediately is that where previously I had three modes of the scale with perfectly just M3s and two 56/45, 379-cent M3s, I now have five almost-perfect 5/4s. One of those is in the V mode, which is nice: I can now get an almost perfectly just M3 if I want to do a i-iv-V progression.

My number of one-step intervals sizes is, unsurprisingly, now 5 instead of 6.

I played with a variety of other commas, too, without knowing whether they were relevant, and didn't see much that was interesting. I did notice that when I tempered out 32805/32768, 1029/1024, and 225/224, I got a mode of 41 equal that's less than 5 cents off of all of my notes. Since I'm limiting myself to 12 notes because that's a limitation of my music-making abilities, I don't know that I'll take advantage of other aspects of 41 equal, but at least for those 12 notes I'm getting what I want.

I've talked too much and done too little, so I'm leaving it there for right now. :)

Regards,
Jake

--- In tuning@yahoogroups.com, Jake Freivald <jdfreivald@...> wrote:

> So this looks like what I'm dealing with now.
>
> 15/14 119.443 major diatonic semitone
> 8/7 231.174 septimal whole tone
> 7/6 266.871 septimal minor third
> 9/7 435.084 septimal major third, BP third
> 4/3 498.045 perfect fourth
> 10/7 617.488 Euler's tritone
> 3/2 701.955 perfect fifth
> 14/9 764.916
> 12/7 933.129 septimal major sixth
> 7/4 968.826 harmonic seventh
> 28/15 1080.557
> 2/1 1200.000 octave

> Here's something new I learned: If you tell Scala Analyze / Show
> Interval Differences, you get lists of commas and their prime factors.
> So, for Canton (before tempering), you get:
> 10648/10647, 0.1626 cents 11.11.11/3.3.7.13.13
> 364/363, 4.7627 cents 7.13/3.11.11
> 352/351, 4.9253 cents 11/3.3.3.13

Etc. If you look at these three commas, they actually boil down to two, since the third is the ratio between them. So you get the same temperament I mentioned before:

http://xenharmonic.wikispaces.com/cantonpenta

> So I tried the same thing with this new septimal scale I got:

Excellent idea, but you do miss the possibility of higher limit commas. Here, instead of just 225/224 you can temper out both 225/224 and 385/384 (or equivalently, 540/539), which is 11-limit marvel.

> So it looks like I could temper 225/224. And Scala helps you do that,
> too, with Modify / Temper. When I do that, I get this Wedgie: <<1 2 -2
> -5|| -- I only sort of know what that is conceptually

One thing it is is a unique way of labeling a temperament, but for a corank one, ie single comma, temperament, all the same information is in the comma itself; this is the monzo written backwards with sign changes, which doesn't help much.

> I
> got a mode of 41 equal that's less than 5 cents off of all of my notes.
> Since I'm limiting myself to 12 notes because that's a limitation of my
> music-making abilities, I don't know that I'll take advantage of other
> aspects of 41 equal, but at least for those 12 notes I'm getting what I
> want.

You might try 72 rather than 41.

Me:
> > So I tried the same thing [finding commas in Scala] with this new
> > septimal scale I got:

Gene:
> Excellent idea, but you do miss the possibility of higher limit
> commas. Here, instead of just 225/224 you can temper out both
> 225/224 and 385/384 (or equivalently, 540/539), which is 11-limit
> marvel.

If I'm focused on making a 7-limit scale, what benefit do I gain by looking at higher-limit commas?

The xenharmonic wiki says that 225/224 is the "marvel comma", so if I understand things correctly, in effect I'm getting 7-limit marvel by just tempering out that one comma. Marvel has "less complexity for 5/4" (which I understand as a measure, but not intuitively yet), and that the tuning gives (5/4)*(5/4)*(9/7) = 2/1 vs. (5/4)^3 = 2/1 like in 12tET. That's interesting, because stacked major thirds tend to annoy me in 12tET. Although there's no opportunity to play with that in this scale (no stacked major thirds), I'll have to see if I like the augmented triad more in marvel than in 12tET. But with this scale, at the moment I'm just tempering stuff to see what pops out, e.g., the smoothing out of major thirds among the modes of this scale.

This is part of the reason that I keep posting stuff that seems so stupid: I just can't seem to be able to figure out, from the xenharmonic wiki or other reading that I've done, why I would temper some commas rather than others. When you showed me that cantonpenta took the canton 13/11 and 14/11 and made them add up to a 3/2, that gave me a functional reason for the temperament. When I tempered 225/224 from this septimal scale, I see the smoothing of the major thirds. But I don't have any a priori sense of what tonal quality tempering a given comma should impart, or what functional difference they'd make to a scale.

> > When I do that, I get this Wedgie: <<1 2 -2 -5|| -- I only sort
> > of know what that is conceptually
>
> One thing it is is a unique way of labeling a temperament, but for
> a corank one, ie single comma, temperament, all the same
> information is in the comma itself; this is the monzo written
> backwards with sign changes, which doesn't help much.

I think I understand that.

> > I got a mode of 41 equal that's less than 5 cents off of all of
> > my notes. Since I'm limiting myself to 12 notes because that's a
> > limitation of my music-making abilities, I don't know that I'll
> > take advantage of other aspects of 41 equal, but at least for
> > those 12 notes I'm getting what I want.
>
> You might try 72 rather than 41.

Why 72? Does that have to do with the 11-limit marvel suggestion above?

Thanks,
Jake

--- In tuning@yahoogroups.com, "Jake" <jdfreivald@...> wrote:
>
> Me:
> > > So I tried the same thing [finding commas in Scala] with this new
> > > septimal scale I got:
>
> Gene:
> > Excellent idea, but you do miss the possibility of higher limit
> > commas. Here, instead of just 225/224 you can temper out both
> > 225/224 and 385/384 (or equivalently, 540/539), which is 11-limit
> > marvel.
>
> If I'm focused on making a 7-limit scale, what benefit do I gain by looking at higher-limit commas?

The scale you posted had some good 11-limit approximations whether you wanted them or not.

> > You might try 72 rather than 41.
>
> Why 72? Does that have to do with the 11-limit marvel suggestion above?

72 is a more accurate marvel tuning than 41, and is also better in the 11-limit. In either case, you can also look at what you are getting in terms of miracle and see if that helps.

Me:
> > If I'm focused on making a 7-limit scale, what benefit
> > do I gain by looking at higher-limit commas?

Gene:
> The scale you posted had some good 11-limit approximations
> whether you wanted them or not.

Fair enough. So I should start looking at approximations for higher-limit tones whether or not I'm building the scale for them, in case I get some that come along for the ride.

> > > You might try 72 rather than 41.
> >
> > Why 72? Does that have to do with the 11-limit marvel suggestion above?
>
> 72 is a more accurate marvel tuning than 41, and is also better in the
> 11-limit. In either case, you can also look at what you are getting in
> terms of miracle and see if that helps.

So it's both 11-limit and more accurate. I'll take a look at that and miracle.

Thanks as always for your patience.

Regards,
Jake