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== Theory ==
== Theory ==
The tuning is pretty meaningless as far as pitches go, since there's only 7-8 steps of it in the entire human hearing range, but it has a real life equivalence to astronomy - each step of it is known as the Pogson's ratio and it has an application of being the factor which sets two stars being 1 magnitude apart.  
The tuning is pretty meaningless as far as pitches go, since there's only 7-8 steps of it in the entire human hearing range, but it has a real life equivalence to astronomy - each step of it is known as the Pogson's ratio and it has an application of being the factor which sets two stars being 1 magnitude apart. This means subdividing it makes a meaningful major tenth of 1594.525 cents.


===Scale tree===
{| class="wikitable"
|-
! colspan="3" |Generator
! |<span style="display: block; text-align: center;">'''Generator size (cents) '''</span>
!<span style="display: block; text-align: center;">'''Generator size (ed4\3)'''</span>
! |Pentachord steps
! |Comments
|-
| |4\35
| |
| |
| |<u>911.157…</u>
|''914.285…''
| |<nowiki>1 1|1 0</nowiki>
| |
|-
|25\220
|
|
|<u>905.980…</u>
|''909.{{Overline|09}}''
|<nowiki>6 6|6 1</nowiki>
|
|-
|
|71\625
|
|<u>905.690…</u>
|''908.8''
|<nowiki>17 17|17 3</nowiki>
|
|-
|
|46\405
|
|<u>905.532…</u>
|''908.641…''
|<nowiki>11 11|11 2</nowiki>
|
|-
|
|67\590
|
|<u>905.366…</u>
|''908.474…''
|<nowiki>16 16|16 3</nowiki>
|
|-
| |21\185
| |
| |
| |<u>905.000…</u>
|''908.{{Overline|108}}''
| |<nowiki>5 5|5 1</nowiki>
| |
|-
|
|80\705
|
|<u>904.695…</u>
|''907.801…''
|<nowiki>19 19|19 4</nowiki>
|
|-
|
|59\520
|
|<u>904.586…</u>
|''907.692…''
|<nowiki>14 14|14 3</nowiki>
|
|-
|
|38\335
|
|<u>904.357…</u>
|''907.462…''
|<nowiki>9 9|9 2</nowiki>
|
|-
|
|55\485
|
|<u>904.112…</u>
|''907.216…''
|<nowiki>13 13|13 3</nowiki>
|
|-
|
|72\635
|
|<u>903.982…</u>
|''907.086…''
|<nowiki>17 17|17 4</nowiki>
|
|-
|
|89\785
|
|<u>903.902…</u>
|''907.006…''
|<nowiki>21 21|21 5</nowiki>
|
|-
| |17\30
| |
| |
| |<u>903.564…</u>
|''906.{{Overline|6}}''
| |<nowiki>4 4|4 1</nowiki>
| |L/s = 4
|-
|
|115\1015
|
|<u>903.302…</u>
|''906.403…''
|<nowiki>27 27|27 7</nowiki>
|
|-
|
|98\865
|
|<u>903.257…</u>
|''906.358…''
|<nowiki>23 23|23 6</nowiki>
|
|-
|
|81\715
|
|<u>903.192…</u>
|''906.293…''
|<nowiki>19 19|19 5</nowiki>
|
|-
|
|64\565
|
|<u>903.094…</u>
|''906.194…''
|<nowiki>15 15|15 4</nowiki>
|
|-
|
|47\415
|
|<u>902.924…</u>
|''906.024…''
|<nowiki>11 11|11 3</nowiki>
|
|-
| |
| |30\265
| |
| |<u>902.561…</u>
|''905.660…''
| |<nowiki>7 7|7 2</nowiki>
| |
|-
|
|
|73\645
|<u>902.328…</u>
|''905.426…''
|<nowiki>17 17|17 5</nowiki>
|
|-
| |
| |43\380
| |
| |<u>902.165…</u>
|''905.263…''
| |<nowiki>10 10|10 3</nowiki>
| |
|-
| |
| |56\495
| |
| |<u>901.953…</u>
|''905.{{Overline|05}}''
| |<nowiki>13 13|13 4</nowiki>
| |
|-
| |
| |69\610
| |
| |<u>901.921…</u>
|''904.918…''
| |<nowiki>16 16|16 5</nowiki>
| |
|-
| |
| |82\725
| |
| |<u>901.731…</u>
|''904.827…''
| |<nowiki>19 19|19 6</nowiki>
| |
|-
| |
| |95\840
| |
| |<u>901.666…</u>
|''904.761…''
| |<nowiki>22 22|22 7</nowiki>
| |
|-
| |
| |
| |
| |<u>901.662…</u>
|''904.758…''
| |<nowiki>π π|π 1</nowiki>
| |L/s = π
|-
| |
| |108\955
| |
| |<u>901.616…</u>
|''904.712…''
| |<nowiki>25 25|25 8</nowiki>
| |
|-
| |
| |121\1070
| |
| |<u>901.577…</u>
|''904.672…''
| |<nowiki>28 28|28 9</nowiki>
| |28;9 Superdiatonic 1/28-tone
|-
| |
| |134\1185
| |
| |<u>901.546…</u>
|''904.642…''
| |<nowiki>31 31|31 10</nowiki>
| |
|-
| |13\115
| |
| |
| |<u>901.235…</u>
|''904.347…''
| |<nowiki>3 3|3 1</nowiki>
| |Terra Rubra 1/3-tone
|-
| |
| |126\1115
| |
| |<u>900.942…</u>
|''904.035…''
| |<nowiki>29 29|29 10</nowiki>
| |Terra Rubra <span style="font-size: 12.8000001907349px;"><big>1/29-tone</big></span>
|-
| |
| |113\1000
| |
| |<u>900.906…</u>
|''904''
| |<nowiki>26 26|26 9</nowiki>
| |Terra Rubra <span style="font-size: 12.8000001907349px;"><big>1/26-tone</big></span>
|-
| |
| |100\885
| |
| |<u>900.861…</u>
|''903.954…''
| |<nowiki>23 23|23 8</nowiki>
| |
|-
| |
| |87\770
| |
| |<u>900.803…</u>
|''903.896…''
| |<nowiki>20 20|20 7</nowiki>
| |
|-
| |
| |74\655
| |
| |<u>900.724…</u>
|''903.816…''
| |<nowiki>17 17|17 6</nowiki>
| |Terra Rubra 1/17-tone
|-
| |
| |61\840
| |
| |<u>915</u>
|''900.611…''
| |<nowiki>14 14|14 5</nowiki>
| |Terra Rubra 1/14-tone
|-
| |
| |
| |109\965
| |<u>914.685…</u>
|''900.535…''
| |<nowiki>25 25|25 9</nowiki>
| |Terra Rubra 1/25-tone
|-
| |
| |48\425
| |
| |<u>900.437…</u>
|''903.529…''
| |<nowiki>11 11|11 4</nowiki>
| |Terra Rubra 1/11-tone
|-
| |
| |
| |
| |<u>900.324…</u>
|''903.415…''
| |<nowiki>e e|e 1</nowiki>
| |L/s = e
|-
| |
| |35\310
| |
| |<u>900.135…</u>
|''903.225…''
| |<nowiki>8 8|8 3</nowiki>
| |Terra Rubra 1/8-tone
|-
| |
| |
| |92\815
| |<u>899.977…</u>
|''903.067…''
| |<nowiki>21 21|21 8</nowiki>
| |21;8 Superdiatonic 1/21-tone
|-
| |
| |
| |
| |<u>899.950…</u>
|''903.040…''
| |<span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif; font-size: small;"><nowiki>φ+1 φ+1|φ+1 1</nowiki></span>
| |Split φ superdiatonic relation (34;13 - 55;21 - 89;34 - 144;55 - 233;89 - 377;144 - 610;233..)
|-
| |
| |
| |57\505
| |<u>899.880…</u>
|''902.970…''
| |<nowiki>13 13|13 5</nowiki>
| |13;5 Superdiatonic 1/13-tone
|-
|
|
|79\700
|<u>899.767…</u>
|''902.857…''
|<nowiki>18 18|18 7</nowiki>
|
|-
| |
| |22\195
| |
| |<u>899.475…</u>
|''902.564…''
| |<nowiki>5 5|5 2</nowiki>
| |Terra Rubra 1/5-tone
|-
| |
| |
| |75\665
| |<u>899.168…</u>
|''902.255…''
| |<nowiki>17 17|17 7</nowiki>
| |17;7 Superdiatonic 1/17-tone
|-
| |
| |
| |53\470
| |<u>899.040…</u>
|''902.127…''
| |<nowiki>12 12|12 5</nowiki>
| |
|-
| |
| |31\275
| |
| |<u>898.732…</u>
|''901.{{Overline|81}}''
| |<nowiki>7 7|7 3</nowiki>
| |7;3 Superdiatonic 1/7-tone
|-
|
|
|71\630
|<u>898.502…</u>
|''901.587…''
|<nowiki>16 16|16 7</nowiki>
|
|-
| |
| |40\355
| |
| |<u>898.324…</u>
|''901.408…''
| |<nowiki>9 9|9 4</nowiki>
| |9;4 Superdiatonic 1/9-tone
|-
| |
| |49\435
| |
| |<u>898.066…</u>
|''901.149…''
| |<nowiki>11 11|11 5</nowiki>
| |11;5 Superdiatonic 1/11-tone
|-
| |
| |58\515
| |
| |<u>897.888…</u>
|''900.970…''
| |<nowiki>13 13|13 6</nowiki>
| |13;6 Superdiatonic 1/13-tone
|-
|
|67\595
|
|<u>897.758…</u>
|''900.840…''
|<nowiki>15 15|15 7</nowiki>
|
|-
|
|76\675
|
|<u>897.658…</u>
|''900.{{Overline|740}}''
|<nowiki>17 17|17 7</nowiki>
|
|-
|
|85\755
|
|<u>897.580…</u>
|''900.662…''
|<nowiki>19 19|19 9</nowiki>
|
|-
|
|94\835
|
|<u>897.517…</u>
|''900.598…''
|<nowiki>21 21|21 10</nowiki>
|
|-
|
|103\915
|
|<u>897.465…</u>
|''900.564…''
|<nowiki>23 23|23 11</nowiki>
|
|-
|
|112\995
|
|<u>897.421…</u>
|''900.502…''
|<nowiki>25 25|25 12</nowiki>
|
|-
|
|121\1075
|
|<u>897.384…</u>
|''900.465…''
|<nowiki>27 27|27 13</nowiki>
|
|-
| |9\80
| |
| |
| |<u>896.920…</u>
|''900''
| |<nowiki>2 2|2 1</nowiki>
| |<span style="display: block; text-align: left;">'''[BOUNDARY OF PROPRIETY: smaller generators are strictly proper]'''</span>
|-
|
|230\2045
|
|<u>896.676…</u>
|''899.755…''
|<nowiki>51 51|51 26</nowiki>
|
|-
|
|221\1965
|
|<u>896.667…</u>
|''899.745…''
|<nowiki>49 49|49 25</nowiki>
|
|-
|
|212\1885
|
|<u>896.656…</u>
|''899.734…''
|<nowiki>47 47|47 24</nowiki>
|
|-
|
|203\1805
|
|<u>896.644…</u>
|''899.722…''
|<nowiki>45 45|45 23</nowiki>
|
|-
|
|194\1725
|
|<u>896.631…</u>
|''899.710…''
|<nowiki>43 43|43 22</nowiki>
|
|-
|
|185\1645
|
|<u>896.617…</u>
|''899.696…''
|<nowiki>41 41|41 21</nowiki>
|
|-
|
|176\1565
|
|<u>896.602…</u>
|''899.680…''
|<nowiki>39 39|39 20</nowiki>
|
|-
|
|167\1485
|
|<u>896.585…</u>
|''899.663…''
|<nowiki>37 37|37 19</nowiki>
|
|-
|
|158\1405
|
|<u>896.565…</u>
|''899.644…''
|<nowiki>35 35|35 18</nowiki>
|
|-
|
|149\1325
|
|<u>896.544…</u>
|''899.622…''
|<nowiki>33 33|33 17</nowiki>
|
|-
|
|140\1245
|
|<u>896.520…</u>
|''899.598…''
|<nowiki>31 31|31 16</nowiki>
|
|-
|
|131\1165
|
|<u>896.492…</u>
|''899.570…''
|<nowiki>29 29|29 15</nowiki>
|
|-
|
|122\1085
|
|<u>896.461…</u>
|''899.539…''
|<nowiki>27 27|27 14</nowiki>
|
|-
|
|113\1005
|
|<u>896.424…</u>
|''899.502…''
|<nowiki>25 25|25 13</nowiki>
|
|-
|
|104\925
|
|<u>896.381…</u>
|''899.459…''
|<nowiki>23 23|23 12</nowiki>
|
|-
|
|95\845
|
|<u>896.330…</u>
|''899.408…''
|<nowiki>21 21|21 11</nowiki>
|
|-
|
|86\765
|
|<u>896.269…</u>
|''899.346…''
|<nowiki>19 19|19 10</nowiki>
|
|-
|
|77\685
|
|<u>896.193…</u>
|''899.270…''
|<nowiki>17 17|17 9</nowiki>
|
|-
|
|68\605
|
|<u>896.096…</u>
|''899.173…''
|<nowiki>15 15|15 8</nowiki>
|
|-
| |
| |59\525
| |
| |<u>895.971…</u>
|''899.047…''
| |<nowiki>13 13|13 7</nowiki>
| |Terra Rubra 1/13-tone
|-
| |
| |50\445
| |
| |<u>895.800…</u>
|''898.876…''
| |<nowiki>11 11|11 6</nowiki>
| |Terra Rubra 1/11-tone
|-
| |
| |41\365
| |
| |<u>895.555…</u>
|''898.630…''
| |<nowiki>9 9|9 5</nowiki>
| |Terra Rubra 1/9-tone
|-
| |
| |32\285
| |
| |<u>895.172…</u>
|''898.245…''
| |<nowiki>7 7|7 4</nowiki>
| |Terra Rubra 1/7-tone <span style="font-size: 12.8000001907349px;">(the 'Commatic' version of Terra Rubra, because its high accuracy of the 16/15 interval, the note '2b')</span>
|-
| |
| |
| |
| |<u>895.030…</u>
|''898.102…''
| |<span style="background-color: #ffffff;"><nowiki>√3 √3|√3 1</nowiki></span>
| |
|-
| |
| |
| |55\490
| |<u>894.886…</u>
|''897.959…''
| |<nowiki>12 12|12 7</nowiki>
| |
|-
| |
| |
| |78\695
| |<u>894.769…</u>
|''897.841…''
| |<nowiki>17 17|17 10</nowiki>
| |Terra Rubra 1/17-tone
|-
| |
| |23\205
| |
| |<u>894.489…</u>
|''897.560…''
| |<nowiki>5 5|5 3</nowiki>
| |5;3 Golden Terra Rubra 1/5-tone
|-
|
|
|83\740
|<u>894.227…</u>
|''897.{{Overline|297}}''
|<nowiki>18 18|18 11</nowiki>
|
|-
| |
| |
| |60\535
| |<u>894.126…</u>
|''897.196…''
| |<nowiki>13 13|13 8</nowiki>
| |13;8 Golden Terra Rubra 1/13-tone
|-
| |
| |
| |
| |<u>894.064…</u>
|''897.133…''
| |<span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif; font-size: small;"><nowiki>φ φ|φ 1</nowiki></span>
| |GOLDEN Terra Rubra (L/s = <span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif; font-size: small;">φ)</span>
|-
| |
| |
| |97\865
| |<u>894.040…</u>
|''897.109…''
| |<nowiki>21 21|21 13</nowiki>
| |21;13 Golden Terra Rubra 1/21-tone
|-
| |
| |37\330
| |
| |<u>893.900…</u>
|''896.{{Overline|96}}''
| |<nowiki>8 8|8 5</nowiki>
| |8;5 Golden Terra Rubra 1/8-tone
|-
|
|
|88\785
|<u>893.746…</u>
|''896.815…''
|<nowiki>19 19|19 12</nowiki>
|
|-
| |
| |51\455
| |
| |<u>893.635…</u>
|''896.703…''
| |<nowiki>11 11|11 7</nowiki>
| |11;7 Superdiatonic 1/11-tone
|-
| |
| |
| |
| |<u>893.629…</u>
|''896.697…''
| |<nowiki>π π|π 2</nowiki>
| |
|-
| |
| |
| |116\1035
| |<u>893.550…</u>
|''896.618…''
| |<nowiki>25 25|25 16</nowiki>
| |25;16 Superdiatonic 1/25-tone
|-
| |
| |65\580
| |
| |<u>893.484…</u>
|''896.551…''
| |<nowiki>14 14|14 9</nowiki>
| |14;9 Superdiatonic 1/14-tone
|-
| |
| |79\705
| |
| |<u>893.386…</u>
|''896.453…''
| |<nowiki>17 17|17 11</nowiki>
| |17;11 Superdiatonic 1/17-tone
|-
| |
| |93\830
| |
| |<u>893.318…</u>
|''896.385…''
| |<nowiki>20 20|20 13</nowiki>
| |
|-
| |
| |107\955
| |
| |<u>893.268…</u>
|''896.335…''
| |<nowiki>23 23|23 15</nowiki>
| |
|-
| |
| |121\1080
| |
| |<u>893.229…</u>
|''896.{{Overline|296}}''
| |<nowiki>26 26|26 17</nowiki>
| |26;17 Superdiatonic 1/26-tone
|-
| |
| |135\1205
| |
| |<u>893.198…</u>
|''896.265…''
| |<nowiki>29 29|29 19</nowiki>
| |29;19 Superdiatonic 1/29-tone
|-
| |14\25
| |
| |
| |<u>892.934…</u>
|''896''
| |<nowiki>3 3|3 2</nowiki>
| |3;2 Golden Terra Rubra 1/3-tone
|-
| |
| |145\1295
| |
| |<u>892.688…</u>
|''895.752…''
| |<nowiki>31 31|31 21</nowiki>
| |31;21 Superdiatonic 1/31-tone
|-
| |
| |131\1170
| |
| |<u>892.661…</u>
|''895.726…''
| |<nowiki>28 28|28 19</nowiki>
| |28;19 Superdiatonic 1/28-tone
|-
| |
| |117\1045
| |
| |<u>892.629…</u>
|''895.693…''
| |<nowiki>25 25|25 17</nowiki>
| |
|-
| |
| |103\920
| |
| |<u>892.587…</u>
|''895.652…''
| |<nowiki>22 22|22 15</nowiki>
| |
|-
| |
| |89\795
| |
| |<u>892.533…</u>
|''895.579…''
| |<nowiki>19 19|19 13</nowiki>
| |
|-
| |
| |75\670
| |
| |<u>892.458…</u>
|''895.522…''
| |<nowiki>16 16|16 11</nowiki>
| |
|-
| |
| |61\545
| |
| |<u>892.349…</u>
|''895.412…''
| |<nowiki>13 13|13 9</nowiki>
| |
|-
| |
| |47\420
| |
| |<u>892.174…</u>
|''895.238…''
| |<nowiki>10 10|10 7</nowiki>
| |
|-
|
|
|80\715
|<u>892.042…</u>
|''895.104…''
|<nowiki>17 17|17 12</nowiki>
|
|-
| |
| |33\295
| |
| |<u>891.853…</u>
|''894.915…''
| |<nowiki>7 7|7 5</nowiki>
| |
|-
|
|
|85\760
|<u>891.675…</u>
|''894.736…''
|<nowiki>18 18|18 13</nowiki>
|
|-
|
|52\465
|
|<u>891.562…</u>
|''894.623…''
|<nowiki>11 11|11 8</nowiki>
|
|-
|
|71\635
|
|<u>891.427…</u>
|''894.488…''
|<nowiki>15 15|15 11</nowiki>
|
|-
| |19\170
| |
| |
| |<u>891.058…</u>
|''894.117…''
| |<nowiki>4 4|4 3</nowiki>
| |
|-
|
|62\555
|
|<u>890.635…</u>
|''893.{{Overline|693}}''
|<nowiki>13 13|13 10</nowiki>
|
|-
|
|43\385
|
|<u>890.449…</u>
|''893.506…''
|<nowiki>9 9|9 7</nowiki>
|
|-
|
|67\600
|
|<u>890.276…</u>
|''893.{{Overline|3}}''
|<nowiki>14 14|14 11</nowiki>
|
|-
|24\215
|
|
|<u>889.967…</u>
|''893.023…''
|<nowiki>5 5|5 4</nowiki>
|
|-
|
|53\475
|
|<u>889.577…</u>
|''892.631…''
|<nowiki>11 11|11 9</nowiki>
|
|-
|29\260
|
|
|<u>889.254…</u>
|''892.307…''
|<nowiki>6 6|6 5</nowiki>
|
|-
| |5\[[9edX|9]]
| |
| |
| |<u>885.847…</u>
|''888.{{Overline|8}}''
| |<nowiki>1 1|1 1</nowiki>
| |
|}
=== Tempered commas ===
=== Tempered commas ===
Astronomers might sometimes round 2.512 to 2.5, which leads to confusion with the other 2.5 - that is the logarithm multiplier required for base-10 logarithm conversion, with the formula being involved 2.5 log10(m1 - m2). The factor of 2.5 is simply coincidentally close enough to the 5th root of 100.  
Astronomers might sometimes round 2.512 to 2.5, which leads to confusion with the other 2.5 - that is the logarithm multiplier required for base-10 logarithm conversion, with the formula being involved 2.5 log10(m1 - m2). The factor of 2.5 is simply coincidentally close enough to the 5th root of 100.  
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